Introduction
Information geometry is an interdisciplinary field in which probability theory and statistics are studied with the tools of differential geometry. If Ξ \Xi Ξ
B ‾ = { P ( x , Θ ) : Θ = ( Θ 1 , … , Θ n ) ∈ Ω ⊂ R n } \overline{\mathcal{B}} = \{P(x,\Theta) : \Theta= (\Theta_{1},\ldots,\Theta_{n}) \in\Omega\subset\mathbb{R}^{n}\} B = { P ( x , Θ ) : Θ = ( Θ 1 , … , Θ n ) ∈ Ω ⊂ R n } is a smooth n n n Ξ \Xi Ξ
g i j ( Θ ) = ∫ Ξ ∂ log P ( x , Θ ) ∂ Θ i ∂ log P ( x , Θ ) ∂ Θ j P ( x , Θ ) d x g_{ij}(\Theta) = \int_{\Xi} \frac{\partial\log\mathcal{P}(x,\Theta)}{\partial\Theta_i} \frac{\partial\log\mathcal{P}(x,\Theta)}{\partial\Theta_j} \mathcal{P}(x,\Theta)\,dx g ij ( Θ ) = ∫ Ξ ∂ Θ i ∂ log P ( x , Θ ) ∂ Θ j ∂ log P ( x , Θ ) P ( x , Θ ) d x defines a Riemannian metric on B \mathbb{B} B B \mathbb{B} B α \alpha α ∇ ( 1 ) \nabla^{(1)} ∇ ( 1 ) ∇ ( − 1 ) \nabla^{(-1)} ∇ ( − 1 ) [3 ] and Lauritzen [25 ] as a statistical manifold . Affine (dually flat) geometry and the hyperbolic geometry of statistical models are closely related, which explains why curvature properties of statistical manifolds carry statistical meaning. Applications range from the ergodic hierarchy of statistical models on curved manifolds to the asymptotic behaviour of the information geometric complexity of Gaussian models in the presence of microcorrelations [2 , 20 ] .
From the viewpoint of differential geometry, a statistical manifold is a Riemannian manifold ( N , g ) (N,g) ( N , g ) ∇ \nabla ∇ ∇ g \nabla g ∇ g ∇ ∗ \nabla^{*} ∇ ∗ 1 2 ( ∇ + ∇ ∗ ) \frac{1}{2}(\nabla+\nabla^{*}) 2 1 ( ∇ + ∇ ∗ ) ∇ \nabla ∇ statistical curvature tensor and the sectional curvature of Opozda [33 , 34 ] . The submanifold theory of statistical manifolds, with dual second fundamental forms h h h h ∗ h^{*} h ∗ [41 ] and has since produced many curvature inequalities (Chen-type, Wintgen-type and Casorati-type) for statistical submanifolds [28 , 38 ] . Additionally, statistical warped product submersions were introduced and investigated by Siddiqui et al. [37 ] .
Ricci solitons and their variants
A second source of inspiration is the theory of geometric flows. Hamilton’s Ricci flow [21 ] deforms a metric in the direction of its Ricci tensor, and Ricci solitons are its self-similar solutions: a triple ( g , ζ , λ ) (g,\zeta,\lambda) ( g , ζ , λ )
Ric + 1 2 L ζ g + λ g = 0. \operatorname{Ric}+\frac{1}{2}\mathcal{L}_{\zeta}g+\lambda g=0. Ric + 2 1 L ζ g + λ g = 0. They generalise Einstein metrics (the case ζ = 0 \zeta=0 ζ = 0 ζ \zeta ζ λ \lambda λ Yamabe flow instead preserves the conformal class of the metric and its fixed points are metrics of constant scalar curvature; Yamabe solitons are the corresponding self-similar solutions. For dim = 2 \dim=2 dim = 2 ω η ⊗ η \omega\eta\otimes\eta ω η ⊗ η almost and quasi-Yamabe solitons [15 , 22 ] , η \eta η
These notions have been studied intensively on almost contact manifolds, and in particular on Kenmotsu manifolds [23 ] , see, for instance, [29 , 6 , 9 ] . Another fruitful direction is to study solitons on submanifolds . Here, concircular vector fields [17 ] , whose integral curves are geodesics and for which ∇ E v = ∂ E \nabla_{E}v=\partial E ∇ E v = ∂ E [12 ] for Ricci solitons on Riemannian submanifolds: the tangential part of a concircular field of the ambient space is again concircular on a submanifold precisely under an umbilicity condition, and the soliton equation then expresses the Ricci tensor of the submanifold in terms of its second fundamental form.
Solitons in the statistical setting
The soliton equation involves the Ricci tensor and the covariant derivative of the potential field, and both make sense for an arbitrary linear connection. This observation of Crasmareanu [16 ] allows one to speak about a ∇ \nabla ∇ ∇ \nabla ∇ ∇ \nabla ∇ ∇ ∗ \nabla^{*} ∇ ∗ ( ζ , λ ) (\zeta, \lambda) ( ζ , λ ) statistical soliton . The condition is genuinely stronger than a soliton condition for a single connection, and it couples the geometry of the two dual connections. As a consequence, existence questions become rigid: obstructions appear, for instance, from the difference tensor T = ∇ 0 − ∇ ∗ \mathcal{T} = \nabla^{0} - \nabla^{*} T = ∇ 0 − ∇ ∗ § 13 ). In addition, several necessary and sufficient conditions for some particular couples ( g , ∇ ) (g, \nabla) ( g , ∇ ) [8 ] .
Warped products and Kenmotsu statistical manifolds
Two families of manifolds provide the natural testing ground for this theory. The first consists of warped products [5 ] . They are among the most fruitful generalisations of Riemannian products, are the standard building blocks of space-time models in general relativity [11 , 13 ] , and carry a natural dualistic structure [39 ] ; Einstein statistical warped products and Wintgen-type inequalities for their submanifolds have been studied in [4 , 28 ] . Since the soliton equation on a warped product N 1 × f N 2 N_{1} \times_{f} N_{2} N 1 × f N 2 f f f § 11 ).
The second family consists of Kenmotsu statistical manifolds , introduced by Furuhata, Hasegawa, Okuyama and Sato [19 ] by imposing a compatible statistical structure on a Kenmotsu manifold; they are obtained as warped products of a holomorphic statistical manifold [18 ] and a line. Kenmotsu geometry has applications in the mechanics of time-dependent Hamiltonian systems, geometrical optics, and thermodynamics, and its statistical counterpart is of comparable importance. The Reeb field ξ \xi ξ ξ \xi ξ η \eta η [1 ] .
Aim and contents of this chapter
The purpose of this chapter is to give a unified and reasonably self-contained account of these developments. We shall explain how
the statistical soliton equation is formulated and why it is rigid (§ 7 );
gradient solitons decide when Hessian or Ricci metrics, together with a suitable connection, form (nearly) statistical structures, and the connection ∇ θ \nabla^{\theta} ∇ θ 1 1 1 § 8 );
on statistical warped products R × f N 2 \mathbb{R} \times_{f} N_{2} R × f N 2 N 1 × f R N_{1} \times_{f} \mathbb{R} N 1 × f R § 11 );
on submanifolds of Kenmotsu statistical manifolds, the position of ξ \xi ξ η \eta η
character of the submanifold, and the scalar curvature relation for almost quasi-Yamabe solitons (§ 12 –§ 15 );
We include the key steps of the proofs, remarks on the role of the hypotheses and on sign conventions, explicit examples (§ 17 ), and a list of open problems (§ 18 ).
Organisation. Section 2 collects the basic notions on statistical manifolds, statistical curvature and statistical submanifolds. Section 3 introduces statistical solitons, almost quasi-Yamabe solitons and concircular vector fields. Section 4 reviews gradient solitons and statistical structures after Blaga and Chen. Section 5 recalls statistical warped products and Section 6 describes statistical solitons on them. Section 7 reviews Kenmotsu statistical manifolds, and Sections 8–10 study solitons on their submanifolds, with and without concircular fields. Section 11 describes the Chen inequality for statistical warped product submanifolds. Section 12 contains explicit examples, and Section 13 concludes with remarks and open problems.
Conventions. All manifolds are smooth and connected; Γ ( T N ) \Gamma(TN) Γ ( T N ) ∇ ~ 0 \widetilde{\nabla}^{0} ∇ 0 ∇ ~ ∗ \widetilde{\nabla}^{*} ∇ ∗ ∇ \nabla ∇ ∇ ∗ \nabla^{*} ∇ ∗ ∇ \nabla ∇ ∇ ∗ \nabla^{*} ∇ ∗ T ↦ − T \mathcal{T} \mapsto-\mathcal{T} T ↦ − T
Preliminaries
Statistical manifolds
Let ( N ~ , g ~ ) (\widetilde{N},\widetilde{g}) ( N , g ) ∇ ~ \widetilde{\nabla} ∇ N ~ \widetilde{N} N ∇ ~ ∗ \widetilde{\nabla}^{*} ∇ ∗ ∇ ~ \widetilde{\nabla} ∇ g ~ \widetilde{g} g
E g ~ ( F , G ) = g ~ ( ∇ ~ E F , G ) + g ~ ( F , ∇ ~ E ∗ G ) , E , F , G ∈ Γ ( T N ~ ) , (1) E\widetilde{g}(F,G)=\widetilde{g}(\widetilde{\nabla}_{E}F,G)+\widetilde{g}(F,\widetilde{\nabla}^{*}_{E}G), \qquad E,F,G\in\Gamma(T\widetilde{N}), \tag*{(1)} E g ( F , G ) = g ( ∇ E F , G ) + g ( F , ∇ E ∗ G ) , E , F , G ∈ Γ ( T N ) , (1) where Γ ( T N ~ ) \Gamma(T\widetilde{N}) Γ ( T N ) T N ~ T\widetilde{N} T N N ~ \widetilde{N} N
Definition 2.1. The triple ( N ~ , ∇ ~ , g ~ ) (\widetilde{N},\widetilde{\nabla},\widetilde{g}) ( N , ∇ , g ) statistical manifold if
the Codazzi equation
( ∇ ~ E g ~ ) ( F , G ) = ( ∇ ~ F g ~ ) ( E , G ) (\widetilde{\nabla}_{E}\widetilde{g})(F,G)=(\widetilde{\nabla}_{F}\widetilde{g})(E,G) ( ∇ E g ) ( F , G ) = ( ∇ F g ) ( E , G ) holds, and
∇ ~ \widetilde{\nabla} ∇
In this case ( ∇ ~ ∗ , g ~ ) (\widetilde{\nabla}^{*},\widetilde{g}) ( ∇ ∗ , g )
( ∇ ~ ∗ ) ∗ = ∇ ~ (\widetilde{\nabla}^{*})^{*}=\widetilde{\nabla} ( ∇ ∗ ) ∗ = ∇ and
∇ ~ 0 = 1 2 ( ∇ ~ + ∇ ~ ∗ ) \widetilde{\nabla}^{0}=\frac{1}{2}(\widetilde{\nabla}+\widetilde{\nabla}^{*}) ∇ 0 = 2 1 ( ∇ + ∇ ∗ ) is the Levi-Civita connection of g ~ \widetilde{g} g
Equivalently, a statistical structure is obtained as
∇ ~ = ∇ ~ 0 + T , \widetilde{\nabla}=\widetilde{\nabla}^{0}+\mathcal{T}, ∇ = ∇ 0 + T , where T ∈ Γ ( T ( 1 , 2 ) N ~ ) \mathcal{T}\in\Gamma(T^{(1,2)}\widetilde{N}) T ∈ Γ ( T ( 1 , 2 ) N ) difference tensor satisfying
T ( E , F ) = T ( F , E ) , g ~ ( T ( E , F ) , G ) = g ~ ( F , T ( E , G ) ) , (2) \mathcal{T}(E,F)=\mathcal{T}(F,E),\qquad\widetilde{g}(\mathcal{T}(E,F),G)=\widetilde{g}(F,\mathcal{T}(E,G)), \tag*{(2)} T ( E , F ) = T ( F , E ) , g ( T ( E , F ) , G ) = g ( F , T ( E , G )) , (2) and then ∇ ~ ∗ = ∇ ~ 0 − T \widetilde{\nabla}^{*}=\widetilde{\nabla}^{0}-\mathcal{T} ∇ ∗ = ∇ 0 − T
Statistical curvature
Let Rie \operatorname{Rie} Rie Rie ∗ \operatorname{Rie}^{*} Rie ∗ Rie 0 \operatorname{Rie}^{0} Rie 0 ∇ \nabla ∇ ∇ ∗ \nabla^{*} ∇ ∗ ∇ 0 \nabla^{0} ∇ 0 statistical curvature tensor is
S = 1 2 ( Rie + Rie ∗ ) = Rie 0 + [ T E , T F ] , (3) S=\frac{1}{2}(\operatorname{Rie}+\operatorname{Rie}^{*})=\operatorname{Rie}^{0}+[\mathcal{T}_{E},\mathcal{T}_{F}], \tag*{(3)} S = 2 1 ( Rie + Rie ∗ ) = Rie 0 + [ T E , T F ] , (3) (see [19 ] ). Following Opozda [33 , 34 ] , the statistical sectional curvature of the plane spanned by orthonormal E , F E,F E , F
Sec ( E ∧ F ) = g ( S ( E , F ) F , E ) = 1 2 ( g ( Rie ( E , F ) F , E ) + g ( Rie ∗ ( E , F ) F , E ) ) , (4) \operatorname{Sec}(E\wedge F)=g(S(E,F)F,E)=\frac{1}{2}\bigl(g(\operatorname{Rie}(E,F)F,E)+g(\operatorname{Rie}^{*}(E,F)F,E)\bigr), \tag*{(4)} Sec ( E ∧ F ) = g ( S ( E , F ) F , E ) = 2 1 ( g ( Rie ( E , F ) F , E ) + g ( Rie ∗ ( E , F ) F , E ) ) , (4) and for an orthonormal basis { e 1 , … , e m } \{e_{1},\ldots,e_{m}\} { e 1 , … , e m }
σ ∇ , ∇ ∗ = ∑ i < j Sec ( e i ∧ e j ) . \sigma^{\nabla,\nabla^{*}}=\sum_{i<j}\operatorname{Sec}(e_{i}\wedge e_{j}). σ ∇ , ∇ ∗ = i < j ∑ Sec ( e i ∧ e j ) . The Ricci tensors Ric \operatorname{Ric} Ric Ric ∗ \operatorname{Ric}^{*} Ric ∗ ∇ \nabla ∇ ∇ ∗ \nabla^{*} ∇ ∗ E ↦ Rie ( E , F ) G E\mapsto\operatorname{Rie}(E,F)G E ↦ Rie ( E , F ) G E ↦ Rie ∗ ( E , F ) G E\mapsto\operatorname{Rie}^{*}(E,F)G E ↦ Rie ∗ ( E , F ) G
A statistical manifold ( N ~ , ∇ ~ , g ~ ) (\widetilde{N},\widetilde{\nabla},\widetilde{g}) ( N , ∇ , g ) constant curvature c ~ \widetilde{c} c
S ( E , F ) G = c ~ ( g ( F , G ) E − g ( E , G ) F ) . S(E,F)G=\widetilde{c}\bigl(g(F,G)E-g(E,G)F\bigr). S ( E , F ) G = c ( g ( F , G ) E − g ( E , G ) F ) . Statistical submanifolds
