An introduction to tensor analysis by Leonard Lovering Barrett

By Leonard Lovering Barrett

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For a Hermitian metric g we set ρ(X, Y ) = g(JX, Y ). Then the skew symmetric bilinear form ρ is called a K¨ ahlerian form for (J, g), and, using a holomorhic coordinate system, we have √ gi¯j dz i ∧ d¯ zj . 5. Let g be a Hermitian metric on a complex manifold M . Then the following conditions are equivalent. (1) g is a K¨ ahlerian metric. (2) The K¨ ahlerian form ρ is closed; dρ = 0. Let (M, D) be a flat manifold and let T M be the tangent bundle over M with projection π : T M −→ M . 2) where ξ i = xi ◦ π and ξ n+i = dxi .

1) Let Ω = Rn and ϕ = i (2) Let Ω = R+ = {x ∈ R | x > 0} and ϕ = log x−1 . We then have 1 g = 2 dx2 . 2). Then T R+ is identified with a half plane {(ξ 1 , ξ 2 ) | ξ 1 > 0}, and the K¨ ahlerian metric g T on T R+ induced by g is expressed by gT = (dξ 1 )2 + (dξ 2 )2 . (ξ 1 )2 Thus g T is the Poincar´e metric on the half plane. 3 (2) is extended to regular convex cones as follows. 4. Let Ω be a regular convex cone in Rn , and let ψ be the characteristic function. Then (D, g = Dd log ψ) is a Hessian structure on Ω (cf.

On the other hand, a Riemannian metric on a complex manifold is said to be a K¨ ahlerian metric if it can be locally given by the complex Hessian with respect to a holomorphic coordinate system. This suggests that the following set of analogies exists between Hessian structures and K¨ ahlerian structures: Flat manifolds ←→ Complex manifolds Affine coordinate systems ←→ Holomorphic coordinate systems Hessian metrics ←→ K¨ ahlerian metrics In this section we show that the tangent bundle over a Hessian manifold admits a K¨ ahlerian metric induced by the Hessian metric.

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