Sasakian statistical manifolds II

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Sasakian statistical manifolds II

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Sasakian statistical manifolds II Hitoshi Furuhata Department of Mathematics, Hokkaido University Sapporo 060-0810, Japan furuhata@math.sci.hokudai.ac.jp 1 Introduction This article is a digest of [2] and [3] with additional remarks on invariant sub- manifolds of Sasakian statistical manifolds. We set Ω = {1, . . . , n+1} as a sample space, and denote by P+ (Ω) the set of positive probability densities, that is, P+ (Ω) = {p : Ω → R+ | ∑ x∈Ω p(x) = 1 }, where R+ is the set of positive real numbers. Let M be a smooth manifold as a parameter space, and s : M ∋ u 7→ p(·, u) ∈ P+ (Ω) an injection with the property that p(x, ·) : M → R+ is smooth for each x ∈ Ω. Consider a family of positive probability densities on Ω parametrized by M in this manner. We define a (0, 2)-tensor field on M by gu(X, Y ) = ∑ x∈Ω {X log p(x, ·)}{Y log p(x, ·)}p(x, u) for tangent vectors X, Y ∈ TuM. We say that an injection s : M → P+ (Ω) is a statistical model if gu is nondegenerate for each u ∈ M, namely, if g is a Rieman- nian metric on M, which is called the Fisher information metric for s. Define φ : M → Rn+1 for a statistical model s by φ(u) = t [2 √ p(1, u), . . . , 2 √ p(n + 1, u)]. It is known that the metric on M induced by φ from the Euclidean metric on Rn+1 coincides with the Fisher information metric g. Since the image φ(M) lies on the n-dimensional hypersphere Sn (2) of radius 2, the Fisher information met- ric is considered as the Riemannian metric induced from the standard metric of the hypersphere. For example, we set M = {u = t [u1 , . . . , un ] ∈ Rn | uj > 0, n ∑ l=1 ul < 1 }, s : M ∋ u 7→ p(x, u) = { uk , x = k ∈ {1, . . . , n}, 1 − ∑n l=1 ul , x = n + 1. Then φ(M) = Sn (2) ∩ (R+)n+1 and the Fisher information metric is the re- striction of the standard metric of Sn (2). It shows that a hypersphere with the standard metric plays an important role in information geometry. It is an inter- esting question whether a whole hypersphere plays another part there. In this article, we give a certain statistical structure on an odd-dimensional hypersphere, and explain its background. 2 2 Sasakian statistical structures Throughout this paper, M denotes a smooth manifold, and Γ(E) denotes the set of sections of a vector bundle E → M. All the objects are assumed to be smooth. For example, Γ(TM(p,q) ) means the set of all the C∞ tensor fields on M of type (p, q). At first, we will review the basic notion of Sasakian manifolds, which is a classical topic in differential geometry (See [5] for example). Let g ∈ Γ(TM(0,2) ) be a Riemannian metric, and denote by ∇g the Levi-Civita connection of g. Take ϕ ∈ Γ(TM(1,1) ) and ξ ∈ Γ(TM). A triple (g, ϕ, ξ) is called an almost contact metric structure on M if the following equations hold for any X, Y ∈ Γ(TM): ϕ ξ = 0, g(ξ, ξ) = 1, ϕ2 X = −X + g(X, ξ)ξ, g(ϕX, Y ) + g(X, ϕY ) = 0. An almost contact metric structure on M is called a Sasakian structure if (∇g Xϕ)Y = g(Y, ξ)X − g(Y, X)ξ (1) holds for any X, Y ∈ Γ(TM). We call a manifold equipped with a Sasakian structure a Sasakian manifold. It is known that on a Sasakian manifold the formula ∇g Xξ = ϕX (2) holds for X ∈ Γ(TM). A typical example of a Sasakian manifold is a hypersphere of odd dimension as mentioned below. We now review the basic notion of statistical manifolds to fix the notation (See [1] and references therein). Let ∇ be an affine connection