Remarks on symplectic sectional curvature

TL;DR

This paper introduces the concept of constant symplectic sectional curvature, characterizes it through curvature tensors, and explores its implications for symplectic geometry and submanifold structures.

Contribution

It defines and characterizes constant symplectic sectional curvature, extending previous work and analyzing its geometric consequences in symplectic connections.

Findings

Characterization of constant symplectic sectional curvature

Relations between curvature conditions and symplectic submanifold geometry

Insights into curvature tensor properties in symplectic connections

Abstract

In [11], I. M. Gelfand, V. Retakh, and M. Shubin defined the symplectic sectional curvature of a torsion-free connection preserving a symplectic form. The present article defines the corresponding notion of constant symplectic sectional curvature and characterizes this notion in terms of the curvature tensor of the symplectic connection and its covariant derivatives. Some relations between various more general conditions on the symplectic sectional curvature and the geometry of the symplectic connection or that induced on a symplectic submanifold are explored as well.