Geometry of Hessian manifolds

✍ Scribed by Hirohiko Shima; Katsumi Yagi

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 906 KB
• Volume: 7
• Category: Article
• ISSN: 0926-2245
• DOI: 10.1016/s0926-2245(96)00057-5

✦ Synopsis

Let M be an aftine manifold with a flat affme connection D. A Riemannian metric g on M is said to be a Hessian metric if g has a local expression g = D'u. A manifold M provided with such a pair (D, g) is called a Hessian manifold. We first prove the convexity of a Hessian manifold. We give a Weitzenbock type formula for the Laplacian reflecting (D, g). As an application we characterize D-parallel vector fields, and obtain a structure theorem of a compact Hessian manifold which is an extension of Calabi's theorem. We also discuss the Albanese variety of a compact affine manifold.