Let N N N ( N ~ , ∇ ~ , g ~ ) (\widetilde{N},\widetilde{\nabla},\widetilde{g}) ( N , ∇ , g ) ( ∇ , g ) (\nabla,g) ( ∇ , g ) [41 ]
∇ ~ E F = ∇ E F + h ( E , F ) , ∇ ~ E ∗ F = ∇ E ∗ F + h ∗ ( E , F ) , (5) \widetilde{\nabla}_{E}F=\nabla_{E}F+h(E,F),\qquad\widetilde{\nabla}^{*}_{E}F=\nabla^{*}_{E}F+h^{*}(E,F), \tag*{(5)} ∇ E F = ∇ E F + h ( E , F ) , ∇ E ∗ F = ∇ E ∗ F + h ∗ ( E , F ) , (5) ∇ ~ E U = − A U E + D E ⊥ U , ∇ ~ E ∗ U = − A U ∗ E + D E ⊥ ∗ U , (6) \widetilde{\nabla}_{E}U=-A_{U}E+D_{E}^{\perp}U,\qquad\widetilde{\nabla}^{*}_{E}U=-A^{*}_{U}E+D_{E}^{\perp*}U, \tag*{(6)} ∇ E U = − A U E + D E ⊥ U , ∇ E ∗ U = − A U ∗ E + D E ⊥ ∗ U , (6) for E , F ∈ Γ ( T N ) E,F\in\Gamma(TN) E , F ∈ Γ ( T N ) U ∈ Γ ( T ⊥ N ) U\in\Gamma(T^{\perp}N) U ∈ Γ ( T ⊥ N ) h , h ∗ h,h^{*} h , h ∗
g ~ ( h ( E , F ) , U ) = g ( A U ∗ E , F ) , g ~ ( h ∗ ( E , F ) , U ) = g ( A U E , F ) . (7) \widetilde{g}(h(E,F),U)=g(A^{*}_{U}E,F),\qquad\widetilde{g}(h^{*}(E,F),U)=g(A_{U}E,F). \tag*{(7)} g ( h ( E , F ) , U ) = g ( A U ∗ E , F ) , g ( h ∗ ( E , F ) , U ) = g ( A U E , F ) . (7) The Gauss equation takes the form
g ~ ( Rie ~ ( E , F ) G , H ) = g ( Rie ( E , F ) G , H ) + g ~ ( h ( E , G ) , h ∗ ( F , H ) ) − g ~ ( h ∗ ( E , H ) , h ( F , G ) ) , (8) \widetilde{g}(\widetilde{\operatorname{Rie}}(E,F)G,H)=g(\operatorname{Rie}(E,F)G,H)+\widetilde{g}(h(E,G),h^{*}(F,H))-\widetilde{g}(h^{*}(E,H),h(F,G)), \tag*{(8)} g ( Rie ( E , F ) G , H ) = g ( Rie ( E , F ) G , H ) + g ( h ( E , G ) , h ∗ ( F , H )) − g ( h ∗ ( E , H ) , h ( F , G )) , (8) together with its dual. The mean curvature vectors are
H = 1 m tr h and H ∗ = 1 m tr h ∗ , H=\frac{1}{m}\operatorname{tr}h\qquad\text{and}\qquad H^{*}=\frac{1}{m}\operatorname{tr}h^{*}, H = m 1 tr h and H ∗ = m 1 tr h ∗ , and N N N minimal if H = H ∗ = 0 H=H^{*}=0 H = H ∗ = 0 totally geodesic if h = h ∗ = 0 h=h^{*}=0 h = h ∗ = 0 totally umbilical if h ( E , F ) = g ( E , F ) H h(E,F)=g(E,F)H h ( E , F ) = g ( E , F ) H h ∗ ( E , F ) = g ( E , F ) H ∗ h^{*}(E,F)=g(E,F)H^{*} h ∗ ( E , F ) = g ( E , F ) H ∗ N N N U U U U U U A U = f I A_{U}=fI A U = f I A U ∗ = f I A^{*}_{U}=fI A U ∗ = f I f f f
Note that A U 0 : = 1 2 ( A U + A U ∗ ) A^{0}_{U}:=\frac{1}{2}(A_{U}+A^{*}_{U}) A U 0 := 2 1 ( A U + A U ∗ ) N N N h 0 = 1 2 ( h + h ∗ ) h^{0}=\frac{1}{2}(h+h^{*}) h 0 = 2 1 ( h + h ∗ )
Statistical Solitons, Yamabe-Type Solitons and Concircular Fields
Ricci solitons for general connections
A Ricci soliton ( g , ζ , λ ) (g,\zeta,\lambda) ( g , ζ , λ ) shrinking , steady or expanding according to whether λ < 0 \lambda< 0 λ < 0 λ = 0 \lambda= 0 λ = 0 λ > 0 \lambda> 0 λ > 0 Ric + 1 2 L ζ g + λ g = 0 \operatorname{Ric}+\frac{1}{2}\mathcal{L}_{\zeta}g+\lambda g=0 Ric + 2 1 L ζ g + λ g = 0 [16 ] observed that for a manifold with an arbitrary linear connection ∇ ~ \widetilde{\nabla} ∇
∇ ~ ζ + Q ~ + λ I = 0 , g ~ ( Q ~ E , F ) = Ric ~ ( E , F ) , \widetilde{\nabla}\zeta+\widetilde{Q}+\lambda I=0,\qquad\widetilde{g}(\widetilde{Q}E,F)=\widetilde{\operatorname{Ric}}(E,F), ∇ ζ + Q + λ I = 0 , g ( Q E , F ) = Ric ( E , F ) , where Q ~ \widetilde{Q} Q ( ∇ ~ ζ ) ( E ) = ∇ ~ E ζ (\widetilde{\nabla}\zeta)(E)=\widetilde{\nabla}_{E}\zeta ( ∇ ζ ) ( E ) = ∇ E ζ Ricci-symmetric if Q ~ \widetilde{Q} Q Q ~ ∗ \widetilde{Q}^{*} Q ∗
Definition 3.1 (Statistical soliton). A pair ( ζ , λ ) (\zeta,\lambda) ( ζ , λ ) statistical soliton on a Ricci-symmetric statistical manifold ( N ~ , ∇ ~ , g ~ ) (\widetilde{N},\widetilde{\nabla},\widetilde{g}) ( N , ∇ , g ) ( g ~ , ζ , λ ) (\widetilde{g},\zeta,\lambda) ( g , ζ , λ ) ∇ ~ \widetilde{\nabla} ∇ ∇ ~ ∗ \widetilde{\nabla}^{*} ∇ ∗
∇ ~ ζ + Q ~ + λ I = 0 , ∇ ~ ∗ ζ + Q ~ ∗ + λ I = 0. (9) \widetilde{\nabla}\zeta+\widetilde{Q}+\lambda I=0,\qquad\widetilde{\nabla}^{*}\zeta+\widetilde{Q}^{*}+\lambda I=0. \tag*{(9)} ∇ ζ + Q + λ I = 0 , ∇ ∗ ζ + Q ∗ + λ I = 0. (9) In particular, if ∇ ~ = ∇ ~ ∗ = ∇ ~ 0 \widetilde{\nabla}=\widetilde{\nabla}^{*}=\widetilde{\nabla}^{0} ∇ = ∇ ∗ = ∇ 0 (9) reads
g ( ∇ E ζ , F ) + Ric ( E , F ) + λ g ( E , F ) = 0 , g ( ∇ E ∗ ζ , F ) + Ric ∗ ( E , F ) + λ g ( E , F ) = 0. g(\nabla_{E}\zeta,F)+\operatorname{Ric}(E,F)+\lambda g(E,F)=0,\qquad g(\nabla^{*}_{E}\zeta,F)+\operatorname{Ric}^{*}(E,F)+\lambda g(E,F)=0. g ( ∇ E ζ , F ) + Ric ( E , F ) + λ g ( E , F ) = 0 , g ( ∇ E ∗ ζ , F ) + Ric ∗ ( E , F ) + λ g ( E , F ) = 0. Adding the two equations and using that ∇ 0 = 1 2 ( ∇ + ∇ ∗ ) \nabla^{0}=\frac{1}{2}(\nabla+\nabla^{*}) ∇ 0 = 2 1 ( ∇ + ∇ ∗ )
1 2 ( L ζ g ) + 1 2 ( Ric + Ric ∗ ) + λ g = 0 , \frac{1}{2}(\mathcal{L}_{\zeta}g)+\frac{1}{2}(\operatorname{Ric}+\operatorname{Ric}^{*})+\lambda g=0, 2 1 ( L ζ g ) + 2 1 ( Ric + Ric ∗ ) + λ g = 0 , since L ζ g ( E , F ) = g ( ∇ E ζ , F ) + g ( E , ∇ F ∗ ζ ) \mathcal{L}_{\zeta}g(E,F)=g(\nabla_{E}\zeta,F)+g(E,\nabla^{*}_{F}\zeta) L ζ g ( E , F ) = g ( ∇ E ζ , F ) + g ( E , ∇ F ∗ ζ )
Almost quasi-Yamabe solitons
The Yamabe flow deforms the metric within its conformal class, and its fixed points have constant scalar curvature; a Yamabe soliton satisfies
L ζ g = 2 ( R − λ ) g . \mathcal{L}_{\zeta}g=2(R-\lambda)g. L ζ g = 2 ( R − λ ) g . Following Chen and Deshmukh [15 ] (see also [22 ] ), when λ \lambda λ ω η ⊗ η \omega\eta\otimes\eta ω η ⊗ η
Definition 3.2 (Almost quasi-Yamabe soliton). Let ( B , g ) (B,g) ( B , g ) > 2 >2 > 2 ζ \zeta ζ η \eta η λ , ω \lambda,\omega λ , ω ( g , ζ , λ , ω ) (g,\zeta,\lambda,\omega) ( g , ζ , λ , ω ) almost quasi-Yamabe soliton if
1 2 L ζ g + ( λ − R ) g + ω η ⊗ η = 0. (10) \frac{1}{2}\mathcal{L}_{\zeta}g+(\lambda-R)g+\omega\eta\otimes\eta=0. \tag*{(10)} 2 1 L ζ g + ( λ − R ) g + ω η ⊗ η = 0. (10) where R R R
Concircular and concurrent vector fields
A vector field v v v concircular [17 ] if ∇ E v = δ E \nabla_{E}v=\delta E ∇ E v = δ E δ \delta δ concurrent if δ = 1 \delta=1 δ = 1 v v v
∇ ~ E v = δ E , ∇ ~ E ∗ v = δ E . (11) \widetilde{\nabla}_{E}v=\delta E,\qquad\widetilde{\nabla}^{*}_{E}v=\delta E. \tag*{(11)} ∇ E v = δ E , ∇ E ∗ v = δ E . (11) Chen [12 ] used concircular fields to investigate Ricci solitons on Riemannian submanifolds; the statistical version is developed in § 14 .
Gradient Solitons and Statistical Structures
The soliton equation can also be read in the opposite direction. Instead of asking which solitons live on a given statistical manifold, one asks for which pairs ( h , ∇ ) (h, \nabla) ( h , ∇ ) [8 ] (see also [7 ] ); in this section we review their main results, in the notation of the present chapter.
Throughout, ( M , g ) (M, g) ( M , g ) n n n ∇ 0 \nabla^{0} ∇ 0 Ric 0 \operatorname{Ric}^{0} Ric 0 Q 0 Q^{0} Q 0 g ( Q 0 X , Y ) = Ric 0 ( X , Y ) g(Q^{0}X, Y) = \operatorname{Ric}^{0}(X, Y) g ( Q 0 X , Y ) = Ric 0 ( X , Y ) scal 0 \operatorname{scal}^{0} scal 0 f f f Hess 0 f ( X , Y ) = g ( ∇ X 0 grad f , Y ) \operatorname{Hess}^{0}f(X, Y) = g(\nabla^{0}_{X}\operatorname{grad} f, Y) Hess 0 f ( X , Y ) = g ( ∇ X 0 grad f , Y ) Δ 0 f = tr g Hess 0 f = div ( grad f ) \Delta_{0}f = \operatorname{tr}_{g}\operatorname{Hess}^{0}f = \operatorname{div}(\operatorname{grad} f) Δ 0 f = tr g Hess 0 f = div ( grad f )
Note that this Laplacian carries the sign + + +
Solitons are written here in the form of [8 ] ,
∇ ζ + J = λ I , (12) \nabla\zeta+ J = \lambda I, \tag*{(12)} ∇ ζ + J = λ I , (12) so that compared with (9) the soliton function is replaced by its negative; the vector field is called gradient if ζ = grad f \zeta= \operatorname{grad} f ζ = grad f ξ \xi ξ ζ \zeta ζ θ = g ( ζ , ⋅ ) \theta= g(\zeta, \cdot) θ = g ( ζ , ⋅ )
Nearly statistical and quasi-statistical structures
Let ∇ \nabla ∇ Tor ∇ \operatorname{Tor}^{\nabla} Tor ∇ h h h ( 0 , 2 ) (0,2) ( 0 , 2 )
( d ∇ h ) ( X , Y , Z ) = ( ∇ X h ) ( Y , Z ) − ( ∇ Y h ) ( X , Z ) + h ( Tor ∇ ( X , Y ) , Z ) . (d^{\nabla}h)(X,Y,Z)=(\nabla_{X}h)(Y,Z)-(\nabla_{Y}h)(X,Z)+h(\operatorname{Tor}^{\nabla}(X,Y),Z). ( d ∇ h ) ( X , Y , Z ) = ( ∇ X h ) ( Y , Z ) − ( ∇ Y h ) ( X , Z ) + h ( Tor ∇ ( X , Y ) , Z ) . The pair ( h , ∇ ) (h,\nabla) ( h , ∇ ) quasi-statistical structure if d ∇ h = 0 d^{\nabla}h = 0 d ∇ h = 0 [26 ] ), and a nearly statistical structure if h h h ∇ \nabla ∇ d ∇ h = 0 d^{\nabla}h = 0 d ∇ h = 0 ( ∇ X h ) ( Y , Z ) = ( ∇ Y h ) ( X , Z ) (\nabla_{X}h)(Y,Z) = (\nabla_{Y}h)(X,Z) ( ∇ X h ) ( Y , Z ) = ( ∇ Y h ) ( X , Z ) ( g , ∇ ) (g,\nabla) ( g , ∇ ) g g g Ric ∇ = 0 \operatorname{Ric}^{\nabla} = 0 Ric ∇ = 0 Ric ∇ ∗ = 0 \operatorname{Ric}^{\nabla^{*}} = 0 Ric ∇ ∗ = 0 Hessian (dually flat) structure [36 ] .
Hessian metrics and Ricci metrics
If Hess 0 f \operatorname{Hess}^{0}f Hess 0 f Ric 0 \operatorname{Ric}^{0} Ric 0
2 g ( ∇ X Ric 0 Y − ∇ X 0 Y , Q 0 Z ) = g ( ( ∇ Y 0 Q 0 ) X , Z ) + g ( ( ∇ X 0 Q 0 ) Y , Z ) − g ( ( ∇ Z 0 Q 0 ) X , Y ) . 2g\left(\nabla^{\operatorname{Ric}^{0}}_{X}Y-\nabla^{0}_{X}Y,Q^{0}Z\right)=g\left((\nabla^{0}_{Y}Q^{0})X,Z\right)+g\left((\nabla^{0}_{X}Q^{0})Y,Z\right)-g\left((\nabla^{0}_{Z}Q^{0})X,Y\right). 2 g ( ∇ X Ric 0 Y − ∇ X 0 Y , Q 0 Z ) = g ( ( ∇ Y 0 Q 0 ) X , Z ) + g ( ( ∇ X 0 Q 0 ) Y , Z ) − g ( ( ∇ Z 0 Q 0 ) X , Y ) . From this one reads off the following facts [7 ] :
( Ric 0 , ∇ 0 ) (\operatorname{Ric}^{0},\nabla^{0}) ( Ric 0 , ∇ 0 ) ( ∇ X 0 Q 0 ) Y = ( ∇ Y 0 Q 0 ) X (\nabla^{0}_{X}Q^{0})Y=(\nabla^{0}_{Y}Q^{0})X ( ∇ X 0 Q 0 ) Y = ( ∇ Y 0 Q 0 ) X X , Y X,Y X , Y
if ( g , ∇ Ric 0 ) (g,\nabla^{\operatorname{Ric}^{0}}) ( g , ∇ Ric 0 ) D : = ∇ Ric 0 − ∇ 0 D:=\nabla^{\operatorname{Ric}^{0}}-\nabla^{0} D := ∇ Ric 0 − ∇ 0
g ( X , D ( Y , Z ) ) = g ( Y , D ( X , Z ) ) ; g(X,D(Y,Z))=g(Y,D(X,Z)); g ( X , D ( Y , Z )) = g ( Y , D ( X , Z )) ; if ( g , ∇ ) (g,\nabla) ( g , ∇ ) ( Ric 0 , ∇ ) (\operatorname{Ric}^{0},\nabla) ( Ric 0 , ∇ )
g ( ( ∇ X Q 0 ) Z , Y ) = g ( ( ∇ Y Q 0 ) Z , X ) . g((\nabla_{X}Q^{0})Z,Y)=g((\nabla_{Y}Q^{0})Z,X). g (( ∇ X Q 0 ) Z , Y ) = g (( ∇ Y Q 0 ) Z , X ) . Theorem 4.1 ([8 ] ). ( Hess 0 f , ∇ 0 ) (\operatorname{Hess}^{0} f,\nabla^{0}) ( Hess 0 f , ∇ 0 ) M M M f f f Rie 0 ( X , Y ) grad f = 0 \operatorname{Rie}^{0}(X,Y)\operatorname{grad} f=0 Rie 0 ( X , Y ) grad f = 0 X , Y ∈ Γ ( T M ) X,Y\in\Gamma(TM) X , Y ∈ Γ ( T M )
Proof. Since ∇ 0 \nabla^{0} ∇ 0 ∇ 0 g = 0 \nabla^{0}g=0 ∇ 0 g = 0
( ∇ X 0 Hess 0 f ) ( Y , Z ) − ( ∇ Y 0 Hess 0 f ) ( X , Z ) (\nabla^{0}_{X}\operatorname{Hess}^{0}f)(Y,Z)-(\nabla^{0}_{Y}\operatorname{Hess}^{0}f)(X,Z) ( ∇ X 0 Hess 0 f ) ( Y , Z ) − ( ∇ Y 0 Hess 0 f ) ( X , Z ) gives
g ( ( ∇ 0 ) X , Y 2 grad f − ( ∇ 0 ) Y , X 2 grad f , Z ) = g ( Rie 0 ( X , Y ) grad f , Z ) . g\left((\nabla^{0})^{2}_{X,Y}\operatorname{grad}f-(\nabla^{0})^{2}_{Y,X}\operatorname{grad}f,Z\right)=g(\operatorname{Rie}^{0}(X,Y)\operatorname{grad}f,Z). g ( ( ∇ 0 ) X , Y 2 grad f − ( ∇ 0 ) Y , X 2 grad f , Z ) = g ( Rie 0 ( X , Y ) grad f , Z ) . where ∇ X , Y 2 = ∇ X ∇ Y − ∇ ∇ X Y \nabla^{2}_{X,Y}=\nabla_{X}\nabla_{Y}-\nabla_{\nabla_{X}Y} ∇ X , Y 2 = ∇ X ∇ Y − ∇ ∇ X Y
Remark 4.2. The term radial curvature goes back to Klingenberg [24 ] .
For a general statistical structure, the same computation is slightly less rigid. Let ( g , ∇ ) (g,\nabla) ( g , ∇ ) C = ∇ g C=\nabla g C = ∇ g Hess ∇ f ( X , Y ) = g ( ∇ X grad f , Y ) \operatorname{Hess}^{\nabla}f(X,Y)=g(\nabla_{X}\operatorname{grad}f,Y) Hess ∇ f ( X , Y ) = g ( ∇ X grad f , Y ) ∇ = ∇ 0 + T \nabla=\nabla^{0}+T ∇ = ∇ 0 + T g ( T ( ⋅ , ⋅ ) , ⋅ ) g(T(\cdot,\cdot),\cdot) g ( T ( ⋅ , ⋅ ) , ⋅ ) (2) .
