Remarks on symplectic sectional curvature

Abstract

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.

Affine surfaces which are Kähler, para-Kähler, or nilpotent Kähler

“…Let M be an affine surface equipped with a parallel volume form Ω. Since dΩ = 0 and ∇Ω = 0, M is a Fedosov manifold [16] and there is a notion of symplectic sectional curvature (see [15,16]). A symplectic surface (M, ∇, Ω) has zero symplectic sectional curvature if and only if the Ω-Ricci operator Ω(Ric Ω (X), Y ) = ρ(X, Y ) is a nilpotent Kähler structure.…”

“…A symplectic surface (M, ∇, Ω) has zero symplectic sectional curvature if and only if the Ω-Ricci operator Ω(Ric Ω (X), Y ) = ρ(X, Y ) is a nilpotent Kähler structure. Moreover the symplectic sectional curvature is positive definite (resp., negative definite) if and only if Ric Ω is a Kähler (resp., para-Kähler) structure [15].…”

“…Moreover, since a symplectic surface has constant symplectic sectional curvature if and only if the Ricci tensor is parallel [15], it must be locally symmetric and thus locally homogeneous [13]. Hence the cases of non zero constant symplectic curvature correspond to affine structure defined by the Levi-Civita connections of the sphere, the hyperbolic plane and the Lorentzian hyperbolic plane.…”