of M, and g ∈ Γ(TM(0,2) ) a Riemannian metric. The pair (∇, g) is called a statistical structure on M if (i) ∇XY −∇Y X −[X, Y ] = 0 and (ii) (∇Xg)(Y, Z) = (∇Y g)(X, Z) hold for any X, Y, Z ∈ Γ(TM). By definition, (∇g , g) is a statistical structure on M. We denote by R∇ the curvature tensor field of ∇, and by ∇∗ the dual con- nection of ∇ with respect to g, and set S = S(∇,g) ∈ Γ(TM(1,3) ) as the mean of the curvature tensor fields of ∇ and of ∇∗ , that is, for X, Y, Z ∈ Γ(TM), R∇ (X, Y )Z = ∇X∇Y Z − ∇Y ∇XZ − ∇[X,Y ]Z, Xg(Y, Z) = g(∇XY, Z) + g(Y, ∇∗ XZ), S(X, Y )Z = 1 2 {R∇ (X, Y )Z + R∇∗ (X, Y )Z}. (3) A statistical manifold (M, ∇, g) is called a Hessian manifold if R∇ = 0. If so, we have R∇∗ = S = 0 automatically. 3 For a statistical structure (∇, g) on M, we set K = ∇ − ∇g . Then the following hold: K ∈ Γ(TM(1,2) ), KXY = KY X, g(KXY, Z) = g(Y, KXZ) (4) for any X, Y, Z ∈ Γ(TM). Conversely, if K satisfies (4), the pair (∇ = ∇g +K, g) is a statistical structure on M. The formula S(X, Y )Z = Rg (X, Y )Z + [KX, KY ]Z (5) holds, where Rg = R∇g is the curvature tensor field of the Levi-Civita connection of g. For a statistical structure (∇, g), we often use the expression like (∇ = ∇g + K, g), and write KXY by K(X, Y ). Definition 1. A quadruplet (∇ = ∇g +K, g, ϕ, ξ) is called a Sasakian statistical structure on M if (i) (g, ϕ, ξ) is a Sasakian structure and (ii) (∇, g) is a statistical structure on M, and (iii) K ∈ Γ(TM(1,2) ) for (∇, g) satisfies K(X, ϕY ) + ϕK(X, Y ) = 0 for X, Y ∈ Γ(TM). (6) These three conditions are paraphrased in the following three conditions([3, Theorem 2.17]: (i’) (g, ϕ, ξ) is an almost contact metric structure and (ii) (∇, g) is a statistical structure on M, and (iii’) they satisfy ∇X(ϕY ) − ϕ∇∗ XY = g(ξ, Y )X − g(X, Y )ξ, (7) ∇Xξ = ϕX + g(∇Xξ, ξ)ξ. (8) We get the following formulas for a Sasakian statistical manifold: K(X, ξ) = λg(X, ξ)ξ, g(K(X, Y ), ξ) = λg(X, ξ)g(Y, ξ), (9) where λ = g(K(ξ, ξ), ξ). (10) Proposition 2. For a Sasakian statistical manifold (M, ∇, g, ϕ, ξ), S(X, Y )ξ = g(Y, ξ)X − g(X, ξ)Y (11) holds for X, Y ∈ Γ(TM). Proof. By (9), we have [KX, KY ]ξ = 0, from which (5) implies S = Rg . It is known that Rg is written as the right hand side of (11) (See [5]). A quadruplet (f M, e ∇ = ∇e g + e K, e g, e J) is called a holomorphic statistical man- ifold if (e g, e J) is a Kähler structure, (e ∇, e g) is a statistical structure on f M, and e K(X, e JY ) + e J e K(X, Y ) = 0 (12) 4 holds for X, Y ∈ Γ(T f M). The notion of Sasakian statistical manifold can be also expressed in the following: The cone over M defined below is a holomorphic statistical manifold. Let (M, ∇ = ∇g + K, g, ϕ, ξ) be a statistical manifold with an almost contact metric structure. Set f M as M ×R+, and define a Riemannian metric e g = r2 g + (dr)2 on f M. Take a vector field Ψ = r ∂ ∂r ∈ Γ(T f M), and define e J ∈ Γ(T f M(1,1) ) by e JΨ = ξ and e JX = ϕX − g(X, ξ)Ψ for any X ∈ Γ(TM). Then, (e g, e J) is an almost Hermitian structure on f M, and furthermore, (g, ϕ, ξ) is a Sasakian structure on M if and only if (e g, e J) is a Kähler structure on f M. We construct connection e ∇ on f M by      e ∇Ψ Ψ = −λξ + Ψ, e ∇XΨ = e ∇Ψ X = X − λg(X, ξ)Ψ, e ∇XY = ∇XY − g(X, Y )Ψ, that is, e K(Ψ, Ψ) = −λξ, e K(X, Ψ) = −λg(X, ξ)Ψ, e K(X, Y ) = K(X, Y ) for X, Y ∈ Γ(TM), where λ is in (10). We then have that (M, ∇, g, ϕ, ξ) is a Sasakian statistical manifold if and only if (f M, e ∇, e g, e J) is a holomorphic statis- tical manifold (A general statement is given as [2, Proposition 4.8 and Theorem 