Theorem 4.3. Let ( g , ∇ ) (g,\nabla) ( g , ∇ ) ( Hess ∇ f , ∇ ) (\operatorname{Hess}^{\nabla}f,\nabla) ( Hess ∇ f , ∇ )
g ( Rie ∇ ( X , Y ) grad f , Z ) = C ( Y , ∇ X grad f , Z ) − C ( X , ∇ Y grad f , Z ) , X , Y , Z ∈ Γ ( T M ) . g(\operatorname{Rie}^{\nabla}(X,Y)\operatorname{grad}f,Z)=C(Y,\nabla_{X}\operatorname{grad}f,Z)-C(X,\nabla_{Y}\operatorname{grad}f,Z),\qquad X,Y,Z\in\Gamma(TM). g ( Rie ∇ ( X , Y ) grad f , Z ) = C ( Y , ∇ X grad f , Z ) − C ( X , ∇ Y grad f , Z ) , X , Y , Z ∈ Γ ( T M ) . In particular , if ∇ g = 0 \nabla g=0 ∇ g = 0 ∇ = ∇ 0 \nabla=\nabla^{0} ∇ = ∇ 0 Rie ∇ ( X , Y ) grad f = 0 \operatorname{Rie}^{\nabla}(X,Y)\operatorname{grad}f=0 Rie ∇ ( X , Y ) grad f = 0
Proof. With H = Hess ∇ f H=\operatorname{Hess}^{\nabla}f H = Hess ∇ f
( ∇ X H ) ( Y , Z ) = C ( X , ∇ Y grad f , Z ) + g ( ∇ X , Y 2 grad f , Z ) . (\nabla_{X}H)(Y,Z)=C(X,\nabla_{Y}\operatorname{grad}f,Z)+g(\nabla^{2}_{X,Y}\operatorname{grad}f,Z). ( ∇ X H ) ( Y , Z ) = C ( X , ∇ Y grad f , Z ) + g ( ∇ X , Y 2 grad f , Z ) . Antisymmetrising in X , Y X,Y X , Y Tor ∇ = 0 \operatorname{Tor}^{\nabla}=0 Tor ∇ = 0
( d ∇ H ) ( X , Y , Z ) = C ( X , ∇ Y grad f , Z ) − C ( Y , ∇ X grad f , Z ) + g ( Rie ∇ ( X , Y ) grad f , Z ) . (d^{\nabla}H)(X,Y,Z)=C(X,\nabla_{Y}\operatorname{grad}f,Z)-C(Y,\nabla_{X}\operatorname{grad}f,Z)+g(\operatorname{Rie}^{\nabla}(X,Y)\operatorname{grad}f,Z). ( d ∇ H ) ( X , Y , Z ) = C ( X , ∇ Y grad f , Z ) − C ( Y , ∇ X grad f , Z ) + g ( Rie ∇ ( X , Y ) grad f , Z ) . Remark 4.4. In [8 ] the statement is given as “Rie ∇ ( X , Y ) grad f = 0 \operatorname{Rie}^{\nabla}(X,Y)\operatorname{grad}f=0 Rie ∇ ( X , Y ) grad f = 0 ∇ \nabla ∇ C C C ∇ g \nabla g ∇ g ∇ g = 0 \nabla g=0 ∇ g = 0
Gradient almost solitons
Let f , λ f,\lambda f , λ ( g , f , λ ) (g,f,\lambda) ( g , f , λ )
gradient almost Ricci soliton if Hess 0 f + Ric 0 = λ g \operatorname{Hess}^{0}f+\operatorname{Ric}^{0}=\lambda g Hess 0 f + Ric 0 = λ g
gradient almost Einstein soliton if Hess 0 f + Ric 0 = ( λ + 1 2 scal 0 ) g \operatorname{Hess}^{0}f+\operatorname{Ric}^{0}=(\lambda+\frac{1}{2}\operatorname{scal}^{0})g Hess 0 f + Ric 0 = ( λ + 2 1 scal 0 ) g
gradient almost Yamabe soliton if Hess 0 f = ( λ − scal 0 ) g \operatorname{Hess}^{0}f=(\lambda-\operatorname{scal}^{0})g Hess 0 f = ( λ − scal 0 ) g
When λ \lambda λ
( ∇ X 0 Hess 0 f ) ( Y , Z ) + ( ∇ X 0 Ric 0 ) ( Y , Z ) = X ( λ ) g ( Y , Z ) (\nabla^{0}_{X}\operatorname{Hess}^{0}f)(Y,Z)+(\nabla^{0}_{X}\operatorname{Ric}^{0})(Y,Z)=X(\lambda)g(Y,Z) ( ∇ X 0 Hess 0 f ) ( Y , Z ) + ( ∇ X 0 Ric 0 ) ( Y , Z ) = X ( λ ) g ( Y , Z ) with λ \lambda λ λ + 1 2 scal 0 \lambda+\frac{1}{2}\operatorname{scal}^{0} λ + 2 1 scal 0
( ∇ X 0 Hess 0 f ) ( Y , Z ) = X ( λ − scal 0 ) g ( Y , Z ) (\nabla^{0}_{X}\operatorname{Hess}^{0}f)(Y,Z)=X(\lambda-\operatorname{scal}^{0})g(Y,Z) ( ∇ X 0 Hess 0 f ) ( Y , Z ) = X ( λ − scal 0 ) g ( Y , Z ) for Yamabe solitons. This yields:
Proposition 4.5 ([8 ] ). (i) If ( g , f , λ ) (g,f,\lambda) ( g , f , λ ) ( Hess 0 f , ∇ 0 ) (\operatorname{Hess}^{0} f,\nabla^{0}) ( Hess 0 f , ∇ 0 ) ( Ric 0 , ∇ 0 ) (\operatorname{Ric}^{0},\nabla^{0}) ( Ric 0 , ∇ 0 )
(ii) If ( g , f , λ ) (g,f,\lambda) ( g , f , λ ) scal 0 \operatorname{scal}^{0} scal 0
(iii) If ( g , f , λ ) (g,f,\lambda) ( g , f , λ ) scal 0 \operatorname{scal}^{0} scal 0 ( Hess 0 f , ∇ 0 ) (\operatorname{Hess}^{0} f,\nabla^{0}) ( Hess 0 f , ∇ 0 )
Proposition 4.6 ([8 ] ). Let Ric 0 \operatorname{Ric}^{0} Ric 0
(i) If ( g , f , λ ) (g,f,\lambda) ( g , f , λ ) ( Ric 0 , ∇ 0 ) (\operatorname{Ric}^{0},\nabla^{0}) ( Ric 0 , ∇ 0 )
g ( Rie 0 ( X , Y ) grad f , Z ) = X ( λ ) g ( Y , Z ) − Y ( λ ) g ( X , Z ) . g(\operatorname{Rie}^{0}(X,Y)\operatorname{grad} f,Z)=X(\lambda)g(Y,Z)-Y(\lambda)g(X,Z). g ( Rie 0 ( X , Y ) grad f , Z ) = X ( λ ) g ( Y , Z ) − Y ( λ ) g ( X , Z ) . (ii) If ( g , f , λ ) (g,f,\lambda) ( g , f , λ ) ( Ric 0 , ∇ 0 ) (\operatorname{Ric}^{0},\nabla^{0}) ( Ric 0 , ∇ 0 )
g ( Rie 0 ( X , Y ) grad f , Z ) = X ( λ + scal 0 2 ) g ( Y , Z ) − Y ( λ + scal 0 2 ) g ( X , Z ) . g(\operatorname{Rie}^{0}(X,Y)\operatorname{grad} f,Z)=X\left(\lambda+\frac{\operatorname{scal}^{0}}{2}\right)g(Y,Z)-Y\left(\lambda+\frac{\operatorname{scal}^{0}}{2}\right)g(X,Z). g ( Rie 0 ( X , Y ) grad f , Z ) = X ( λ + 2 scal 0 ) g ( Y , Z ) − Y ( λ + 2 scal 0 ) g ( X , Z ) . These follow by combining the differentiated soliton equations with the identity
d ∇ 0 Hess 0 f ( X , Y , Z ) = g ( Rie 0 ( X , Y ) grad f , Z ) d^{\nabla^{0}}\operatorname{Hess}^{0}f(X,Y,Z)=g(\operatorname{Rie}^{0}(X,Y)\operatorname{grad} f,Z) d ∇ 0 Hess 0 f ( X , Y , Z ) = g ( Rie 0 ( X , Y ) grad f , Z ) from the proof of Theorem 4.1 . Analogous conditions for a statistical connection ∇ ≠ ∇ 0 \nabla\ne\nabla^{0} ∇ = ∇ 0 [8 Prop. 2.6] .
Almost ( ∇ , J , ζ , λ ) (\nabla,J,\zeta,\lambda) ( ∇ , J , ζ , λ )
The three soliton equations above are of the form ∇ 0 ζ + J = λ ′ I \nabla^{0}\zeta+J=\lambda'I ∇ 0 ζ + J = λ ′ I ζ = grad f \zeta=\operatorname{grad} f ζ = grad f J = Q 0 J=Q^{0} J = Q 0 λ ′ = λ \lambda'=\lambda λ ′ = λ J = Q 0 J=Q^{0} J = Q 0 λ ′ = λ + 1 2 scal 0 \lambda'=\lambda+\frac{1}{2}\operatorname{scal}^{0} λ ′ = λ + 2 1 scal 0 J = 0 J=0 J = 0 λ ′ = λ − scal 0 \lambda'=\lambda-\operatorname{scal}^{0} λ ′ = λ − scal 0 ( ∇ , J , ζ , λ ) (\nabla,J,\zeta,\lambda) ( ∇ , J , ζ , λ ) ∇ \nabla ∇ ( 1 , 1 ) (1,1) ( 1 , 1 ) J J J ζ \zeta ζ λ \lambda λ (12) . Put Ω = g ( J ⋅ , ⋅ ) \Omega=g(J\cdot,\cdot) Ω = g ( J ⋅ , ⋅ )
Lemma 4.7. For an almost ( ∇ , J , ζ , λ ) (\nabla,J,\zeta,\lambda) ( ∇ , J , ζ , λ ) Ω \Omega Ω ∇ ζ \nabla\zeta ∇ ζ g g g
g ( ∇ X ζ , Y ) = g ( X , ∇ Y ζ ) . g(\nabla_X\zeta,Y)=g(X,\nabla_Y\zeta). g ( ∇ X ζ , Y ) = g ( X , ∇ Y ζ ) . Lemma 4.8 ([7 ] ). For torsion-free ∇ \nabla ∇ ( ∇ X Ω ) ( Y , Z ) = ( ∇ Y Ω ) ( X , Z ) (\nabla_X\Omega)(Y,Z)=(\nabla_Y\Omega)(X,Z) ( ∇ X Ω ) ( Y , Z ) = ( ∇ Y Ω ) ( X , Z ) ( ∇ X g ) ( J Y , Z ) − ( ∇ Y g ) ( J X , Z ) = g ( ( ∇ Y J ) X − ( ∇ X J ) Y , Z ) (\nabla_Xg)(JY,Z)-(\nabla_Yg)(JX,Z)=g((\nabla_YJ)X-(\nabla_XJ)Y,Z) ( ∇ X g ) ( J Y , Z ) − ( ∇ Y g ) ( J X , Z ) = g (( ∇ Y J ) X − ( ∇ X J ) Y , Z ) Ω \Omega Ω ∇ 0 \nabla^{0} ∇ 0 J J J ∇ 0 \nabla^{0} ∇ 0
Remark 4.9. If Ω \Omega Ω ∇ 0 \nabla^{0} ∇ 0 J J J ( ∇ X 0 J ) X = 0 ( \nabla^{0}_{X}J)X=0 ( ∇ X 0 J ) X = 0 ∇ 0 J = 0 \nabla^{0}J=0 ∇ 0 J = 0
Proposition 4.10 ([8 ] ). Let ( ∇ , J , ζ , λ ) (\nabla,J,\zeta,\lambda) ( ∇ , J , ζ , λ ) ( M , g ) (M,g) ( M , g ) Ω \Omega Ω ∇ \nabla ∇ ( Ω , ∇ ) (\Omega,\nabla) ( Ω , ∇ )
g ( Rie ∇ ( X , Y ) ζ , Z ) = ( ∇ X g ) ( J Y , Z ) − ( ∇ Y g ) ( J X , Z ) + g ( X ( λ ) Y − Y ( λ ) X , Z ) . g(\operatorname{Rie}^{\nabla}(X,Y)\zeta,Z)=(\nabla_Xg)(JY,Z)-(\nabla_Yg)(JX,Z)+g(X(\lambda)Y-Y(\lambda)X,Z). g ( Rie ∇ ( X , Y ) ζ , Z ) = ( ∇ X g ) ( J Y , Z ) − ( ∇ Y g ) ( J X , Z ) + g ( X ( λ ) Y − Y ( λ ) X , Z ) . Proof. From (12) , Rie ∇ ( X , Y ) ζ = ( ∇ Y J ) X − ( ∇ X J ) Y + X ( λ ) Y − Y ( λ ) X \operatorname{Rie}^{\nabla}(X,Y)\zeta=(\nabla_YJ)X-(\nabla_XJ)Y+X(\lambda)Y-Y(\lambda)X Rie ∇ ( X , Y ) ζ = ( ∇ Y J ) X − ( ∇ X J ) Y + X ( λ ) Y − Y ( λ ) X Lemma 4.8 .
Corollary 4.11. For ∇ = ∇ 0 \nabla=\nabla^{0} ∇ = ∇ 0 Ω \Omega Ω ( Ω , ∇ 0 ) (\Omega,\nabla^{0}) ( Ω , ∇ 0 )
Rie 0 ( X , Y ) ζ = X ( λ ) Y − Y ( λ ) X . \operatorname{Rie}^{0}(X,Y)\zeta=X(\lambda)Y-Y(\lambda)X. Rie 0 ( X , Y ) ζ = X ( λ ) Y − Y ( λ ) X . If λ \lambda λ ζ = grad f \zeta=\operatorname{grad} f ζ = grad f ( Ω , ∇ 0 ) (\Omega,\nabla^{0}) ( Ω , ∇ 0 ) Rie 0 ( X , Y ) grad f = 0 \operatorname{Rie}^{0}(X,Y)\operatorname{grad} f=0 Rie 0 ( X , Y ) grad f = 0 ( Hess 0 f , ∇ 0 ) (\operatorname{Hess}^{0}f,\nabla^{0}) ( Hess 0 f , ∇ 0 )
Connections defined by 1-forms
Inspired by projective equivalence, for an arbitrary 1-form θ = g ( ζ , ⋅ ) \theta= g(\zeta,\cdot) θ = g ( ζ , ⋅ ) ( M , g ) (M,g) ( M , g )
∇ θ : = ∇ 0 + θ ⊗ I + I ⊗ θ + g ⊗ ζ , i.e. ∇ X θ Y = ∇ X 0 Y + θ ( X ) Y + θ ( Y ) X + g ( X , Y ) ζ . (13) \nabla^\theta:= \nabla^0 + \theta\otimes I + I \otimes\theta+ g \otimes\zeta,\qquad\text{i.e.}\quad\nabla^\theta_XY = \nabla^0_XY + \theta(X)Y + \theta(Y)X + g(X,Y)\zeta. \tag*{(13)} ∇ θ := ∇ 0 + θ ⊗ I + I ⊗ θ + g ⊗ ζ , i.e. ∇ X θ Y = ∇ X 0 Y + θ ( X ) Y + θ ( Y ) X + g ( X , Y ) ζ . (13) Then Tor ∇ θ = 0 \operatorname{Tor}^{\nabla^\theta}=0 Tor ∇ θ = 0 ( ∇ X θ g ) ( Y , Z ) = − 2 { θ ( X ) g ( Y , Z ) + θ ( Y ) g ( X , Z ) + θ ( Z ) g ( X , Y ) } (\nabla^\theta_Xg)(Y,Z)=-2\{\theta(X)g(Y,Z)+\theta(Y)g(X,Z)+\theta(Z)g(X,Y)\} ( ∇ X θ g ) ( Y , Z ) = − 2 { θ ( X ) g ( Y , Z ) + θ ( Y ) g ( X , Z ) + θ ( Z ) g ( X , Y )} d ∇ θ g = 0 d^{\nabla^\theta}g=0 d ∇ θ g = 0
Proposition 4.12 ([8 ] ). For every 1-form θ \theta θ ( g , ∇ θ ) (g,\nabla^\theta) ( g , ∇ θ ) ∇ − θ \nabla^{-\theta} ∇ − θ ∇ ζ θ = ∇ ζ 0 + 3 ∣ ζ ∣ 2 ζ \nabla^\theta_\zeta=\nabla^0_\zeta+3|\zeta|^2\zeta ∇ ζ θ = ∇ ζ 0 + 3∣ ζ ∣ 2 ζ
(i) ζ \zeta ζ ∇ θ \nabla^\theta ∇ θ ( ∇ 0 , J = 2 θ ⊗ ζ , ζ , λ = − ∣ ζ ∣ 2 ) (\nabla^0,J=2\theta\otimes\zeta,\zeta,\lambda=-|\zeta|^2) ( ∇ 0 , J = 2 θ ⊗ ζ , ζ , λ = − ∣ ζ ∣ 2 )
(ii) θ \theta θ ∇ θ \nabla^\theta ∇ θ ( ∇ 0 , J = − 2 θ ⊗ ζ , ζ , λ = ∣ ζ ∣ 2 ) (\nabla^0,J=-2\theta\otimes\zeta,\zeta,\lambda=|\zeta|^2) ( ∇ 0 , J = − 2 θ ⊗ ζ , ζ , λ = ∣ ζ ∣ 2 )
Proof. By direct calculation,
g ( ∇ X θ Y , Z ) + g ( Y , ∇ X − θ Z ) = X g ( Y , Z ) . g(\nabla^\theta_XY,Z)+g(Y,\nabla^{-\theta}_XZ)=Xg(Y,Z). g ( ∇ X θ Y , Z ) + g ( Y , ∇ X − θ Z ) = X g ( Y , Z ) . For (i), ∇ X θ ζ = ∇ X 0 ζ + 2 θ ( X ) ζ + ∣ ζ ∣ 2 X \nabla^\theta_X\zeta=\nabla^0_X\zeta+2\theta(X)\zeta+|\zeta|^2X ∇ X θ ζ = ∇ X 0 ζ + 2 θ ( X ) ζ + ∣ ζ ∣ 2 X ( ∇ X θ θ ) ( Y ) = ( ∇ X 0 θ ) ( Y ) − 2 θ ( X ) θ ( Y ) − ∣ ζ ∣ 2 g ( X , Y ) (\nabla^\theta_X\theta)(Y)=(\nabla^0_X\theta)(Y)-2\theta(X)\theta(Y)-|\zeta|^2g(X,Y) ( ∇ X θ θ ) ( Y ) = ( ∇ X 0 θ ) ( Y ) − 2 θ ( X ) θ ( Y ) − ∣ ζ ∣ 2 g ( X , Y )
Remark 4.13. The signs of λ \lambda λ [8 ] ; the signs above are the ones obtained from the two displayed computations.
The curvature of ∇ θ \nabla^\theta ∇ θ
( Rie θ − Rie 0 ) ( X , Y ) Z = [ g ( Y , ∇ X 0 ζ ) − g ( X , ∇ Y 0 ζ ) ] Z + [ θ ( Y ) θ ( Z ) + g ( Y , Z ) ∣ ζ ∣ 2 − g ( Z , ∇ ζ 0 ) ] X − [ θ ( X ) θ ( Z ) + g ( X , Z ) ∣ ζ ∣ 2 − g ( Z , ∇ X 0 ζ ) ] Y + g ( Y , Z ) ∇ X 0 ζ − g ( X , Z ) ∇ Y 0 ζ + [ θ ( X ) g ( Y , Z ) − θ ( Y ) g ( X , Z ) ] ζ . \begin{aligned}
(\operatorname{Rie}^{\theta}-\operatorname{Rie}^{0})(X,Y)Z
={}&[g(Y,\nabla^0_X\zeta)-g(X,\nabla^0_Y\zeta)]Z\\
&+[\theta(Y)\theta(Z)+g(Y,Z)|\zeta|^2-g(Z,\nabla^0_\zeta)]X\\
&-[\theta(X)\theta(Z)+g(X,Z)|\zeta|^2-g(Z,\nabla^0_X\zeta)]Y\\
&+g(Y,Z)\nabla^0_X\zeta-g(X,Z)\nabla^0_Y\zeta\\
&+[\theta(X)g(Y,Z)-\theta(Y)g(X,Z)]\zeta.
