By Michael Spivak

ISBN-10: 0914098713

ISBN-13: 9780914098713

Booklet by means of Michael Spivak, Spivak, Michael

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**Example text**

Then g is locally expressed by ϕ RicD , n = dim M, g = Ddϕ + n−1 where RicD is the Ricci tensor of D and ϕ is a local function (cf. [Nomizu and Simon (1992)]). Proof. For the proof of this proposition the reader may refer to the above literature. 1 we introduced the difference tensor γ = ∇ − D on a Hessian manifold (M, D, g). The covariant differential Q = Dγ of γ is called the Hessian curvature tensor for (D, g). It reflects the properties of the Hessian structure (D, g), and performs a variety of important roles.

2 (3), where Ω = x ∈ Rn | xn > 2 i=1 1 2 n−1 (xi )2 . 2). Consider the following holomorphic transformation defined by wj = z j z n − wn = z n − 1 4 1 4 n−1 (z k )2 + 1 −1 1 ≤ j ≤ n − 1, , k=1 n−1 k=1 (z k )2 − 1 zn − 1 4 n−1 (z k )2 + 1 −1 . 3 k=1 |wk |2 < 1 . Dual Hessian structures In this section we will establish the duality that exists for Hessian structures. Let R∗n be the dual vector space of Rn . 1) Let Ω be a domain in Rn with a Hessian structure (D, g = Ddϕ). We call this domain a Hessian domain, and denote it by (Ω, D, g = Ddϕ).

Let Ω be a domain in Rn equipped with a convex function ϕ, that is, the Hessian g = Ddϕ is positive definite on Ω. Then the pair (D, g = Ddϕ) is a Hessian structure on Ω. Important examples of these structures include: n 1 (xi )2 , then gij = δij (Kronecker’s delta) 2 i=1 and g is a Euclidean metric. (1) Let Ω = Rn and ϕ = n (2) Let Ω = {x ∈ Rn | x1 > 0, · · · , xn > 0} and ϕ = then gij = δij 1 . xi (3) Let Ω = x ∈ Rn | xn > 1 2 i=1 (xi log xi − xi ), n−1 i (xi )2 and ϕ = − log xn − 1 δij f + xi xj −xi , where f = xn − −xj 1 2 1 Then [gij ] = 2 f n n (4) Let Ω = R and ϕ = log 1+ n e i=1 xi 1 2 n−1 (xi )2 .

### A Comprehensive Introduction to Differential Geometry by Michael Spivak

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