4.10]). It is derived from the fact that the formula (12) holds if and only if both (6) and (9) hold. Example 3. Let S2n−1 be a unit hypersphere in the Euclidean space R2n . Let J be a standard almost complex structure on R2n considered as Cn , and set ξ = −JN, where N is a unit normal vector field of S2n−1 . Define ϕ ∈ Γ(T(S2n−1 )(1,1) ) by ϕ(X) = JX−⟨JX, N⟩N. Denote by g the standard metric of the hypersphere. Then such a (g, ϕ, ξ) is known as a standard Sasakian structure on S2n−1 . We set K(X, Y ) = g(X, ξ)g(Y, ξ)ξ (13) for any X, Y ∈ Γ(TS2n−1 ). Since K satisfies (4) and (6), we have a Sasakian statistical structure (∇ = ∇g + K, g, ϕ, ξ) on S2n−1 . Proposition 4. Let (M, g, ϕ, ξ) be a Sasakian manifold. Set ∇ as ∇g + fK for f ∈ C∞ (M), where K is given in (13). Then (∇, g, ϕ, ξ) is a Sasakian statistical structure on M. Conversely, we define ∇XY = ∇g XY + L(X, Y )V for some unit vector field V and L ∈ Γ(TM(0,2) ). If (∇, g, ϕ, ξ) is a Sasakian statistical structure, then L ⊗ V is written as L(X, Y )V = fg(X, ξ)g(Y, ξ)ξ for some f ∈ C∞ (M), as above. Proof. The first half is obtained by direct calculation. To get the second half, we have by (4), 0 = L(X, Y )V − L(Y, X)V = {L(X, Y ) − L(Y, X)}V, 0 = g(L(X, Y )V, Z) − g(Y, L(X, Z)V ) = g(L(X, Y )Z − L(X, Z)Y, V ). (14) 5 Substituting V for Z in (14), we have L(X, Y ) = L(V, V )g(X, V )g(Y, V ). Accordingly, we get by (6), 0 = L(X, ϕY )V + ϕ{L(X, Y )V } = L(V, V )g(X, V ){−g(Y, ϕV )V + g(Y, V )ϕV }, which implies that ϕV = 0 if L(V, V ) ̸= 0, and hence V = ±ξ. 3 Invariant submanifolds Let (f M, e g, e ϕ, e ξ) be a Sasakian manifold, and M a submanifold of f M. We say that M is an invariant submanifold of f M if (i) e ξu ∈ TuM, (ii) e ϕX ∈ TuM for any X ∈ TuM and u ∈ M. Let g ∈ Γ(TM(0,2) ), ϕ ∈ Γ(TM(1,1) ) and ξ ∈ Γ(TM) be the restriction of e g, e ϕ and e ξ, respectively. Then it is shown that (g, ϕ, ξ) is a Sasakian structure on M. A typical example of an invariant submanifold of a Sasakian manifold S2n−1 in Example 3 is an odd dimensional unit sphere. Furthermore, we have the following example. Let ι : Q → CPn−1 be a complex hyperquadric in the complex projective space, and e Q the principal fiber bundle over Q induced by ι from the Hopf fibration π : S2n−1 → CPn−1 . We denote the induced homomorphism by e ι : e Q → S2n−1 . Then it is known that e ι( e Q) is an invariant submanifold (See [4], [5]). We briefly review the statistical submanifold theory to study invariant sub- manifolds of a Sasakian statistical manifold. Let (f M, e ∇, e g) be a statistical mani- fold, and M a submanifold of f M. Let g be the metric on M induced from e g, and consider the orthogonal decomposition with respect to e g: Tu f M = TuM ⊕TuM⊥ . According to this decomposition, we define an affine connection ∇ on M, B ∈ Γ(TM⊥ ⊗TM(0,2) ), A ∈ Γ((TM⊥ )(0,1) ⊗TM(1,1) ), and a connection ∇⊥ of the vector bundle TM⊥ by e ∇XY = ∇XY + B(X, Y ), e ∇XN = −AN X + ∇⊥ XN (15) for X, Y ∈ Γ(TM) and N ∈ Γ(TM⊥ ). Then (∇, g) is a statistical structure on M. In the same fashion, we define an affine connection ∇∗ on M, B∗ ∈ Γ(TM⊥ ⊗ TM(0,2) ), A∗ ∈ Γ((TM⊥ )(0,1) ⊗ TM(1,1) ), and a connection (∇⊥ )∗ of TM⊥ by using th dual connection e ∇∗ instead of e ∇ in (15). We remark that e g(B(X, Y ), N) = g(A∗ N X, Y ) for X, Y ∈ Γ(TM) and N ∈ Γ(TM⊥ ), and remark that ∇∗ coincides with the dual connection of ∇ with respect to g. See [1] for example. Theorem 5. Let (f M, e ∇, e g, e ϕ, e ξ) be a Sasakian statistical manifold, and M an invariant submanifold of f M with g, ϕ, ξ, ∇, B, A, ∇⊥ , ∇∗ , B∗ , A∗ , (∇⊥ )∗ defined as above. Then the following hold: (i) A quintuplet (M, ∇, g, ϕ, ξ) is a Sasakian statistical manifold. 