\end{aligned} ( Rie θ − Rie 0 ) ( X , Y ) Z = [ g ( Y , ∇ X 0 ζ ) − g ( X , ∇ Y 0 ζ )] Z + [ θ ( Y ) θ ( Z ) + g ( Y , Z ) ∣ ζ ∣ 2 − g ( Z , ∇ ζ 0 )] X − [ θ ( X ) θ ( Z ) + g ( X , Z ) ∣ ζ ∣ 2 − g ( Z , ∇ X 0 ζ )] Y + g ( Y , Z ) ∇ X 0 ζ − g ( X , Z ) ∇ Y 0 ζ + [ θ ( X ) g ( Y , Z ) − θ ( Y ) g ( X , Z )] ζ . In particular, if ζ \zeta ζ g g g ∇ 0 \nabla^0 ∇ 0 [40 ] ), then ( Rie θ − Rie 0 ) ( X , Y ) Z ∈ ker θ (\operatorname{Rie}^{\theta}-\operatorname{Rie}^{0})(X,Y)Z\in\ker\theta ( Rie θ − Rie 0 ) ( X , Y ) Z ∈ ker θ
Example 4.14. Let ( M , φ , ξ , η , g ) (M,\varphi,\xi,\eta,g) ( M , φ , ξ , η , g ) ∇ η = ∇ 0 + η ⊗ I + I ⊗ η + g ⊗ ξ \nabla^\eta=\nabla^0+\eta\otimes I+I\otimes\eta+g\otimes\xi ∇ η = ∇ 0 + η ⊗ I + I ⊗ η + g ⊗ ξ ∇ 0 ξ = I − η ⊗ ξ \nabla^0\xi=I-\eta\otimes\xi ∇ 0 ξ = I − η ⊗ ξ ∇ η ξ = 2 I + η ⊗ ξ \nabla^\eta\xi=2I+\eta\otimes\xi ∇ η ξ = 2 I + η ⊗ ξ ( ∇ η , J = − η ⊗ ξ , ξ , λ = 2 ) (\nabla^\eta,J=-\eta\otimes\xi,\xi,\lambda=2) ( ∇ η , J = − η ⊗ ξ , ξ , λ = 2 ) M M M
Divergence, Laplacian and Hessian. Tracing (13) gives div θ Z = div 0 Z + ( n + 2 ) θ ( Z ) \operatorname{div}^{\theta}Z=\operatorname{div}^{0}Z+(n+2)\theta(Z) div θ Z = div 0 Z + ( n + 2 ) θ ( Z ) Δ θ = div θ ∘ grad \Delta_\theta=\operatorname{div}^{\theta}\circ\operatorname{grad} Δ θ = div θ ∘ grad
Δ θ f ~ = Δ 0 f ~ + ( n + 2 ) θ ( grad f ~ ) , \Delta_\theta\widetilde f=\Delta_0\widetilde f+(n+2)\theta(\operatorname{grad}\widetilde f), Δ θ f = Δ 0 f + ( n + 2 ) θ ( grad f ) , Hess θ f ~ ( X , Y ) = Hess 0 f ~ ( X , Y ) + θ ( grad f ~ ) g ( X , Y ) + θ ( X ) d f ~ ( Y ) + θ ( Y ) d f ~ ( X ) . \operatorname{Hess}^{\theta}\widetilde f(X,Y)=\operatorname{Hess}^{0}\widetilde f(X,Y)+\theta(\operatorname{grad}\widetilde f)g(X,Y)+\theta(X)d\widetilde f(Y)+\theta(Y)d\widetilde f(X). Hess θ f ( X , Y ) = Hess 0 f ( X , Y ) + θ ( grad f ) g ( X , Y ) + θ ( X ) d f ( Y ) + θ ( Y ) d f ( X ) . For ζ = grad f \zeta=\operatorname{grad}f ζ = grad f θ = d f \theta=df θ = df
Hess d f f = Hess 0 f + ∣ grad f ∣ 2 g + 2 d f ⊗ d f , Δ d f f = Δ 0 f + ( n + 2 ) ∣ grad f ∣ 2 . (14) \operatorname{Hess}^{df}f=\operatorname{Hess}^{0}f+|\operatorname{grad}f|^2g+2\,df\otimes df,\qquad\Delta_{df}f=\Delta_0f+(n+2)|\operatorname{grad}f|^2. \tag*{(14)} Hess df f = Hess 0 f + ∣ grad f ∣ 2 g + 2 df ⊗ df , Δ df f = Δ 0 f + ( n + 2 ) ∣ grad f ∣ 2 . (14) Consequently, if f f f Δ 0 \Delta_0 Δ 0 Δ d f f ≥ 0 \Delta_{df}f\geq0 Δ df f ≥ 0 ∣ grad f ∣ 2 ≥ 0 |\operatorname{grad}f|^2\geq0 ∣ grad f ∣ 2 ≥ 0 f f f Δ d f \Delta_{df} Δ df f f f Δ d f \Delta_{df} Δ df f ~ \widetilde f f Δ 0 \Delta_0 Δ 0 grad f ⊥ grad f ~ \operatorname{grad}f\perp\operatorname{grad}\widetilde f grad f ⊥ grad f div d f \operatorname{div}^{df} div df M M M
∫ M div d f X d μ g = ( n + 2 ) ∫ M g ( grad f , X ) d μ g . \int_M \operatorname{div}^{df}X\,d\mu_g=(n+2)\int_M g(\operatorname{grad}f,X)\,d\mu_g. ∫ M div df X d μ g = ( n + 2 ) ∫ M g ( grad f , X ) d μ g . Ricci tensor. Tracing the curvature formula yields
Ric θ ( Y , Z ) = Ric 0 ( Y , Z ) + g ( Y , Z ) { n ∣ ζ ∣ 2 + div 0 ζ } + ( n − 2 ) θ ( Y ) θ ( Z ) + g ( Y , ∇ Z 0 ζ ) − ( n + 1 ) g ( Z , ∇ Y 0 ζ ) . \operatorname{Ric}^{\theta}(Y,Z)=\operatorname{Ric}^{0}(Y,Z)+g(Y,Z)\{n|\zeta|^{2}+\operatorname{div}^{0}\zeta\}+(n-2)\theta(Y)\theta(Z)+g(Y,\nabla^{0}_{Z}\zeta)-(n+1)g(Z,\nabla^{0}_{Y}\zeta). Ric θ ( Y , Z ) = Ric 0 ( Y , Z ) + g ( Y , Z ) { n ∣ ζ ∣ 2 + div 0 ζ } + ( n − 2 ) θ ( Y ) θ ( Z ) + g ( Y , ∇ Z 0 ζ ) − ( n + 1 ) g ( Z , ∇ Y 0 ζ ) . Proposition 4.15. ∇ θ \nabla^{\theta} ∇ θ equiaffine if and only if ∇ 0 ζ \nabla^{0}\zeta ∇ 0 ζ g g g θ \theta θ ∇ d f \nabla^{df} ∇ df
Indeed, an equiaffine torsion-free connection is characterised by symmetry of its Ricci tensor [30 ] , and the only non-symmetric part of Ric θ \operatorname{Ric}^{\theta} Ric θ g ( Y , ∇ Z 0 ζ ) − ( n + 1 ) g ( Z , ∇ Y 0 ζ ) g(Y,\nabla^{0}_{Z}\zeta)-(n+1)g(Z,\nabla^{0}_{Y}\zeta) g ( Y , ∇ Z 0 ζ ) − ( n + 1 ) g ( Z , ∇ Y 0 ζ ) ( n + 2 ) (n+2) ( n + 2 ) g ( Y , ∇ Z 0 ζ ) g(Y,\nabla^{0}_{Z}\zeta) g ( Y , ∇ Z 0 ζ )
Remark 4.16 (Conjugate Ricci symmetry). Recall that a statistical manifold is conjugate Ricci-symmetric if Ric ∇ = Ric ∇ ∗ \operatorname{Ric}^{\nabla}=\operatorname{Ric}^{\nabla^{*}} Ric ∇ = Ric ∇ ∗ [27 ] . For ( g , ∇ θ , ∇ − θ ) (g,\nabla^{\theta},\nabla^{-\theta}) ( g , ∇ θ , ∇ − θ ) Ric θ − Ric − θ \operatorname{Ric}^{\theta}-\operatorname{Ric}^{-\theta} Ric θ − Ric − θ ζ \zeta ζ 2 { g ( Y , Z ) div 0 ζ + g ( Y , ∇ Z 0 ζ ) − ( n + 1 ) g ( Z , ∇ Y 0 ζ ) } 2\{g(Y,Z)\operatorname{div}^{0}\zeta+g(Y,\nabla^{0}_{Z}\zeta)-(n+1)g(Z,\nabla^{0}_{Y}\zeta)\} 2 { g ( Y , Z ) div 0 ζ + g ( Y , ∇ Z 0 ζ ) − ( n + 1 ) g ( Z , ∇ Y 0 ζ )} ( n + 2 ) d θ (n+2)d\theta ( n + 2 ) d θ 2 div 0 ζ g − n L ζ g 2\operatorname{div}^{0}\zeta\,g-n\mathcal{L}_{\zeta}g 2 div 0 ζ g − n L ζ g ( M , g , ∇ θ , ∇ − θ ) (M,g,\nabla^{\theta},\nabla^{-\theta}) ( M , g , ∇ θ , ∇ − θ ) θ \theta θ L ζ g = 2 n ( div 0 ζ ) g \mathcal{L}_{\zeta}g=\frac{2}{n}(\operatorname{div}^{0}\zeta)g L ζ g = n 2 ( div 0 ζ ) g ζ \zeta ζ ζ = grad f \zeta=\operatorname{grad}f ζ = grad f Hess 0 f = 1 n ( Δ 0 f ) g \operatorname{Hess}^{0}f=\frac{1}{n}(\Delta_{0}f)g Hess 0 f = n 1 ( Δ 0 f ) g
In the gradient case ζ = grad f \zeta=\operatorname{grad}f ζ = grad f θ = d f \theta=df θ = df
Ric d f = Ric 0 + { n ∣ grad f ∣ 2 + Δ 0 f } g + ( n − 2 ) d f ⊗ d f − n Hess 0 f , (15) \operatorname{Ric}^{df}=\operatorname{Ric}^{0}+\{n|\operatorname{grad}f|^{2}+\Delta_{0}f\}g+(n-2)df\otimes df-n\operatorname{Hess}^{0}f, \tag*{(15)} Ric df = Ric 0 + { n ∣ grad f ∣ 2 + Δ 0 f } g + ( n − 2 ) df ⊗ df − n Hess 0 f , (15) and by tracing,
scal d f = scal 0 + ( n − 1 ) ( n + 2 ) ∣ grad f ∣ 2 ; \operatorname{scal}^{df}=\operatorname{scal}^{0}+(n-1)(n+2)|\operatorname{grad}f|^{2}; scal df = scal 0 + ( n − 1 ) ( n + 2 ) ∣ grad f ∣ 2 ; hence scal d f ≥ scal 0 \operatorname{scal}^{df}\geq\operatorname{scal}^{0} scal df ≥ scal 0 ∣ grad f ∣ 2 ≥ 0 |\operatorname{grad}f|^{2}\geq0 ∣ grad f ∣ 2 ≥ 0
Q d f = Q 0 + { n ∣ grad f ∣ 2 + Δ 0 f } I + ( n − 2 ) d f ⊗ grad f − n ∇ 0 grad f . Q^{df}=Q^{0}+\{n|\operatorname{grad}f|^{2}+\Delta_{0}f\}I+(n-2)df\otimes\operatorname{grad}f-n\nabla^{0}\operatorname{grad}f. Q df = Q 0 + { n ∣ grad f ∣ 2 + Δ 0 f } I + ( n − 2 ) df ⊗ grad f − n ∇ 0 grad f . Proposition 4.17. Let ζ = grad f \zeta=\operatorname{grad}f ζ = grad f θ = d f \theta=df θ = df
(i) ( ∇ d f , Q 0 , ζ , λ ) (\nabla^{df},Q^{0},\zeta,\lambda) ( ∇ df , Q 0 , ζ , λ )
( ∇ 0 , Q 0 + 2 θ ⊗ ζ , ζ , λ − ∣ ζ ∣ 2 ) (\nabla^{0},Q^{0}+2\theta\otimes\zeta,\zeta,\lambda-|\zeta|^{2}) ( ∇ 0 , Q 0 + 2 θ ⊗ ζ , ζ , λ − ∣ ζ ∣ 2 ) is;
(ii) ( ∇ 0 , Q d f , ζ , λ ) (\nabla^{0},Q^{df},\zeta,\lambda) ( ∇ 0 , Q df , ζ , λ )
( ∇ 0 , 1 1 − n { Q 0 + ( n − 2 ) θ ⊗ ζ } , ζ , 1 1 − n { λ − n ∣ ζ ∣ 2 − Δ 0 f } ) \left(\nabla^{0},\frac{1}{1-n}\{Q^{0}+(n-2)\theta\otimes\zeta\},\zeta,\frac{1}{1-n}\{\lambda-n|\zeta|^{2}-\Delta_{0}f\}\right) ( ∇ 0 , 1 − n 1 { Q 0 + ( n − 2 ) θ ⊗ ζ } , ζ , 1 − n 1 { λ − n ∣ ζ ∣ 2 − Δ 0 f } ) is;
(iii) ( ∇ d f , Q d f , ζ , λ ) (\nabla^{df},Q^{df},\zeta,\lambda) ( ∇ df , Q df , ζ , λ )
( ∇ 0 , 1 1 − n ( Q 0 + n θ ⊗ ζ ) , ζ , 1 1 − n { λ − ( n + 1 ) ∣ ζ ∣ 2 − Δ 0 f } ) \left(\nabla^{0},\frac{1}{1-n}(Q^{0}+n\theta\otimes\zeta),\zeta,\frac{1}{1-n}\{\lambda-(n+1)|\zeta|^{2}-\Delta_{0}f\}\right) ( ∇ 0 , 1 − n 1 ( Q 0 + n θ ⊗ ζ ) , ζ , 1 − n 1 { λ − ( n + 1 ) ∣ ζ ∣ 2 − Δ 0 f } ) is.
These follow by inserting
∇ d f ζ = ∇ 0 ζ + 2 θ ⊗ ζ + ∣ ζ ∣ 2 I \nabla^{df}\zeta=\nabla^{0}\zeta+2\theta\otimes\zeta+|\zeta|^{2}I ∇ df ζ = ∇ 0 ζ + 2 θ ⊗ ζ + ∣ ζ ∣ 2 I and the formula for Q d f Q^{df} Q df
Hess 0 f + Ric 0 = ( λ − ∣ grad f ∣ 2 ) g − 2 d f ⊗ d f , \operatorname{Hess}^{0}f+\operatorname{Ric}^{0}=(\lambda-|\operatorname{grad}f|^{2})g-2df\otimes df, Hess 0 f + Ric 0 = ( λ − ∣ grad f ∣ 2 ) g − 2 df ⊗ df , a gradient almost η \eta η [6 ] .
Proposition 4.18. Let ζ = grad f \zeta= \operatorname{grad} f ζ = grad f ∇ 0 ζ = θ ⊗ ζ \nabla^{0}\zeta= \theta\otimes\zeta ∇ 0 ζ = θ ⊗ ζ Hess 0 f = d f ⊗ d f \operatorname{Hess}^{0}f = df \otimes df Hess 0 f = df ⊗ df Δ 0 f = ∣ ζ ∣ 2 \Delta_{0}f = |\zeta|^{2} Δ 0 f = ∣ ζ ∣ 2
Then:
(i) ( ∇ 0 , Q d f , ζ , λ ) (\nabla^{0},Q^{df},\zeta,\lambda) ( ∇ 0 , Q df , ζ , λ )
− Hess 0 f + Ric 0 = { λ − ( n + 1 ) ∣ ζ ∣ 2 } g ; -\operatorname{Hess}^{0}f+\operatorname{Ric}^{0}=\{\lambda-(n+1)|\zeta|^{2}\}g; − Hess 0 f + Ric 0 = { λ − ( n + 1 ) ∣ ζ ∣ 2 } g ; in this case
scal d f = n λ − ∣ ζ ∣ 2 ; \operatorname{scal}^{df}=n\lambda-|\zeta|^{2}; scal df = nλ − ∣ ζ ∣ 2 ; (ii) ( ∇ d f , Q d f , ζ , λ ) (\nabla^{df},Q^{df},\zeta,\lambda) ( ∇ df , Q df , ζ , λ )
( g , f , λ − ( n + 2 ) ∣ ζ ∣ 2 ) (g,f,\lambda-(n+2)|\zeta|^{2}) ( g , f , λ − ( n + 2 ) ∣ ζ ∣ 2 ) is a gradient almost Ricci soliton; in this case
scal 0 = n λ − ( n + 1 ) 2 ∣ ζ ∣ 2 . \operatorname{scal}^{0}=n\lambda-(n+1)^{2}|\zeta|^{2}. scal 0 = nλ − ( n + 1 ) 2 ∣ ζ ∣ 2 . Volume formulas and bounds for the Ricci tensor
In this subsection M M M ∣ grad f ∣ g |\operatorname{grad} f|_{g} ∣ grad f ∣ g
∫ M ∣ grad f ∣ 2 Δ 0 f d μ g = 0 , \int_{M}|\operatorname{grad} f|^{2}\Delta_{0}f\,d\mu_{g}=0, ∫ M ∣ grad f ∣ 2 Δ 0 f d μ g = 0 , Hess 0 f ( grad f , grad f ) = 1 2 grad f ( ∣ grad f ∣ 2 ) = 0 , \operatorname{Hess}^{0}f(\operatorname{grad} f,\operatorname{grad} f)=\frac{1}{2}\operatorname{grad} f(|\operatorname{grad} f|^{2})=0, Hess 0 f ( grad f , grad f ) = 2 1 grad f ( ∣ grad f ∣ 2 ) = 0 , and Bochner’s formula
1 2 Δ 0 ∣ grad f ∣ 2 = ∣ Hess 0 f ∣ 2 + Ric 0 ( grad f , grad f ) + grad f ( Δ 0 f ) \frac{1}{2}\Delta_{0}|\operatorname{grad} f|^{2}=|\operatorname{Hess}^{0}f|^{2}+\operatorname{Ric}^{0}(\operatorname{grad} f,\operatorname{grad} f)+\operatorname{grad} f(\Delta_{0}f) 2 1 Δ 0 ∣ grad f ∣ 2 = ∣ Hess 0 f ∣ 2 + Ric 0 ( grad f , grad f ) + grad f ( Δ 0 f ) reduces to
Ric 0 ( grad f , grad f ) = − ∣ Hess 0 f ∣ 2 − grad f ( Δ 0 f ) . \operatorname{Ric}^{0}(\operatorname{grad} f,\operatorname{grad} f)=-|\operatorname{Hess}^{0}f|^{2}-\operatorname{grad} f(\Delta_{0}f). Ric 0 ( grad f , grad f ) = − ∣ Hess 0 f ∣ 2 − grad f ( Δ 0 f ) . Together with (15) this gives:
Proposition 4.19 ([8 ] ).
