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Geometry of Hessian Structures

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. . . . . . . . . .. . . Geometry of Hessian Structures Hirohiko Shima h-shima@c-able.ne.jp Yamaguchi University 2013/8/29 Hirohiko Shima (Yamaguchi University) Geometry of Hessian Structures 2013/8/29 1 / 27 . . . . . . . ..1 Hessian Structures . ..2 Hessian Structures and K¨ahlerian Structures . ..3 Dual Hessian Structures . ..4 Hessian Curvature Tensor . ..5 Regular Convex Cones . ..6 Hessian Structures and Affine Differential Geometry . ..7 Hessian Structures and Information Geometry . ..8 Invariant Hessian Structures Hirohiko Shima (Yamaguchi University) Geometry of Hessian Structures 2013/8/29 2 / 27 . . . . . . Preface In 1964 Prof. Koszul was sent to Japan by the French Government and gave lectures at Osaka University. I was a student in those days and attended the lectures together with the late Professor Matsushima and Murakami. The topics of the lectures were a theory of flat manifolds with flat connection D and closed 1-form α such that Dα is positive definite. α being a closed 1-form it is locally expressed as α = dφ, and so Dα = Ddφ is just a Hessian metric in our view point. This is the ultimate origin of the notion of Hessian structures and the starting point of my research. Hirohiko Shima (Yamaguchi University) Geometry of Hessian Structures 2013/8/29 3 / 27 . . . . . . 1. Hessian structures . Definition (Hessian metric) .. . . .. . . M : manifold with flat connection D A Riemannian metric g on M is said to be a Hessian metric if g can be locally expressed by g = Ddφ, gij = ∂2 φ ∂xi ∂xj , where {x1 , · · · , xn } is an affine coordinate system w.r.t. D. (D, g) : Hessian structure on M (M, D, g) : Hessian manifold The function φ is called a potential of (D, g). Hirohiko Shima (Yamaguchi University) Geometry of Hessian Structures 2013/8/29 4 / 27 . . . . . . . Definition (difference tensor γ) .. . . .. . . Let γ be the difference tensor between the Levi-Civita connection ∇ for g and the flat connection D; γ = ∇ − D, γX Y = ∇X Y − DX Y . γi jk(component of γ)=Γi jk(Christoffel symbol for g) . Proposition (characterizations of Hessian metric) .. . . . Let (M, D) be a flat manifold and g a Riemannian metric on M. The following conditions are equivalent. (1) g is a Hessian metric. (2) (DX g)(Y , Z) = (DY g)(X, Z) Codazzi equation i.e. the covariant tensor Dg is symmetric. (3) g(γX Y , Z) = g(Y , γX Z) Hirohiko Shima (Yamaguchi University) Geometry of Hessian Structures 2013/8/29 5 / 27 . . . . . . 2. Hessian Structures and K¨ahlerian Structures . Definition (K¨ahlerian metric) .. . . .. . . A complex manifold with Hermitian metric g = ∑ i,j gij dzi d¯zj is called a K¨ahlerian metric if g is expressed by complex Hessian gij = ∂2 ψ ∂zi ∂¯zj , where {z1 , · · · , zn } is a holomorphic coordinate system. Chen and Yau called Hessian metrics affine K¨ahlerian metrics. . Proposition .. . . .. . . Let TM be the tangent bundle over Hessian manifold (M, D, g). Then TM is a complex manifold with K¨ahlerian metric gT = ∑n i,j=1 gij dzi d¯zj where zi = xi + √ −1dxi . Hirohiko Shima (Yamaguchi University) Geometry of Hessian Structures 2013/8/29 6 / 27 . . . . . . . Example (tangent bundle of paraboloid) .. . . .. . . Ω = { x ∈ Rn | xn − 1 2 n−1∑ i=1 (xi )2 > 0 } paraboloid φ = log { xn − 1 2 n−1∑ i=1 (xi )2 }−1 g = Ddφ : Hessian metric on Ω TΩ ∼= Ω + √ −1Rn ⊂ Cn : tube domain over Ω TΩ ∼