Manifolds and Lie Groups pp 385-392 | Cite as

# Hessian Manifolds and Convexity

## Abstract

Let M be a *flat affine manifold*, that is, M admits open charts (U_{i}, x _{i} ^{1} ,..., x _{i} ^{n} such that M =U U_{i} and whose coordinate changes are all affine functions. Such local coordinate systems { _{i} ^{n} ,...,x _{i} ^{n} } will be called *affine local coordinate systems*. Throughout this note the local expressions for geometric concepts on M will be given in terms of affine local coordinate systems. A Riemannian metric g on M is said to be *Hessian* if for each point p∈M there exists a C^{∞}-function (⌽ defined on a neighborhood of p such that g_{ij}=∂^{2}ø/∂x^{i} ∂x^{j}. Such a function ⌽ is called a *primitive* of g on a neighborhood of p. Using the flat affine structure we define the exterior differentiation dl for tensor bundle valued forms on M. Let g be the cotangent bundle valued 1-form on M corresponding to a Riemannian metric g on M. Then we know that g is Hessian if and only if g^{0} is dl-closed. A flat affine manifold provided with a Hessian metric is called a *Hessian manifold*[4], [5]. Koszul dealt with the case where g^{0} is dl-exact [1], [2], [3].

## Keywords

Local Coordinate System Convex Domain Affine Function Open Chart Exterior Differentiation## Preview

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## References

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