On Topology and Geometry of Hessian Manifolds

Abstract: For a differentiable manifold $M$, a pair $(M, \nabla)$ is termed an affine manifold, if $\nabla$ is a flat and torsion-free connection on the tangent bundle $TM\rightarrow M$. A Riemannian metric $g$ on $M$ is called a Hessian metric on affine manifold $(M, \nabla)$, if $g$ can be expressed locally as the Hessian quadratic form of some smooth potential $f$. An affine manifold equipped with a Hessian metric is termed a Hessian manifold. We study the topology and geometry of Hessian manifolds. Key topological constraints are derived, including the vanishing of the Euler characteristic for any compact Hessian manifold and the proof that its fundamental group must be infinite and torsion-free. We demonstrate that compact Hessian manifolds with an abelian fundamental group are homeomorphic to the flat torus. The theory of flat line bundles is developed to introduce the canonical and obstruction bundles, which are used to prove that a curved compact orientable Hessian manifold must be a mapping torus. A Hessian manifold $((M, \nabla),g)$ is said to be of Koszul type, if there exists an $1$-form $\eta\in\Omega^1(M)$ such that $g=\nabla\eta$. We place emphasis on Hessian manifolds of Koszul type, establishing an equivalence between their existence and that of positive flat line bundles, space-like Lagrangian immersions, and hyperbolic affine structures. A duality with radiant manifolds, analogous to the classical Legendre transformation, is explored. We also construct a non-trivial family of compact Hessian manifolds of Koszul type with negative scalar curvature.