vol ( M ) = 1 2 ( n − 1 ) ∣ grad f ∣ 4 { ∫ M ∣ Hess 0 f ∣ 2 d μ g + ∫ M grad f ( Δ 0 f ) d μ g + ∫ M ⟨ Ric d f , d f ⊗ d f ⟩ d μ g } . \operatorname{vol}(M)=\frac{1}{2(n-1)|\operatorname{grad} f|^{4}}\left\{\int_{M}|\operatorname{Hess}^{0}f|^{2}\,d\mu_{g}+\int_{M}\operatorname{grad} f(\Delta_{0}f)\,d\mu_{g}+\int_{M}\langle\operatorname{Ric}^{df},df\otimes df\rangle\,d\mu_{g}\right\}. vol ( M ) = 2 ( n − 1 ) ∣ grad f ∣ 4 1 { ∫ M ∣ Hess 0 f ∣ 2 d μ g + ∫ M grad f ( Δ 0 f ) d μ g + ∫ M ⟨ Ric df , df ⊗ df ⟩ d μ g } . Writing Bochner’s formula in terms of Ric d f \operatorname{Ric}^{df} Ric df Hess d f \operatorname{Hess}^{df} Hess df
vol ( M ) = 1 3 ( n + 2 ) ∣ grad f ∣ 4 { ∫ M ∣ Hess d f f ∣ 2 d μ g + ∫ M grad f ( Δ d f f ) d μ g + ∫ M Ric d f ( grad f , grad f ) d μ g } . \operatorname{vol}(M)=\frac{1}{3(n+2)|\operatorname{grad} f|^{4}}\left\{\int_{M}|\operatorname{Hess}^{df}f|^{2}\,d\mu_{g}+\int_{M}\operatorname{grad} f(\Delta_{df}f)\,d\mu_{g}+\int_{M}\operatorname{Ric}^{df}(\operatorname{grad} f,\operatorname{grad} f)\,d\mu_{g}\right\}. vol ( M ) = 3 ( n + 2 ) ∣ grad f ∣ 4 1 { ∫ M ∣ Hess df f ∣ 2 d μ g + ∫ M grad f ( Δ df f ) d μ g + ∫ M Ric df ( grad f , grad f ) d μ g } . (The Levi-Civita analogue is ∫ M ∣ Hess 0 f ∣ 2 + ∫ M grad f ( Δ 0 f ) + ∫ M Ric 0 ( grad f , grad f ) = 0 \int_{M}|\operatorname{Hess}^{0}f|^{2}+\int_{M}\operatorname{grad} f(\Delta_{0}f)+\int_{M}\operatorname{Ric}^{0}(\operatorname{grad} f,\operatorname{grad} f)=0 ∫ M ∣ Hess 0 f ∣ 2 + ∫ M grad f ( Δ 0 f ) + ∫ M Ric 0 ( grad f , grad f ) = 0
Proposition 4.20 ([8 ] ). Under the same assumptions:
(i) if ( ∇ 0 , Q d f , grad f , λ ) (\nabla^{0},Q^{df},\operatorname{grad} f,\lambda) ( ∇ 0 , Q df , grad f , λ )
vol ( M ) = 1 2 ( n − 1 ) ∣ grad f ∣ 4 { ∫ M ∣ Hess 0 f ∣ 2 d μ g + ∣ grad f ∣ 2 ∫ M λ d μ g + n ∫ M grad f ( λ ) d μ g − ∫ M grad f ( scal 0 ) d μ g } . \operatorname{vol}(M)=\frac{1}{2(n-1)|\operatorname{grad} f|^{4}}\left\{\int_M|\operatorname{Hess}^{0}f|^{2}\,d\mu_g+|\operatorname{grad} f|^{2}\int_M\lambda\,d\mu_g+n\int_M\operatorname{grad} f(\lambda)\,d\mu_g-\int_M\operatorname{grad} f(\operatorname{scal}^{0})\,d\mu_g\right\}. vol ( M ) = 2 ( n − 1 ) ∣ grad f ∣ 4 1 { ∫ M ∣ Hess 0 f ∣ 2 d μ g + ∣ grad f ∣ 2 ∫ M λ d μ g + n ∫ M grad f ( λ ) d μ g − ∫ M grad f ( scal 0 ) d μ g } . (ii) if ( ∇ d f , Q 0 , grad f , λ ) (\nabla^{df},Q^{0},\operatorname{grad} f,\lambda) ( ∇ df , Q 0 , grad f , λ ) λ ≠ 3 ∣ grad f ∣ 2 \lambda\ne3|\operatorname{grad} f|^{2} λ = 3∣ grad f ∣ 2
vol ( M ) = 1 ( λ − 3 ∣ grad f ∣ 2 ) ∣ grad f ∣ 2 { − ∫ M ∣ Hess 0 f ∣ 2 d μ g + ∫ M grad f ( scal 0 ) d μ g } . \operatorname{vol}(M)=\frac{1}{(\lambda-3|\operatorname{grad} f|^{2})|\operatorname{grad} f|^{2}}\left\{-\int_M|\operatorname{Hess}^{0}f|^{2}\,d\mu_g+\int_M\operatorname{grad} f(\operatorname{scal}^{0})\,d\mu_g\right\}. vol ( M ) = ( λ − 3∣ grad f ∣ 2 ) ∣ grad f ∣ 2 1 { − ∫ M ∣ Hess 0 f ∣ 2 d μ g + ∫ M grad f ( scal 0 ) d μ g } . Proof. For (i), the equation reads
Hess 0 f + Ric d f = λ g , \operatorname{Hess}^{0}f+\operatorname{Ric}^{df}=\lambda g, Hess 0 f + Ric df = λ g , so
⟨ Ric d f , d f ⊗ d f ⟩ = λ ∣ grad f ∣ 2 − 1 2 grad f ( ∣ grad f ∣ 2 ) , \langle\operatorname{Ric}^{df},df\otimes df\rangle=\lambda|\operatorname{grad} f|^{2}-\frac{1}{2}\operatorname{grad} f\left(|\operatorname{grad} f|^{2}\right), ⟨ Ric df , df ⊗ df ⟩ = λ ∣ grad f ∣ 2 − 2 1 grad f ( ∣ grad f ∣ 2 ) , and its trace gives
Δ 0 f + scal 0 + ( n − 1 ) ( n + 2 ) ∣ grad f ∣ 2 = n λ ; \Delta_{0}f+\operatorname{scal}^{0}+(n-1)(n+2)|\operatorname{grad} f|^{2}=n\lambda; Δ 0 f + scal 0 + ( n − 1 ) ( n + 2 ) ∣ grad f ∣ 2 = nλ ; apply grad f \operatorname{grad} f grad f Proposition 4.19 . For (ii), use (14) :
Hess d f f + Ric 0 = λ g , \operatorname{Hess}^{df}f+\operatorname{Ric}^{0}=\lambda g, Hess df f + Ric 0 = λ g , with trace
Δ 0 f + scal 0 + ( n + 2 ) ∣ grad f ∣ 2 = n λ \Delta_{0}f+\operatorname{scal}^{0}+(n+2)|\operatorname{grad} f|^{2}=n\lambda Δ 0 f + scal 0 + ( n + 2 ) ∣ grad f ∣ 2 = nλ and
Ric 0 ( grad f , grad f ) = ( λ − 3 ∣ grad f ∣ 2 ) ∣ grad f ∣ 2 . \operatorname{Ric}^{0}(\operatorname{grad} f,\operatorname{grad} f)=(\lambda-3|\operatorname{grad} f|^{2})|\operatorname{grad} f|^{2}. Ric 0 ( grad f , grad f ) = ( λ − 3∣ grad f ∣ 2 ) ∣ grad f ∣ 2 . Remark 4.21. In case (i), if λ = 2 ( n − 1 ) ∣ grad f ∣ 2 \lambda=2(n-1)|\operatorname{grad} f|^{2} λ = 2 ( n − 1 ) ∣ grad f ∣ 2 n ≥ 3 n\ge3 n ≥ 3
vol ( M ) = 1 ( n − 1 ) ( n − 2 ) ∣ grad f ∣ 2 ∫ M scal 0 d μ g ; \operatorname{vol}(M)=\frac{1}{(n-1)(n-2)|\operatorname{grad} f|^{2}}\int_M\operatorname{scal}^{0}\,d\mu_g; vol ( M ) = ( n − 1 ) ( n − 2 ) ∣ grad f ∣ 2 1 ∫ M scal 0 d μ g ; if scal 0 \operatorname{scal}^{0} scal 0 f f f Δ 0 \Delta_{0} Δ 0
scal 0 = ( n − 1 ) ( n − 2 ) ∣ grad f ∣ 2 ≥ 0. \operatorname{scal}^{0}=(n-1)(n-2)|\operatorname{grad} f|^{2}\ge0. scal 0 = ( n − 1 ) ( n − 2 ) ∣ grad f ∣ 2 ≥ 0. If λ \lambda λ
λ ≠ 2 ( n − 1 ) ∣ grad f ∣ 2 , \lambda\ne2(n-1)|\operatorname{grad} f|^{2}, λ = 2 ( n − 1 ) ∣ grad f ∣ 2 , then
vol ( M ) = 1 { 2 ( n − 1 ) ∣ grad f ∣ 2 − λ } ∣ grad f ∣ 2 { ∫ M ∣ Hess 0 f ∣ 2 d μ g − ∫ M grad f ( scal 0 ) d μ g } , \operatorname{vol}(M)=\frac{1}{\{2(n-1)|\operatorname{grad} f|^{2}-\lambda\}|\operatorname{grad} f|^{2}}\left\{\int_M|\operatorname{Hess}^{0}f|^{2}\,d\mu_g-\int_M\operatorname{grad} f(\operatorname{scal}^{0})\,d\mu_g\right\}, vol ( M ) = { 2 ( n − 1 ) ∣ grad f ∣ 2 − λ } ∣ grad f ∣ 2 1 { ∫ M ∣ Hess 0 f ∣ 2 d μ g − ∫ M grad f ( scal 0 ) d μ g } , and the second integral drops out if f f f
λ ≤ 2 ( n − 1 ) ∣ grad f ∣ 2 \lambda\le2(n-1)|\operatorname{grad} f|^{2} λ ≤ 2 ( n − 1 ) ∣ grad f ∣ 2 and then
scal 0 ≤ ( n − 1 ) ( n − 2 ) ∣ grad f ∣ 2 \operatorname{scal}^{0}\le(n-1)(n-2)|\operatorname{grad} f|^{2} scal 0 ≤ ( n − 1 ) ( n − 2 ) ∣ grad f ∣ 2 (strictly if Hess 0 f ≢ 0 \operatorname{Hess}^{0}f\not\equiv0 Hess 0 f ≡ 0
Finally, eliminating the unknown function from the soliton equations bounds the norm of the Ricci tensor (solvability of a quadratic equation for λ \lambda λ
Proposition 4.22 ([8 ] ). (i) If ( ∇ 0 , Q d f , grad f , λ ) (\nabla^0,Q^{df},\operatorname{grad} f,\lambda) ( ∇ 0 , Q df , grad f , λ )
( n − 1 ) 2 ∣ Hess 0 f ∣ 2 + ( n − 1 ) ( n − 2 ) 2 n ∣ grad f ∣ 4 − ( n − 1 ) 2 n ( Δ 0 f ) 2 + 2 ( n − 1 ) ( n − 2 ) n ∣ grad f ∣ 2 Δ 0 f − 2 ( n − 1 ) ( n − 2 ) Hess 0 f ( grad f , grad f ) ≤ ∣ Ric 0 ∣ 2 (n-1)^2|\operatorname{Hess}^0 f|^2+\frac{(n-1)(n-2)^2}{n}|\operatorname{grad} f|^4-\frac{(n-1)^2}{n}(\Delta_0 f)^2+\frac{2(n-1)(n-2)}{n}|\operatorname{grad} f|^2\Delta_0 f-2(n-1)(n-2)\operatorname{Hess}^0 f(\operatorname{grad} f,\operatorname{grad} f)\leq|\operatorname{Ric}^0|^2 ( n − 1 ) 2 ∣ Hess 0 f ∣ 2 + n ( n − 1 ) ( n − 2 ) 2 ∣ grad f ∣ 4 − n ( n − 1 ) 2 ( Δ 0 f ) 2 + n 2 ( n − 1 ) ( n − 2 ) ∣ grad f ∣ 2 Δ 0 f − 2 ( n − 1 ) ( n − 2 ) Hess 0 f ( grad f , grad f ) ≤ ∣ Ric 0 ∣ 2 ≤ ( n − 1 ) 2 ∣ Hess 0 f ∣ 2 − ( n − 1 ) ( n − 2 ) 2 n ∣ grad f ∣ 4 + 1 n ( scal 0 ) 2 + 2 ( n − 2 ) n ∣ grad f ∣ 2 scal 0 − 2 ( n − 2 ) Ric 0 ( grad f , grad f ) . \leq(n-1)^2|\operatorname{Hess}^0 f|^2-\frac{(n-1)(n-2)^2}{n}|\operatorname{grad} f|^4+\frac{1}{n}(\operatorname{scal}^0)^2+\frac{2(n-2)}{n}|\operatorname{grad} f|^2\operatorname{scal}^0-2(n-2)\operatorname{Ric}^0(\operatorname{grad} f,\operatorname{grad} f). ≤ ( n − 1 ) 2 ∣ Hess 0 f ∣ 2 − n ( n − 1 ) ( n − 2 ) 2 ∣ grad f ∣ 4 + n 1 ( scal 0 ) 2 + n 2 ( n − 2 ) ∣ grad f ∣ 2 scal 0 − 2 ( n − 2 ) Ric 0 ( grad f , grad f ) . (ii) If ( ∇ d f , Q 0 , grad f , λ ) (\nabla^{df},Q^0,\operatorname{grad} f,\lambda) ( ∇ df , Q 0 , grad f , λ )
∣ Hess 0 f ∣ 2 + 4 ( n − 1 ) n ∣ grad f ∣ 4 − 1 n ( Δ 0 f ) 2 − 4 n ∣ grad f ∣ 2 Δ 0 f + 4 Hess 0 f ( grad f , grad f ) ≤ ∣ Ric 0 ∣ 2 |\operatorname{Hess}^0 f|^2+\frac{4(n-1)}{n}|\operatorname{grad} f|^4-\frac{1}{n}(\Delta_0 f)^2-\frac{4}{n}|\operatorname{grad} f|^2\Delta_0 f+4\operatorname{Hess}^0 f(\operatorname{grad} f,\operatorname{grad} f)\leq|\operatorname{Ric}^0|^2 ∣ Hess 0 f ∣ 2 + n 4 ( n − 1 ) ∣ grad f ∣ 4 − n 1 ( Δ 0 f ) 2 − n 4 ∣ grad f ∣ 2 Δ 0 f + 4 Hess 0 f ( grad f , grad f ) ≤ ∣ Ric 0 ∣ 2 ≤ ∣ Hess 0 f ∣ 2 − 4 ( n − 1 ) n ∣ grad f ∣ 4 + 1 n ( scal 0 ) 2 + 4 n ∣ grad f ∣ 2 scal 0 − 4 Ric 0 ( grad f , grad f ) . \leq|\operatorname{Hess}^0 f|^2-\frac{4(n-1)}{n}|\operatorname{grad} f|^4+\frac{1}{n}(\operatorname{scal}^0)^2+\frac{4}{n}|\operatorname{grad} f|^2\operatorname{scal}^0-4\operatorname{Ric}^0(\operatorname{grad} f,\operatorname{grad} f). ≤ ∣ Hess 0 f ∣ 2 − n 4 ( n − 1 ) ∣ grad f ∣ 4 + n 1 ( scal 0 ) 2 + n 4 ∣ grad f ∣ 2 scal 0 − 4 Ric 0 ( grad f , grad f ) . Proof. In (i) write
Ric 0 = a g − ( n − 2 ) d f ⊗ d f + ( n − 1 ) Hess 0 f . \operatorname{Ric}^0=ag-(n-2)\,df\otimes df+(n-1)\operatorname{Hess}^0 f. Ric 0 = a g − ( n − 2 ) df ⊗ df + ( n − 1 ) Hess 0 f . and
( n − 1 ) Hess 0 f = Ric 0 − a g + ( n − 2 ) d f ⊗ d f (n-1)\operatorname{Hess}^0 f=\operatorname{Ric}^0-ag+(n-2)\,df\otimes df ( n − 1 ) Hess 0 f = Ric 0 − a g + ( n − 2 ) df ⊗ df with
a = λ − n ∣ grad f ∣ 2 − Δ 0 f , a=\lambda-n|\operatorname{grad} f|^2-\Delta_0 f, a = λ − n ∣ grad f ∣ 2 − Δ 0 f , and compute ∣ Ric 0 ∣ 2 |\operatorname{Ric}^0|^2 ∣ Ric 0 ∣ 2 ∣ Hess 0 f ∣ 2 |\operatorname{Hess}^0 f|^2 ∣ Hess 0 f ∣ 2 a a a ≥ 0 \geq0 ≥ 0
a = λ − ∣ grad f ∣ 2 a=\lambda-|\operatorname{grad} f|^2 a = λ − ∣ grad f ∣ 2 and
Ric 0 = a g − 2 d f ⊗ d f − Hess 0 f . \operatorname{Ric}^0=ag-2\,df\otimes df-\operatorname{Hess}^0 f. Ric 0 = a g − 2 df ⊗ df − Hess 0 f . Statistical Warped Products
Let ( N 1 , g 1 ) (N_1,g_1) ( N 1 , g 1 ) ( N 2 , g 2 ) (N_2,g_2) ( N 2 , g 2 ) f > 0 f>0 f > 0 N 1 N_1 N 1 warped product [5 ] N ~ = N 1 × f N 2 \widetilde{N}=N_1\times_f N_2 N = N 1 × f N 2 N 1 × N 2 N_1\times N_2 N 1 × N 2
g ~ = π ∗ g 1 + ( f ∘ π ) 2 π ′ ∗ g 2 . \widetilde{g}=\pi^*g_1+(f\circ\pi)^2\pi^{\prime*}g_2. g = π ∗ g 1 + ( f ∘ π ) 2 π ′ ∗ g 2 . Warped products are ubiquitous in general relativity [11 , 13 ] . A dualistic structure on a warped product was described by Todjihounde [39 ] .
