Hessian manifolds of nonpositive constant Hessian sectional curvature. (English) Zbl 1277.53010

Furuhata, Hitoshi; Kurose, Takashi

Tohoku Math. J. (2) 65, No. 1, 31-42 (2013).

Let \(M\) be a connected \(C^{\infty}\)-manifold of dimension \(n \geq 2\), \(g\) a Riemannian metric and \(\nabla\) a torsion free connection on \(M\). Denote by \(\hat{\nabla}\) the Levi-Civita connection of \(g\) and by \(K(X,Y):= \nabla_XY - \hat{\nabla}_XY\) the symmetric difference tensor. The triple \((M,\nabla,g)\) is called a Hessian manifold if \(\nabla g\) is totally symmetric. For \(c \in \mathbb{R}\) the triple is called of constant Hessian curvature \(c\) if \[ (\nabla_XK)(Y,Z) = - \tfrac{c}{2} \cdot \left(g(X,Y)Z + g(X,Z)Y \right). \] The authors give examples of Hessian manifolds of constant Hessian curvature and investigate the relations between Riemannian manifolds of constant curvature and Hessian manifolds of constant Hessian curvature.

Reviewer: Udo Simon (Berlin)

MSC:

53A15	Affine differential geometry
53C05	Connections (general theory)
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords:

Hessian manifold; constant curvature