6 (ii) B(X, ξ) = B∗ (X, ξ) = 0 for any X ∈ Γ(TM). (iii) B(X, ϕY ) = B(ϕX, Y ) = e ϕB∗ (X, Y ) for any X, Y ∈ Γ(TM). In particular, trgB = trgB∗ = 0. (iv) If B is parallel with respect to the Van der Weaden-Bortolotti connection e ∇′ for e ∇, then B and B∗ vanish. Namely, if (e ∇′ XB)(Y, Z) = ∇⊥ XB(Y, Z) − B(∇XY, Z) − B(Y, ∇XZ) = 0 for Z ∈ Γ(TM), then B∗ (X, Y ) = 0. (v) e g(e S(X, e ϕX)e ϕX − S(X, ϕX)ϕX, X) = 2e g(B∗ (X, X), B(X, X)) for X ∈ Γ(TM), where S = S(∇,g) and e S = S( e ∇,e g) as in (3). Corollary 6. Let (f M, e ∇, e g, e ϕ, e ξ) be a Sasakian statistical manifold of constant e ϕ-sectional curvature c, and M an invariant submanifold of f M. The induced Sasakian statistical structure on M has constant ϕ-sectional curvature c if and only if e g(B∗ (X, X), B(X, X)) = 0 for any X ∈ Γ(TM) orthogonal to ξ. If we take the Levi-Civita connection as e ∇, the properties above reduce to the ones for an invariant submanifold of a Sasakian manifold. It is known that an invariant submanifold of a Sasakian manifold of constant e ϕ-sectional curvature c is of constant ϕ-sectional curvature c if and only if it is totally geodesic. It is obtained by setting B = B∗ in Corollary 6. It is an interesting question whether there is an interesting invariant submanifold having nonvanishing B with the above property. Outline of Proof of Theorem 5. The proof of (i) can be omitted. By (i) and (8), we calculate that ∇Xξ+B(X, ξ) = e ∇Xξ = e ϕX+e g(e ∇Xξ, e ξ)e ξ = ϕX + g(∇Xξ, ξ)ξ. Comparing the normal components, we have (ii). By (7), we have e g(Y, e ξ)X − e g(Y, X)e ξ = e ∇X(e ϕY ) − e ϕe ∇∗ XY = ∇X(ϕY ) + B(X, ϕY )− e ϕ(∇∗ XY +B∗ (X, Y )) = g(Y, ξ)X−g(Y, X)ξ+B(X, ϕY )− e ϕB∗ (X, Y ). Comparing the normal components, we have (iii). By (i) and (ii), we get that 0 = ∇⊥ XB(Y, ξ) − B(∇XY, ξ) − B(Y, ∇Xξ) = −B(Y, ϕX) = −e ϕB∗ (X, Y ), which implies (iv). To get (v), we use the Gauss equation in the submanifold theory. The tan- gential component of R e ∇ (X, Y )Z is given as R∇ (X, Y )Z − AB(Y,Z)X + AB(X,Z)Y, for X, Y, Z ∈ Γ(TM), which implies that 2e g(e S(X, Y )Z, W) = 2g(S(X, Y )Z, W) −e g(B∗ (X, W), B(Y, Z)) + e g(B∗ (Y, W), B(X, Z)) −e g(B(X, W), B∗ (Y, Z)) + e g(B(Y, W), B∗ (X, Z)). Therefore, we prove (v) from (iii). To get Corollary 6, we have only to review the definition. A Sasakian sta- tistical structure (∇, g, ϕ, ξ) is said to be of constant ϕ-sectional curvature c if the sectional curvature defined by using S equals c for each ϕ-section, the plane 7 spanned by X and ϕX for a unit vector X orthogonal to ξ: g(S(X, ϕX)ϕX, X) = cg(X, X)2 for X ∈ Γ(TM) such that g(X, ξ) = 0. Acknowledgments The author thanks the anonymous reviewers for their careful reading of the manuscript. This work was supported by JSPS KAKENHI Grant Number JP26400058. References 1. H. Furuhata and I. Hasegawa, Submanifold theory in holomorphic statistical mani- folds, S. Dragomir, M.H. Shahid, and F.R. Al-Solamy (eds), Geometry of Cauchy- Riemann Submanifolds, Springer Singapore, 2016, 179–215. 2. H. Furuhata, I. Hasegawa, Y. Okuyama and K. Sato, Kenmotsu statistical mani- folds and warped product, Preprint. 3. H. Furuhata, I. Hasegawa, Y. Okuyama, K. Sato and M.H. Shahid, Sasakian sta- tistical manifolds, J. Geom. Phys. 117(2017), 179–186. 4. K. Kenmotsu, Invariant submanifolds in a Sasakian manifold, Tohoku Math. J. 21(1969), 495–500. 5. K. Yano and M. Kon, Structures on manifolds, World Scientific, 1984.