Proposition 5.1. ([39 ] ). Let ( g i , ∇ N i , ∇ N i ∗ ) (g_i,\nabla^{N_i},\nabla^{N_i*}) ( g i , ∇ N i , ∇ N i ∗ ) N i N_i N i i = 1 , 2 i=1,2 i = 1 , 2 X , Y ∈ χ ( N 1 ) X,Y\in\chi(N_1) X , Y ∈ χ ( N 1 ) U , V ∈ χ ( N 2 ) U,V\in\chi(N_2) U , V ∈ χ ( N 2 ) D , D ∗ D,D^* D , D ∗ N 1 × N 2 N_1\times N_2 N 1 × N 2
D X Y = ∇ X N 1 Y , D X U = D U X = X f f U , D U V = ∇ U N 2 V − g ~ ( U , V ) f grad f . D_XY=\nabla^{N_1}_X Y,\qquad D_XU=D_UX=\frac{Xf}{f}U,\qquad D_UV=\nabla^{N_2}_U V-\frac{\widetilde{g}(U,V)}{f}\operatorname{grad}f. D X Y = ∇ X N 1 Y , D X U = D U X = f X f U , D U V = ∇ U N 2 V − f g ( U , V ) grad f . and similarly for D ∗ D^* D ∗ ( g ~ , D , D ∗ ) (\widetilde{g},D,D^*) ( g , D , D ∗ ) N 1 × f N 2 N_1\times_f N_2 N 1 × f N 2
The curvature and Ricci tensors of ( N ~ , D , D ∗ , g ~ ) (\widetilde{N},D,D^*,\widetilde{g}) ( N , D , D ∗ , g ) [39 , 4 ] :
Lemma 5.2. For X , Y ∈ χ ( N 1 ) X,Y\in\chi(N_1) X , Y ∈ χ ( N 1 ) U , V ∈ χ ( N 2 ) U,V\in\chi(N_2) U , V ∈ χ ( N 2 )
(a) Ric ~ ( X , Y ) = Ric 1 ( X , Y ) − dim ( N 2 ) f − 1 Hess f ( X , Y ) \widetilde{\operatorname{Ric}}(X,Y)=\operatorname{Ric}_1(X,Y)-\dim(N_2)f^{-1}\operatorname{Hess}f(X,Y) Ric ( X , Y ) = Ric 1 ( X , Y ) − dim ( N 2 ) f − 1 Hess f ( X , Y )
(b) Ric ~ ( X , V ) = 0 \widetilde{\operatorname{Ric}}(X,V)=0 Ric ( X , V ) = 0
(c) Ric ~ ( U , V ) = Ric 2 ( U , V ) − [ f Δ f + ( dim N 2 − 1 ) ∥ grad f ∥ 2 ] g 2 ( U , V ) \widetilde{\operatorname{Ric}}(U,V)=\operatorname{Ric}_2(U,V)-\left[f\mathop{\Delta}f+(\dim N_2-1)\lVert\operatorname{grad}f\rVert^2\right]g_2(U,V) Ric ( U , V ) = Ric 2 ( U , V ) − [ f Δ f + ( dim N 2 − 1 ) ∥ grad f ∥ 2 ] g 2 ( U , V )
where Δ f \Delta f Δ f f f f Ric ∗ \operatorname{Ric}^* Ric ∗
Statistical Solitons on Statistical Warped Products
We consider the two basic one-dimensional situations, with R \mathbb{R} R ( ∇ R , d z 2 ) (\nabla^{\mathbb{R}},dz^2) ( ∇ R , d z 2 )
The case N ~ = R × f N 2 \widetilde{N}=\mathbb{R}\times_f N_2 N = R × f N 2
Let dim N 2 = k \dim N_2=k dim N 2 = k
Ric ~ ( ∂ z , ∂ z ) = − k f ′ ′ f , Ric ~ ( ∂ z , V ) = 0 , \widetilde{\operatorname{Ric}}(\partial_z,\partial_z)=-k\frac{f''}{f},\qquad\widetilde{\operatorname{Ric}}(\partial_z,V)=0, Ric ( ∂ z , ∂ z ) = − k f f ′′ , Ric ( ∂ z , V ) = 0 , Ric ~ ( U , V ) = Ric 2 ( U , V ) − [ f f ′ ′ + ( k − 1 ) f ′ 2 ] g 2 ( U , V ) . (16) \widetilde{\operatorname{Ric}}(U,V)=\operatorname{Ric}_2(U,V)-\left[ff''+(k-1)f'^2\right]g_2(U,V). \tag*{(16)} Ric ( U , V ) = Ric 2 ( U , V ) − [ f f ′′ + ( k − 1 ) f ′2 ] g 2 ( U , V ) . (16) Proposition 6.1. Let ζ = ( ∂ z , ζ 2 ) \zeta=(\partial_z,\zeta_2) ζ = ( ∂ z , ζ 2 ) ( ζ , λ ) (\zeta,\lambda) ( ζ , λ ) R × f N 2 \mathbb{R}\times_f N_2 R × f N 2 ∂ z ⊗ ∂ z \partial_z\otimes\partial_z ∂ z ⊗ ∂ z
λ = k f ′ ′ f , equivalently Hess f = λ k f d z 2 . \lambda=k\frac{f''}{f},\qquad\text{equivalently}\qquad\operatorname{Hess}f=\frac{\lambda}{k}f\,dz^2. λ = k f f ′′ , equivalently Hess f = k λ f d z 2 . Proof. Since D ∂ z ∂ z = 0 D_{\partial_z}\partial_z=0 D ∂ z ∂ z = 0 (9) along ( ∂ z , ∂ z ) (\partial_z,\partial_z) ( ∂ z , ∂ z ) Ric ~ ( ∂ z , ∂ z ) + λ = 0 \widetilde{\operatorname{Ric}}(\partial_z,\partial_z)+\lambda=0 Ric ( ∂ z , ∂ z ) + λ = 0 (16) .
Since λ \lambda λ f ′ ′ = ( λ / k ) f f'' = (\lambda/k)f f ′′ = ( λ / k ) f
Theorem 6.2. Let ( ζ , λ ) (\zeta,\lambda) ( ζ , λ ) N ~ = R × f N 2 \widetilde{N} = \mathbb{R} \times_f N_2 N = R × f N 2 ζ = ( ∂ z , ζ 2 ) \zeta= (\partial_z,\zeta_2) ζ = ( ∂ z , ζ 2 )
(a) λ = k f ′ ′ / f \lambda= kf''/f λ = k f ′′ / f
(b) if λ = 0 \lambda= 0 λ = 0 f ( z ) = a z + b f(z) = az + b f ( z ) = a z + b
(c) if λ > 0 \lambda> 0 λ > 0 f f f e ± λ / k z e^{\pm\sqrt{\lambda/k}\,z} e ± λ / k z
f ( z ) = c cosh ( λ / k z + b ) f(z) = c\cosh\left(\sqrt{\lambda/k}\,z + b\right) f ( z ) = c cosh ( λ / k z + b ) for suitable constants b , c b,c b , c λ < 0 \lambda< 0 λ < 0 f f f cos ( ∣ λ ∣ / k z ) \cos\left(\sqrt{|\lambda|/k}\,z\right) cos ( ∣ λ ∣/ k z ) sin ( ∣ λ ∣ / k z ) \sin\left(\sqrt{|\lambda|/k}\,z\right) sin ( ∣ λ ∣/ k z )
∇ N 2 ζ 2 + Ric 2 + ( k − 1 ) ( f f ′ ′ − f ′ 2 ) g 2 + ( lower-order terms, see x r e f ‘ R e m a r k 6.3 ‘ ) = 0 , \nabla^{N_2}\zeta_2 + \operatorname{Ric}_2 + (k-1)(ff''-f'^2)g_2 + (\text{lower-order terms, see }{xref}`Remark 6.3`) = 0, ∇ N 2 ζ 2 + Ric 2 + ( k − 1 ) ( f f ′′ − f ′2 ) g 2 + ( lower-order terms, see x r e f ‘ R e ma r k 6.3‘ ) = 0 , and its dual; consequently ( N 2 , g 2 , ζ 2 , λ 2 ) (N_2,g_2,\zeta_2,\lambda_2) ( N 2 , g 2 , ζ 2 , λ 2 )
λ 2 = ( k − 1 ) ( f f ′ ′ − f ′ 2 ) \lambda_2 = (k-1)(ff''-f'^2) λ 2 = ( k − 1 ) ( f f ′′ − f ′2 ) is a statistical soliton whenever this quantity is constant.
Proof. Part (a) is Proposition 6.1. Eliminating λ \lambda λ
λ f 2 = k f f ′ ′ \lambda f^2 = kff'' λ f 2 = k f f ′′ yields
λ f 2 − ( f f ′ ′ + ( k − 1 ) f ′ 2 ) = ( k − 1 ) ( f f ′ ′ − f ′ 2 ) , \lambda f^2 - (ff'' + (k-1)f'^2) = (k-1)(ff''-f'^2), λ f 2 − ( f f ′′ + ( k − 1 ) f ′2 ) = ( k − 1 ) ( f f ′′ − f ′2 ) , which is (d). The same computation with D ∗ D^* D ∗ Δ ∗ \Delta^* Δ ∗
f ′ ′ = ( λ / k ) f . f'' = (\lambda/k)f. f ′′ = ( λ / k ) f . Remark 6.3. The two projections used above are the base and pure-fibre components of the soliton equation. For completeness, we note that the full tensorial equation also contains the term coming from
D U ∂ z = ( f ′ / f ) U D_U\partial_z = (f'/f)U D U ∂ z = ( f ′ / f ) U and the mixed terms coming from D U ζ 2 D_U\zeta_2 D U ζ 2 ∂ z \partial_z ∂ z − f f ′ g 2 ( U , ζ 2 ) -ff'g_2(U,\zeta_2) − f f ′ g 2 ( U , ζ 2 ) f ′ f g 2 f'f g_2 f ′ f g 2
f ′ g 2 ( ⋅ , ζ 2 ) = 0. f'g_2(\mathord{\cdot},\zeta_2) = 0. f ′ g 2 ( ⋅ , ζ 2 ) = 0. Thus, Theorem 6.2 should be understood as giving necessary conditions on f f f R × e z R k \mathbb{R}\times_{e^z}\mathbb{R}^k R × e z R k ∂ z \partial_z ∂ z λ = k \lambda= k λ = k λ = k − 1 \lambda= k-1 λ = k − 1
The case N ~ = N 1 × f R \widetilde{N}=N_1\times_f\mathbb{R} N = N 1 × f R
Lemma 6.4. Let dim N 1 = k \dim N_1=k dim N 1 = k X , Y ∈ Γ ( T N 1 ) X,Y\in\Gamma(TN_1) X , Y ∈ Γ ( T N 1 )
Ric ~ ( X , Y ) = Ric 1 ( X , Y ) − f − 1 Hess f ( X , Y ) , \widetilde{\operatorname{Ric}}(X,Y)=\operatorname{Ric}_1(X,Y)-f^{-1}\operatorname{Hess}f(X,Y), Ric ( X , Y ) = Ric 1 ( X , Y ) − f − 1 Hess f ( X , Y ) , Ric ~ ( X , ∂ z ) = 0 \widetilde{\operatorname{Ric}}(X,\partial_z)=0 Ric ( X , ∂ z ) = 0 and
Ric ~ ( ∂ z , ∂ z ) = − f Δ f . \widetilde{\operatorname{Ric}}(\partial_z,\partial_z)=-f\Delta f. Ric ( ∂ z , ∂ z ) = − f Δ f . Theorem 6.5. Let ζ = ( ζ 1 , ∂ z ) \zeta=(\zeta_1,\partial_z) ζ = ( ζ 1 , ∂ z ) N ~ = N 1 × f R \widetilde{N}=N_1\times_f\mathbb{R} N = N 1 × f R Hess f = 0 \operatorname{Hess}f=0 Hess f = 0 (9) for ( ζ , λ ) (\zeta,\lambda) ( ζ , λ )
∇ N 1 ζ 1 + Ric 1 + λ g 1 = 0 , ∇ N 1 ∗ ζ 1 + Ric 1 ∗ + λ g 1 = 0 , (17) \nabla^{N_1}\zeta_1+\operatorname{Ric}_1+\lambda g_1=0,\qquad\nabla^{N_1*}\zeta_1+\operatorname{Ric}_1^*+\lambda g_1=0, \tag*{(17)} ∇ N 1 ζ 1 + Ric 1 + λ g 1 = 0 , ∇ N 1 ∗ ζ 1 + Ric 1 ∗ + λ g 1 = 0 , (17) that is, ( ζ 1 , λ ) (\zeta_1,\lambda) ( ζ 1 , λ ) N 1 N_1 N 1 λ = f − 1 Δ f \lambda=f^{-1}\Delta f λ = f − 1 Δ f ( ζ 1 , λ ) (\zeta_1,\lambda) ( ζ 1 , λ ) N 1 N_1 N 1 λ = f − 1 Δ f \lambda=f^{-1}\Delta f λ = f − 1 Δ f N ~ \widetilde{N} N
Proof. For horizontal X , Y X,Y X , Y Ric ~ ( X , Y ) = Ric 1 ( X , Y ) \widetilde{\operatorname{Ric}}(X,Y)=\operatorname{Ric}_1(X,Y) Ric ( X , Y ) = Ric 1 ( X , Y ) Hess f = 0 \operatorname{Hess}f=0 Hess f = 0 D X ∂ z = ( X f / f ) ∂ z D_X\partial_z=(Xf/f)\partial_z D X ∂ z = ( X f / f ) ∂ z (9) restricted to T N 1 TN_1 T N 1 (17) . The relation λ = f − 1 Δ f \lambda=f^{-1}\Delta f λ = f − 1 Δ f ( ∂ z , ∂ z ) (\partial_z,\partial_z) ( ∂ z , ∂ z ) D , Ric 1 , Δ D,\operatorname{Ric}_1,\Delta D , Ric 1 , Δ D ∗ , Ric 1 ∗ , Δ ∗ D^*,\operatorname{Ric}_1^*,\Delta^* D ∗ , Ric 1 ∗ , Δ ∗
Corollary 6.6. If Hess f = ϖ g 1 \operatorname{Hess}f=\varpi g_1 Hess f = ϖ g 1 ϖ ∈ C ∞ ( N 1 ) \varpi\in C^\infty(N_1) ϖ ∈ C ∞ ( N 1 ) ( N 1 , g 1 , ζ 1 , f − 1 ( Δ f − ϖ ) ) (N_1,g_1,\zeta_1,f^{-1}(\Delta f-\varpi)) ( N 1 , g 1 , ζ 1 , f − 1 ( Δ f − ϖ ))
Remark 6.7. When Hess f = 0 \operatorname{Hess}f=0 Hess f = 0 Δ f = 0 \Delta f=0 Δ f = 0 ϖ g 1 \varpi g_1 ϖ g 1 ζ 1 f \zeta_1f ζ 1 f d f df df
Kenmotsu Statistical Manifolds
The results of § 13 –§ 15 follow the approach of [1 ] . Kenmotsu [23 ] identified a class of almost contact metric manifolds ( B , ϕ , ξ , η , g ) (B,\phi,\xi,\eta,g) ( B , ϕ , ξ , η , g )
∇ E g ξ = E − η ( E ) ξ , ( ∇ E g ϕ ) F = g ( ϕ E , F ) ξ − η ( F ) ϕ E , \nabla^\mathsf{g}_E\xi=E-\eta(E)\xi,\qquad(\nabla^\mathsf{g}_E\phi)F=g(\phi E,F)\xi-\eta(F)\phi E, ∇ E g ξ = E − η ( E ) ξ , ( ∇ E g ϕ ) F = g ( ϕE , F ) ξ − η ( F ) ϕE , that arise as warped products R × e α B ~ \mathbb{R}\times_{e^\alpha}\widetilde{B} R × e α B [19 ] introduced the statistical counterpart.
Definition 7.1 ([19 ] ). Let ( B , ∇ g , g , ϕ , ξ ) (B,\nabla^\mathsf{g},g,\phi,\xi) ( B , ∇ g , g , ϕ , ξ ) ∇ = ∇ g + T \nabla=\nabla^\mathsf{g}+\mathcal{T} ∇ = ∇ g + T B B B Kenmotsu statistical structure if
T ( E , ϕ F ) = − ϕ T ( E , F ) for all E , F . (18) \mathcal{T}(E,\phi F)=-\phi\mathcal{T}(E,F)\qquad\text{for all }E,F. \tag*{(18)} T ( E , ϕF ) = − ϕ T ( E , F ) for all E , F . (18) Proposition 7.2 ([19 ] ). Let ( B ~ , ∇ ~ B ~ = ∇ G + T , G , J ) (\widetilde{B},\widetilde{\nabla}^{\widetilde{B}}=\nabla^G+\mathcal{T},G,J) ( B , ∇ B = ∇ G + T , G , J ) B = B ~ × R B=\widetilde{B}\times\mathbb{R} B = B × R g = e 2 α G + d α 2 g=e^{2\alpha}G+d\alpha^2 g = e 2 α G + d α 2 ξ = ∂ α \xi=\partial_\alpha ξ = ∂ α β ∈ C ∞ ( B ) \beta\in C^\infty(B) β ∈ C ∞ ( B )
T ( E 2 , F 2 ) = T ~ ( E 2 , F 2 ) (on B ~ ) , T ( E 2 , ξ ) = T ( ξ , E 2 ) = 0 , T ( ξ , ξ ) = β ξ . \mathcal{T}(E_2,F_2)=\widetilde{\mathcal{T}}(E_2,F_2)\text{ (on }\widetilde{B}\text{)},\qquad\mathcal{T}(E_2,\xi)=\mathcal{T}(\xi,E_2)=0,\qquad\mathcal{T}(\xi,\xi)=\beta\xi. T ( E 2 , F 2 ) = T ( E 2 , F 2 ) (on B ) , T ( E 2 , ξ ) = T ( ξ , E 2 ) = 0 , T ( ξ , ξ ) = β ξ . Then ( ∇ = ∇ g + T , g , ϕ , ξ ) (\nabla=\nabla^\mathsf{g}+\mathcal{T},g,\phi,\xi) ( ∇ = ∇ g + T , g , ϕ , ξ ) B B B
For such a structure one obtains, from
∇ E g ξ = E − η ( E ) ξ \nabla^{g}_{E}\xi= E-\eta(E)\xi ∇ E g ξ = E − η ( E ) ξ and
T ( E , ξ ) = β η ( E ) ξ , \mathcal{T}(E,\xi)=\beta\eta(E)\xi, T ( E , ξ ) = β η ( E ) ξ , ∇ E ξ = E + ( β − 1 ) η ( E ) ξ , ∇ E ∗ ξ = E − ( β + 1 ) η ( E ) ξ . (19) \nabla_{E}\xi=E+(\beta-1)\eta(E)\xi,\qquad\nabla^{*}_{E}\xi=E-(\beta+1)\eta(E)\xi. \tag*{(19)} ∇ E ξ = E + ( β − 1 ) η ( E ) ξ , ∇ E ∗ ξ = E − ( β + 1 ) η ( E ) ξ . (19) The dual statement is obtained from the first one by β ↦ − β \beta\mapsto-\beta β ↦ − β ( B , ∇ , g , ϕ , ξ ) (B,\nabla,g,\phi,\xi) ( B , ∇ , g , ϕ , ξ ) Proposition 7.2 , N N N s s s β \beta β
Statistical Solitons on Submanifolds: the Reeb Field
ξ \xi ξ N N N Let ξ \xi ξ N N N ( ξ , λ ) (\xi,\lambda) ( ξ , λ ) (5) and (19) , comparing tangential and normal parts,
∇ E ξ = E + ( β − 1 ) η ( E ) ξ , h ( E , ξ ) = 0. (20) \nabla_{E}\xi=E+(\beta-1)\eta(E)\xi,\qquad h(E,\xi)=0. \tag*{(20)} ∇ E ξ = E + ( β − 1 ) η ( E ) ξ , h ( E , ξ ) = 0. (20) Inserting this into the ∇ \nabla ∇ g ( ∇ E ξ , F ) + Ric ( E , F ) + λ g ( E , F ) = 0 g(\nabla_{E}\xi,F)+\operatorname{Ric}(E,F)+\lambda g(E,F)=0 g ( ∇ E ξ , F ) + Ric ( E , F ) + λ g ( E , F ) = 0
Ric ( E , F ) = − ( λ + 1 ) g ( E , F ) + ( 1 − β ) η ( E ) η ( F ) . (21) \operatorname{Ric}(E,F)=-(\lambda+1)g(E,F)+(1-\beta)\eta(E)\eta(F). \tag*{(21)} Ric ( E , F ) = − ( λ + 1 ) g ( E , F ) + ( 1 − β ) η ( E ) η ( F ) . (21) Theorem 8.1. If ( g , ξ , λ ) (g,\xi,\lambda) ( g , ξ , λ ) N N N ( B , ∇ , g , ξ ) (B,\nabla,g,\xi) ( B , ∇ , g , ξ ) ξ \xi ξ N N N N N N η \eta η ( B , ∇ ∗ , g , ξ ) (B,\nabla^{*},g,\xi) ( B , ∇ ∗ , g , ξ ) β \beta β − β -\beta − β h ∗ ( E , ξ ) = 0 h^{*}(E,\xi)=0 h ∗ ( E , ξ ) = 0
The soliton constant. Using (20) one computes (torsion-free ∇ \nabla ∇ β \beta β
Rie ( E , F ) ξ = ( β − 1 ) [ η ( F ) E − η ( E ) F ] + ( β − 1 ) d η ( E , F ) ξ , \operatorname{Rie}(E,F)\xi=(\beta-1)\left[\eta(F)E-\eta(E)F\right]+(\beta-1)d\eta(E,F)\xi, Rie ( E , F ) ξ = ( β − 1 ) [ η ( F ) E − η ( E ) F ] + ( β − 1 ) d η ( E , F ) ξ , and taking the trace gives
Ric ( ξ , ξ ) = ( β − 1 ) ( s − 1 ) . \operatorname{Ric}(\xi,\xi)=(\beta-1)(s-1). Ric ( ξ , ξ ) = ( β − 1 ) ( s − 1 ) . On the other hand, (21) gives
Ric ( ξ , ξ ) = − λ − β . \operatorname{Ric}(\xi,\xi)=-\lambda-\beta. Ric ( ξ , ξ ) = − λ − β . Hence
λ = ( s − 1 ) ( 1 − β ) − β , and for ∇ ∗ : λ ∗ = ( s − 1 ) ( 1 + β ) + β . (22) \lambda=(s-1)(1-\beta)-\beta,\qquad\text{and for }\nabla^{*}:\quad\lambda^{*}=(s-1)(1+\beta)+\beta. \tag*{(22)} λ = ( s − 1 ) ( 1 − β ) − β , and for ∇ ∗ : λ ∗ = ( s − 1 ) ( 1 + β ) + β . (22) Theorem 8.2. Under the hypotheses of Theorem 8.1 with β \beta β ( ξ , λ ) (\xi,\lambda) ( ξ , λ ) ∇ \nabla ∇ ∇ ∗ \nabla^{*} ∇ ∗ λ = λ ∗ \lambda=\lambda^{*} λ = λ ∗ s β = 0 s\beta=0 s β = 0 β = 0 \beta=0 β = 0 T ( ξ , ξ ) = 0 \mathcal{T}(\xi,\xi)=0 T ( ξ , ξ ) = 0 λ = s − 1 \lambda=s-1 λ = s − 1
Remark 8.3. Formula (22) shows that the sign of λ \lambda λ β \beta β s s s β = 0 \beta=0 β = 0 s ≥ 2 s\geq2 s ≥ 2 expanding , whereas λ < 0 \lambda<0 λ < 0 β \beta β 1 1 1 ξ \xi ξ β \beta β β > s − 1 s \beta>\frac{s-1}{s} β > s s − 1 λ < 0 \lambda<0 λ < 0 ∇ \nabla ∇
ξ \xi ξ N N N If ξ ⊥ N \xi\perp N ξ ⊥ N η ( E ) = 0 \eta(E)=0 η ( E ) = 0 E E E (19) reduces to ∇ ~ E ξ = E \widetilde{\nabla}_E\xi=E ∇ E ξ = E (6) ,
A ξ = − I , A ξ ∗ = − I , h ( E , F ) = − g ( E , F ) ξ , A_{\xi}=-I,\qquad A_{\xi}^{*}=-I,\qquad h(E,F)=-g(E,F)\xi, A ξ = − I , A ξ ∗ = − I , h ( E , F ) = − g ( E , F ) ξ , i.e. N N N ξ \xi ξ ξ \xi ξ ξ ⊤ = 0 \xi^{\top}=0 ξ ⊤ = 0 Ric = − λ g \operatorname{Ric}=-\lambda g Ric = − λ g Ric ∗ = − λ g \operatorname{Ric}^{*}=-\lambda g Ric ∗ = − λ g
Theorem 8.4. If ξ \xi ξ N ⊂ ( B , ∇ , g , ξ ) N\subset(B,\nabla,g,\xi) N ⊂ ( B , ∇ , g , ξ ) ξ \xi ξ N N N Ric = − λ g \operatorname{Ric}=-\lambda g Ric = − λ g ∇ ∗ \nabla^{*} ∇ ∗ N N N λ = − scal / s \lambda=-\operatorname{scal}/s λ = − scal / s
Statistical Solitons with a Concircular Vector Field
Let v v v ( B , ∇ , g ) (B,\nabla,g) ( B , ∇ , g ) (11) and decompose v = v ⊤ + v ⊥ v=v^{\top}+v^{\perp} v = v ⊤ + v ⊥ N N N (5) –(6) ,
δ E = ∇ E v ⊤ + h ( E , v ⊤ ) − A v ⊥ E + D E ⊥ v ⊥ , \delta E=\nabla_Ev^{\top}+h(E,v^{\top})-A_{v^{\perp}}E+D_E^{\perp}v^{\perp}, δ E = ∇ E v ⊤ + h ( E , v ⊤ ) − A v ⊥ E + D E ⊥ v ⊥ , so comparing components,
∇ E v ⊤ = A v ⊥ E + δ E , h ( E , v ⊤ ) = − D E ⊥ v ⊥ . (23) \nabla_Ev^{\top}=A_{v^{\perp}}E+\delta E,\qquad h(E,v^{\top})=-D_E^{\perp}v^{\perp}. \tag*{(23)} ∇ E v ⊤ = A v ⊥ E + δ E , h ( E , v ⊤ ) = − D E ⊥ v ⊥ . (23) and dually ∇ E ∗ v ⊤ = A v ⊥ ∗ E + δ E \nabla_E^{*}v^{\top}=A_{v^{\perp}}^{*}E+\delta E ∇ E ∗ v ⊤ = A v ⊥ ∗ E + δ E
Lemma 9.1. Let v v v B B B N N N v ⊥ v^{\perp} v ⊥ A v ⊥ = f I A_{v^{\perp}}=fI A v ⊥ = f I v ⊤ v^{\top} v ⊤ N N N ∇ \nabla ∇ ∇ E v ⊤ = ( f + δ ) E \nabla_Ev^{\top}=(f+\delta)E ∇ E v ⊤ = ( f + δ ) E ∇ ∗ \nabla^{*} ∇ ∗ A v ⊥ ∗ A_{v^{\perp}}^{*} A v ⊥ ∗
Proof. ( ⇒ ) (\Rightarrow) ( ⇒ ) (23) , and ( ⇐ ) (\Leftarrow) ( ⇐ ) ∇ E v ⊤ = σ E \nabla_Ev^{\top}=\sigma E ∇ E v ⊤ = σ E (23) gives A v ⊥ E = ( σ − δ ) E A_{v^{\perp}}E=(\sigma-\delta)E A v ⊥ E = ( σ − δ ) E (7) , g ( A v ⊥ E , F ) = g ~ ( h ∗ ( E , F ) , v ⊥ ) g(A_{v^{\perp}}E,F)=\widetilde{g}(h^{*}(E,F),v^{\perp}) g ( A v ⊥ E , F ) = g ( h ∗ ( E , F ) , v ⊥ ) (23) into the soliton equation, we get the following statement.
Theorem 9.2. ( g , v ⊤ , λ ) (g,v^{\top},\lambda) ( g , v ⊤ , λ ) N N N ( B , ∇ , g , ξ ) (B,\nabla,g,\xi) ( B , ∇ , g , ξ )
Ric ( E , F ) = − ( λ + δ ) g ( E , F ) − g ~ ( h ∗ ( E , F ) , v ⊥ ) . (24) \operatorname{Ric}(E,F)=-(\lambda+\delta)g(E,F)-\widetilde{g}(h^{*}(E,F),v^{\perp}). \tag*{(24)} Ric ( E , F ) = − ( λ + δ ) g ( E , F ) − g ( h ∗ ( E , F ) , v ⊥ ) . (24) and dually
Ric ∗ ( E , F ) = − ( λ + δ ) g ( E , F ) − g ~ ( h ( E , F ) , v ⊥ ) . (25) \operatorname{Ric}^{*}(E,F)=-(\lambda+\delta)g(E,F)-\widetilde{g}(h(E,F),v^{\perp}). \tag*{(25)} Ric ∗ ( E , F ) = − ( λ + δ ) g ( E , F ) − g ( h ( E , F ) , v ⊥ ) . (25) If N N N A v ⊥ = f I A_{v^{\perp}}=fI A v ⊥ = f I v ⊤ v^{\top} v ⊤ N N N ∇ E v ⊤ = σ E \nabla_Ev^{\top}=\sigma E ∇ E v ⊤ = σ E σ = f + δ \sigma=f+\delta σ = f + δ (24) –(25) become
Ric = − ( λ + σ ) g , Ric ∗ = − ( λ + σ ) g . (26) \operatorname{Ric}=-(\lambda+\sigma)g,\qquad\operatorname{Ric}^{*}=-(\lambda+\sigma)g. \tag*{(26)} Ric = − ( λ + σ ) g , Ric ∗ = − ( λ + σ ) g . (26) Theorem 9.3. Let ( g , v ⊤ , λ ) (g,v^{\top},\lambda) ( g , v ⊤ , λ ) N N N ( B , ∇ , g , ξ ) (B,\nabla,g,\xi) ( B , ∇ , g , ξ ) ( B , ∇ ∗ , g , ξ ) (B,\nabla^{*},g,\xi) ( B , ∇ ∗ , g , ξ ) N N N Ric = − ( λ + σ ) g \operatorname{Ric}=-(\lambda+\sigma)g Ric = − ( λ + σ ) g
Remark 9.4. A manifold carrying a non-trivial concircular field is locally a warped product I × ρ M I \times_{\rho} M I × ρ M N N N σ \sigma σ
Almost Quasi-Yamabe Solitons on Submanifolds
Let v v v ( B , ∇ , g , ξ ) (B,\nabla,g,\xi) ( B , ∇ , g , ξ ) g g g ( L X g ) ( E , F ) = g ( ∇ E X , F ) + g ( E , ∇ F ∗ X ) (\mathcal{L}_{X}g)(E,F)=g(\nabla_E X,F)+g(E,\nabla^{*}_{F}X) ( L X g ) ( E , F ) = g ( ∇ E X , F ) + g ( E , ∇ F ∗ X ) (23) and its dual,
( L v ⊤ g ) ( E , F ) = 2 δ g ( E , F ) + 2 g ( A v ⊥ 0 E , F ) , A v ⊥ 0 = 1 2 ( A v ⊥ + A v ⊥ ∗ ) . (27) (\mathcal{L}_{v^\top}g)(E,F)=2\delta g(E,F)+2g(A^{0}_{v^\perp}E,F), \qquad A^{0}_{v^\perp}=\frac{1}{2}(A_{v^\perp}+A^{*}_{v^\perp}). \tag*{(27)} ( L v ⊤ g ) ( E , F ) = 2 δ g ( E , F ) + 2 g ( A v ⊥ 0 E , F ) , A v ⊥ 0 = 2 1 ( A v ⊥ + A v ⊥ ∗ ) . (27) Combining (27) with (10) gives:
Theorem 10.1. Let ( g , v ⊤ , λ , ω ) (g,v^\top,\lambda,\omega) ( g , v ⊤ , λ , ω ) N N N ( B , ∇ , g , ξ ) (B,\nabla,g,\xi) ( B , ∇ , g , ξ ) ( B , ∇ ∗ , g , ξ ) (B,\nabla^{*},g,\xi) ( B , ∇ ∗ , g , ξ )
( R − λ − δ ) g ( E , F ) = g ( A v ⊥ 0 E , F ) + ω η ( E ) η ( F ) , (28) (R-\lambda-\delta)g(E,F)=g(A^{0}_{v^\perp}E,F)+\omega\eta(E)\eta(F), \tag*{(28)} ( R − λ − δ ) g ( E , F ) = g ( A v ⊥ 0 E , F ) + ω η ( E ) η ( F ) , (28) where R R R N N N
Taking E = F = ξ E=F=\xi E = F = ξ N N N (28) , and tracing, we obtain the following two consequences.
Theorem 10.2. Under the hypotheses of Theorem 10.1, assume that ξ \xi ξ is tangent to N N N
(i) If g ( A v ⊥ 0 ξ , ξ ) = 0 g(A^{0}_{v^\perp}\xi,\xi)=0 g ( A v ⊥ 0 ξ , ξ ) = 0 h ( ξ , ξ ) = 0 h(\xi,\xi)=0 h ( ξ , ξ ) = 0 (20) ), then R = λ + δ + ω R=\lambda+\delta+\omega R = λ + δ + ω
(ii) If N N N tr A v ⊥ 0 = 0 \operatorname{tr}A^{0}_{v^\perp}=0 tr A v ⊥ 0 = 0 R = λ + δ + ω s R=\lambda+\delta+\frac{\omega}{s} R = λ + δ + s ω
Corollary 10.3. In the concurrent case δ = 1 \delta=1 δ = 1 R = λ + 1 + ω R=\lambda+1+\omega R = λ + 1 + ω R = λ + 1 + ω / s R=\lambda+1+\omega/s R = λ + 1 + ω / s N N N
Remark 10.4. The sign conventions in (10) matter: with the term + ω η ⊗ η +\omega\eta\otimes\eta + ω η ⊗ η (10) , the constant ω \omega ω (28) with a plus sign. The version with ω ↦ − ω \omega\mapsto-\omega ω ↦ − ω ω \omega ω
Chen-Type Inequalities for Statistical Warped Product Submanifolds
Solitons on warped products are related to their extrinsic geometry through the following inequality [38 ] , which generalises the first Chen inequality [10 ] .
Let ϕ : N = N 1 × f N 2 → N ~ ( c ~ ) \phi:N=N_{1}\times_{f}N_{2}\to\widetilde{N}(\widetilde{c}) ϕ : N = N 1 × f N 2 → N ( c ) c ~ \widetilde{c} c dim N 1 = r \dim N_{1}=r dim N 1 = r dim N 2 = k \dim N_{2}=k dim N 2 = k m = r + k m=r+k m = r + k { e 1 , … , e r } \{e_{1},\ldots,e_{r}\} { e 1 , … , e r } { e r + 1 , … , e m } \{e_{r+1},\ldots,e_{m}\} { e r + 1 , … , e m } N 1 N_{1} N 1 N 2 N_{2} N 2
Sec ( E 1 ∧ E 2 ) = − Hess f ( E 1 , E 1 ) f ( E 1 ∈ T N 1 , E 2 ∈ T N 2 unit ) , \operatorname{Sec}(E_{1}\wedge E_{2})=-\frac{\operatorname{Hess}f(E_{1},E_{1})}{f}\qquad(E_{1}\in TN_{1},\ E_{2}\in TN_{2}\ \text{unit}), Sec ( E 1 ∧ E 2 ) = − f Hess f ( E 1 , E 1 ) ( E 1 ∈ T N 1 , E 2 ∈ T N 2 unit ) , so that, with the sign convention Δ f = − tr Hess f \Delta f=-\operatorname{tr}\operatorname{Hess}f Δ f = − tr Hess f
∑ b = 1 r Sec ∇ , ∇ ∗ ( e b ∧ e r + 1 ) = Δ 0 f f . (29) \sum_{b=1}^{r}\operatorname{Sec}^{\nabla,\nabla^{*}}(e_{b}\wedge e_{r+1})=\frac{\Delta^{0}f}{f}. \tag*{(29)} b = 1 ∑ r Sec ∇ , ∇ ∗ ( e b ∧ e r + 1 ) = f Δ 0 f . (29) Theorem 11.1 (Chen-type inequality). Let N = N 1 × f N 2 N=N_1\times_f N_2 N = N 1 × f N 2 m m m n n n c ~ \widetilde{c} c
σ ∇ , ∇ ∗ − Sec ∇ , ∇ ∗ ( e 1 ∧ e r + 1 ) ≥ 2 ( σ 0 − Sec 0 ( e 1 ∧ e r + 1 ) ) − ( m − 2 ) ( m + 1 ) c ~ 2 − m 2 ( m − 2 ) 4 ( m − 1 ) ( ∥ H ∥ 2 + ∥ H ∗ ∥ 2 ) . \sigma^{\nabla,\nabla^*}-\operatorname{Sec}^{\nabla,\nabla^*}(e_1\wedge e_{r+1})\geq2\left(\sigma^0-\operatorname{Sec}^0(e_1\wedge e_{r+1})\right)-\frac{(m-2)(m+1)\widetilde{c}}{2}-\frac{m^2(m-2)}{4(m-1)}\left(\lVert H\rVert^2+\lVert H^*\rVert^2\right). σ ∇ , ∇ ∗ − Sec ∇ , ∇ ∗ ( e 1 ∧ e r + 1 ) ≥ 2 ( σ 0 − Sec 0 ( e 1 ∧ e r + 1 ) ) − 2 ( m − 2 ) ( m + 1 ) c − 4 ( m − 1 ) m 2 ( m − 2 ) ( ∥ H ∥ 2 + ∥ H ∗ ∥ 2 ) . Proof. (1) Use the Gauss equation (8) and 2 h 0 = h + h ∗ 2h^0=h+h^* 2 h 0 = h + h ∗
σ ∇ , ∇ ∗ = 2 σ 0 − m ( m − 1 ) c ~ 2 − 1 2 ∑ a ∑ i < j { h i i a h j j a − ( h i j a ) 2 + h i i ∗ a h j j ∗ a − ( h i j ∗ a ) 2 } . \sigma^{\nabla,\nabla^*}=2\sigma^0-\frac{m(m-1)\widetilde{c}}{2}-\frac{1}{2}\sum_a\sum_{i<j}\left\{h_{ii}^ah_{jj}^a-\left(h_{ij}^a\right)^2+h_{ii}^{*a}h_{jj}^{*a}-\left(h_{ij}^{*a}\right)^2\right\}. σ ∇ , ∇ ∗ = 2 σ 0 − 2 m ( m − 1 ) c − 2 1 a ∑ i < j ∑ { h ii a h j j a − ( h ij a ) 2 + h ii ∗ a h j j ∗ a − ( h ij ∗ a ) 2 } . and the analogous expression
Sec ∇ , ∇ ∗ ( e 1 ∧ e r + 1 ) = 2 Sec 0 ( e 1 ∧ e r + 1 ) − c ~ − 1 2 ∑ a { ⋯ } 1 , r + 1 . \operatorname{Sec}^{\nabla,\nabla^*}(e_1\wedge e_{r+1})=2\operatorname{Sec}^0(e_1\wedge e_{r+1})-\widetilde{c}-\frac{1}{2}\sum_a\{\cdots\}_{1,r+1}. Sec ∇ , ∇ ∗ ( e 1 ∧ e r + 1 ) = 2 Sec 0 ( e 1 ∧ e r + 1 ) − c − 2 1 a ∑ { ⋯ } 1 , r + 1 . (2) Subtract; the terms 1 2 ( h 1 , r + 1 a ) 2 \frac{1}{2}(h_{1,r+1}^a)^2 2 1 ( h 1 , r + 1 a ) 2 ( h i j a ) 2 (h_{ij}^a)^2 ( h ij a ) 2 ( h i j ∗ a ) 2 (h_{ij}^{*a})^2 ( h ij ∗ a ) 2
(3) For each normal index a a a φ a = ∑ i < j h i i a h j j a − h r + 1 , r + 1 a h r + 1 , r + 1 a \varphi_a=\sum_{i<j}h_{ii}^ah_{jj}^a-h_{r+1,r+1}^ah_{r+1,r+1}^a φ a = ∑ i < j h ii a h j j a − h r + 1 , r + 1 a h r + 1 , r + 1 a h 11 a + ⋯ + h m m a = t a h_{11}^a+\cdots+h_{mm}^a=t_a h 11 a + ⋯ + h mm a = t a [31 , 32 ] ). The maximum is attained at ( 0 , α , … , α ) (0,\alpha,\ldots,\alpha) ( 0 , α , … , α ) α = t a / ( m − 1 ) \alpha=t_a/(m-1) α = t a / ( m − 1 )
φ a ≤ m 2 ( m − 2 ) 2 ( m − 1 ) ( H a ) 2 . \varphi_a\leq\frac{m^2(m-2)}{2(m-1)}(H^a)^2. φ a ≤ 2 ( m − 1 ) m 2 ( m − 2 ) ( H a ) 2 . The same bound holds for φ a ∗ \varphi_a^* φ a ∗
Summing over e b e_b e b b = 1 , … , r b=1,\ldots,r b = 1 , … , r (29) one obtains a bound in terms of the warping function.
Theorem 11.2. Under the hypotheses of Theorem 11.1 ,
σ ∇ , ∇ ∗ ≥ 2 σ 0 − Δ 0 f r f − ( m − 2 ) ( m + 1 ) c ~ 2 − m 2 ( m − 2 ) 4 ( m − 1 ) ( ∥ H ∥ 2 + ∥ H ∗ ∥ 2 ) . \sigma^{\nabla,\nabla^*}\geq2\sigma^0-\frac{\Delta^0f}{rf}-\frac{(m-2)(m+1)\widetilde{c}}{2}-\frac{m^2(m-2)}{4(m-1)}\left(\lVert H\rVert^2+\lVert H^*\rVert^2\right). σ ∇ , ∇ ∗ ≥ 2 σ 0 − r f Δ 0 f − 2 ( m − 2 ) ( m + 1 ) c − 4 ( m − 1 ) m 2 ( m − 2 ) ( ∥ H ∥ 2 + ∥ H ∗ ∥ 2 ) . Remark 11.3 (Casorati curvatures). Analogous results hold for the Casorati curvatures C = 1 m ∥ h ∥ 2 C=\frac{1}{m}\lVert h\rVert^2 C = m 1 ∥ h ∥ 2 C ∗ = 1 m ∥ h ∗ ∥ 2 C^*=\frac{1}{m}\lVert h^*\rVert^2 C ∗ = m 1 ∥ h ∗ ∥ 2 δ \delta δ 2 σ ∇ , ∇ ∗ 2\sigma^{\nabla,\nabla^*} 2 σ ∇ , ∇ ∗ N 1 N_1 N 1 N 2 N_2 N 2 k Δ 0 f / f k\Delta^0f/f k Δ 0 f / f [38 ] . Equality cases, in which the second fundamental forms are forced to be of a special type, link these inequalities with minimal immersions of warped products into space forms. For further results on Casorati curvatures in statistical manifolds, we refer to [14 Section 18] .
Examples
Example 12.1. Let B = { ( x , y , z , u , w ) ∈ R 5 : w > 0 } B=\{(x,y,z,u,w)\in\mathbb{R}^5:w>0\} B = {( x , y , z , u , w ) ∈ R 5 : w > 0 }
v 1 = e − w ∂ x , v 2 = e − w ∂ y , v 3 = e − w ∂ z , v_1=e^{-w}\partial_x,\quad v_2=e^{-w}\partial_y,\quad v_3=e^{-w}\partial_z, v 1 = e − w ∂ x , v 2 = e − w ∂ y , v 3 = e − w ∂ z , v 4 = e − w ∂ u , v 5 = ∂ w = ξ , v_4=e^{-w}\partial_u,\quad v_5=\partial_w=\xi, v 4 = e − w ∂ u , v 5 = ∂ w = ξ , with ϕ v 1 = v 3 \phi v_1=v_3 ϕ v 1 = v 3 ϕ v 2 = v 4 \phi v_2=v_4 ϕ v 2 = v 4 ϕ v 3 = − v 1 \phi v_3=-v_1 ϕ v 3 = − v 1 ϕ v 4 = − v 2 \phi v_4=-v_2 ϕ v 4 = − v 2 ϕ ξ = 0 \phi\xi=0 ϕ ξ = 0
∇ v i g v i = − ξ , ∇ v i g v j = 0 ( i ≠ j ≤ 4 ) , \nabla^{g}_{v_i}v_i=-\xi,\qquad\nabla^{g}_{v_i}v_j=0\ (i\ne j\leq4), ∇ v i g v i = − ξ , ∇ v i g v j = 0 ( i = j ≤ 4 ) , ∇ v i g ξ = v i , ∇ ξ g v i = 0 , ∇ ξ g ξ = 0 , \nabla^{g}_{v_i}\xi=v_i,\qquad\nabla^{g}_{\xi}v_i=0,\qquad\nabla^{g}_{\xi}\xi=0, ∇ v i g ξ = v i , ∇ ξ g v i = 0 , ∇ ξ g ξ = 0 , and ( B , g ) (B,g) ( B , g ) H 5 ( − 1 ) \mathbb{H}^{5}(-1) H 5 ( − 1 ) g = e 2 w ∑ d x i 2 + d w 2 g=e^{2w}\sum dx_i^2+dw^2 g = e 2 w ∑ d x i 2 + d w 2
Define T ( E , F ) = a η ( E ) η ( F ) ξ \mathcal{T}(E,F)=a\eta(E)\eta(F)\xi T ( E , F ) = a η ( E ) η ( F ) ξ a ∈ R a\in\mathbb{R} a ∈ R (2) and (18) are readily checked, and ∇ = ∇ g + T \nabla=\nabla^{g}+\mathcal{T} ∇ = ∇ g + T ∇ ∗ = ∇ g − T \nabla^{*}=\nabla^{g}-\mathcal{T} ∇ ∗ = ∇ g − T
∇ v i v i = − ξ , ∇ v i ξ = v i , ∇ ξ ξ = a ξ , \nabla_{v_i}v_i=-\xi,\qquad\nabla_{v_i}\xi=v_i,\qquad\nabla_{\xi}\xi=a\xi, ∇ v i v i = − ξ , ∇ v i ξ = v i , ∇ ξ ξ = a ξ , ∇ v i ∗ v i = − ξ , ∇ v i ∗ ξ = v i , ∇ ξ ∗ ξ = − a ξ , \nabla^{*}_{v_i}v_i=-\xi,\qquad\nabla^{*}_{v_i}\xi=v_i,\qquad\nabla^{*}_{\xi}\xi=-a\xi, ∇ v i ∗ v i = − ξ , ∇ v i ∗ ξ = v i , ∇ ξ ∗ ξ = − a ξ , with all other covariant derivatives of frame fields zero. Thus ( B , ∇ , g , ϕ , ξ ) (B,\nabla,g,\phi,\xi) ( B , ∇ , g , ϕ , ξ ) β = a \beta=a β = a [ T E , T F ] = 0 [\mathcal{T}_{E},\mathcal{T}_{F}]=0 [ T E , T F ] = 0 (3) gives S = Rie g S=\operatorname{Rie}^{g} S = Rie g S ( E , F ) G = − ( g ( F , G ) E − g ( E , G ) F ) S(E,F)G=-(g(F,G)E-g(E,G)F) S ( E , F ) G = − ( g ( F , G ) E − g ( E , G ) F ) − 1 -1 − 1 Rie ∇ = Rie g + ( ∇ E g T ) F − ( ∇ F g T ) E \operatorname{Rie}^{\nabla}=\operatorname{Rie}^{g}+(\nabla^{g}_{E}\mathcal{T})_{F}-(\nabla^{g}_{F}\mathcal{T})_{E} Rie ∇ = Rie g + ( ∇ E g T ) F − ( ∇ F g T ) E
Ric ∇ = − ( 4 + a ) g + 5 a η ⊗ η , Ric ∇ ∗ = − ( 4 − a ) g − 5 a η ⊗ η , \operatorname{Ric}^{\nabla}=-(4+a)g+5a\eta\otimes\eta,\qquad\operatorname{Ric}^{\nabla^{*}}=-(4-a)g-5a\eta\otimes\eta, Ric ∇ = − ( 4 + a ) g + 5 a η ⊗ η , Ric ∇ ∗ = − ( 4 − a ) g − 5 a η ⊗ η , so that B B B η \eta η R ∇ = R ∇ ∗ = − 20 R^{\nabla}=R^{\nabla^{*}}=-20 R ∇ = R ∇ ∗ = − 20 a a a a = 0 a=0 a = 0 Ric = − 4 g \operatorname{Ric}=-4g Ric = − 4 g H 5 ( − 1 ) \mathbb{H}^{5}(-1) H 5 ( − 1 )
Example 12.2. On the manifold H 5 ( − 1 ) \mathbb{H}^{5}(-1) H 5 ( − 1 ) L ξ g = 2 ( g − η ⊗ η ) \mathcal{L}_{\xi}g=2(g-\eta\otimes\eta) L ξ g = 2 ( g − η ⊗ η ) (10) becomes
( 1 + λ − R ) g + ( ω − 1 ) η ⊗ η = 0 , (1+\lambda-R)g+(\omega-1)\eta\otimes\eta=0, ( 1 + λ − R ) g + ( ω − 1 ) η ⊗ η = 0 , and ( g , ξ , λ , ω ) (g,\xi,\lambda,\omega) ( g , ξ , λ , ω ) λ = R − 1 = − 21 \lambda=R-1=-21 λ = R − 1 = − 21 ω = 1 \omega=1 ω = 1 shrinking soliton. This holds for the Levi-Civita metric structure and is therefore valid for all members of the family ( ∇ , ∇ ∗ ) (\nabla,\nabla^{*}) ( ∇ , ∇ ∗ ) R ∇ = R ∇ ∗ = R g = − 20 R^{\nabla}=R^{\nabla^{*}}=R^{g}=-20 R ∇ = R ∇ ∗ = R g = − 20
Example 12.3. Using (19) and the Ricci tensors of Example 12.1, the ∇ \nabla ∇ ξ \xi ξ
( λ − 3 − a ) g + ( 6 a − 1 ) η ⊗ η = 0 , (\lambda-3-a)g+(6a-1)\eta\otimes\eta=0, ( λ − 3 − a ) g + ( 6 a − 1 ) η ⊗ η = 0 , and the ∇ ∗ \nabla^{*} ∇ ∗
( λ − 3 + a ) g − ( 6 a + 1 ) η ⊗ η = 0. (\lambda-3+a)g-(6a+1)\eta\otimes\eta=0. ( λ − 3 + a ) g − ( 6 a + 1 ) η ⊗ η = 0. The first is solvable only for a = 1 6 a=\frac{1}{6} a = 6 1 λ = 19 6 \lambda=\frac{19}{6} λ = 6 19 a = − 1 6 a=-\frac{1}{6} a = − 6 1 ( ξ , λ ) (\xi,\lambda) ( ξ , λ ) ∇ \nabla ∇ ∇ ∗ \nabla^{*} ∇ ∗ a a a B B B ξ \xi ξ β = 0 \beta=0 β = 0 η \eta η (21) , which the Einstein manifold H 5 ( − 1 ) \mathbb{H}^{5}(-1) H 5 ( − 1 ) a = 0 a=0 a = 0 η \eta η
Example 12.4. Consider the Kenmotsu structure on the unit hypersphere S 5 S^{5} S 5
T ( E , F ) = a η ( E ) η ( F ) ξ \mathcal{T}(E,F)=a\eta(E)\eta(F)\xi T ( E , F ) = a η ( E ) η ( F ) ξ and local coordinates ( x i , y i , w ) (x^{i},y^{i},w) ( x i , y i , w ) i = 1 , 2 i=1,2 i = 1 , 2
ξ = ∂ w , ϕ ( ∂ x i ) = ∂ y i , ϕ ( ∂ y i ) = − ∂ x i , ϕ ( ∂ w ) = 0. \xi=\partial_{w},\qquad\phi(\partial_{x^{i}})=\partial_{y^{i}},\qquad\phi(\partial_{y^{i}})=-\partial_{x^{i}},\qquad\phi(\partial_{w})=0. ξ = ∂ w , ϕ ( ∂ x i ) = ∂ y i , ϕ ( ∂ y i ) = − ∂ x i , ϕ ( ∂ w ) = 0. The 3-dimensional submanifold N = { ( x 1 , y 1 , 0 , 0 , w ) } N=\{(x^{1},y^{1},0,0,w)\} N = {( x 1 , y 1 , 0 , 0 , w )} ϕ T N ⊂ T N \phi TN\subset TN ϕT N ⊂ T N ξ \xi ξ N N N s = 3 s=3 s = 3 ( ξ , λ ) (\xi,\lambda) ( ξ , λ ) N N N a = β a=\beta a = β a = 0 a=0 a = 0 λ = 2 \lambda=2 λ = 2
Example 12.5. Statistical Ricci solitons without a contact structure arise already in dimensions two and three; we recall the following examples from the literature.
(a) On the hyperbolic half-plane ( H 2 , g = y − 2 ( d x 2 + d y 2 ) ) (H^2,g=y^{-2}(dx^2+dy^2)) ( H 2 , g = y − 2 ( d x 2 + d y 2 )) ∇ ∂ x ∂ x = 2 y − 1 ∂ y \nabla_{\partial_x}\partial_x=2y^{-1}\partial_y ∇ ∂ x ∂ x = 2 y − 1 ∂ y ∇ ∂ y ∂ y = y − 1 ∂ y \nabla_{\partial_y}\partial_y=y^{-1}\partial_y ∇ ∂ y ∂ y = y − 1 ∂ y ∇ ∂ x ∂ y = ∇ ∂ y ∂ x = 0 \nabla_{\partial_x}\partial_y=\nabla_{\partial_y}\partial_x=0 ∇ ∂ x ∂ y = ∇ ∂ y ∂ x = 0 [18 ] defines a statistical structure of constant curvature 0; it is Ricci-flat, hence a steady soliton , λ = 0 \lambda=0 λ = 0
(b) On ( R 2 , d x 2 + d y 2 ) (\mathbb{R}^2,dx^2+dy^2) ( R 2 , d x 2 + d y 2 ) ∇ ∂ x ∂ x = − ∂ y \nabla_{\partial_x}\partial_x=-\partial_y ∇ ∂ x ∂ x = − ∂ y ∇ ∂ y ∂ y = 0 \nabla_{\partial_y}\partial_y=0 ∇ ∂ y ∂ y = 0 ∇ ∂ x ∂ y = ∇ ∂ y ∂ x = ∂ x \nabla_{\partial_x}\partial_y=\nabla_{\partial_y}\partial_x=\partial_x ∇ ∂ x ∂ y = ∇ ∂ y ∂ x = ∂ x ∂ x \partial_x ∂ x − ∂ x -\partial_x − ∂ x [28 ] give an Einstein statistical manifold (scalar curvature − 2 -2 − 2 shrinking soliton .
(c) On R 3 \mathbb{R}^3 R 3 v 1 , v 2 , v 3 v_1,v_2,v_3 v 1 , v 2 , v 3 ∇ v 1 v 1 = b v 1 \nabla_{v_1}v_1=bv_1 ∇ v 1 v 1 = b v 1 ∇ v 2 v 2 = ∇ v 3 v 3 = b 2 v 1 \nabla_{v_2}v_2=\nabla_{v_3}v_3=\frac{b}{2}v_1 ∇ v 2 v 2 = ∇ v 3 v 3 = 2 b v 1 ∇ v 1 v 2 = ∇ v 2 v 1 = b 2 v 2 \nabla_{v_1}v_2=\nabla_{v_2}v_1=\frac{b}{2}v_2 ∇ v 1 v 2 = ∇ v 2 v 1 = 2 b v 2 ∇ v 1 v 3 = ∇ v 3 v 1 = b 2 v 3 \nabla_{v_1}v_3=\nabla_{v_3}v_1=\frac{b}{2}v_3 ∇ v 1 v 3 = ∇ v 3 v 1 = 2 b v 3 ∇ v 2 v 3 = ∇ v 3 v 2 = 0 \nabla_{v_2}v_3=\nabla_{v_3}v_2=0 ∇ v 2 v 3 = ∇ v 3 v 2 = 0 [4 ] one obtains an Einstein statistical manifold of constant curvature b 2 / 4 b^2/4 b 2 /4 3 b 2 / 2 3b^2/2 3 b 2 /2 expanding soliton .
Example 12.6 (Warped products). We have:
(a) The statistical warped product R × e z R n \mathbb{R}\times_{e^z}\mathbb{R}^n R × e z R n [28 ] has g ~ = d z 2 + e 2 z ( d x 1 2 + ⋯ + d x n 2 ) ; \widetilde{g}=dz^2+e^{2z}(dx_1^2+\cdots+dx_n^2); g = d z 2 + e 2 z ( d x 1 2 + ⋯ + d x n 2 ) ; H n + 1 ( − 1 ) H^{n+1}(-1) H n + 1 ( − 1 ) R × e z R n − 1 \mathbb{R}\times_{e^z}\mathbb{R}^{n-1} R × e z R n − 1 H n + 1 ( − 1 ) H^{n+1}(-1) H n + 1 ( − 1 )
(b) Euclidean space E n + 1 ∖ { 0 } E^{n+1}\setminus\{0\} E n + 1 ∖ { 0 } ( 0 , ∞ ) × t S n ( 1 ) (0,\infty)\times_t S^n(1) ( 0 , ∞ ) × t S n ( 1 ) g ~ = d t 2 + t 2 g S n \widetilde{g}=dt^2+t^2g_{S^n} g = d t 2 + t 2 g S n N ⊂ S n ( 1 ) N\subset S^n(1) N ⊂ S n ( 1 ) C ( N ) C(N) C ( N ) d t 2 + t 2 g N dt^2+t^2g_N d t 2 + t 2 g N C ( N ) C(N) C ( N ) E n + 1 E^{n+1} E n + 1
Both examples satisfy the hypotheses of Theorems 11.1 and 11.2 .
Concluding Remarks and Open Problems
We have presented a unified account of statistical solitons and almost quasi-Yamabe solitons on statistical warped products and on submanifolds of Kenmotsu statistical manifolds. The following themes emerge.
The soliton equation for the pair ( ∇ , ∇ ∗ ) (\nabla,\nabla^*) ( ∇ , ∇ ∗ ) ξ \xi ξ T ( ξ , ξ ) = 0 T(\xi,\xi)=0 T ( ξ , ξ ) = 0 Theorem 8.2 ).
Gradient solitons turn the Codazzi condition for Hessian and Ricci metrics into curvature conditions on grad f \operatorname{grad}f grad f ∇ θ \nabla^\theta ∇ θ θ = d f \theta=df θ = df ∣ Ric ∣ |\operatorname{Ric}| ∣ Ric ∣ § 8 ).
Chen-type inequalities link the scalar curvature, the squared mean curvatures ∥ H ∥ 2 \lVert H\rVert^2 ∥ H ∥ 2 ∥ H ∗ ∥ 2 \lVert H^*\rVert^2 ∥ H ∗ ∥ 2
Open problems.
Classify statistical solitons on R × f N 2 \mathbb{R} \times_f N_2 R × f N 2 N 1 × f R N_1 \times_f \mathbb{R} N 1 × f R
Determine all η \eta η ξ \xi ξ β = 0 \beta= 0 β = 0
Find global conditions under which a totally umbilical statistical soliton with a concircular field is isometric to a sphere (cf. Obata’s theorem).
Extend the theory to η \eta η gradient case.
Relate statistical solitons on the manifold of a parametric family B ‾ \overline{B} B
