Remarks on the Donaldson metric

TL;DR

This paper analyzes the Donaldson metric on symplectic forms, computing its Levi-Civita connection, geodesics, and the covariant Hessian of an energy functional, advancing understanding of the geometric structure of symplectic form spaces.

Contribution

It provides explicit formulas for the Levi-Civita connection, geodesics, and the covariant Hessian related to the Donaldson metric, which were previously not fully understood.

Findings

Derived the Levi-Civita connection for the Donaldson metric

Described geodesics in the space of symplectic forms

Computed the covariant Hessian of an energy functional

Abstract

The Donaldson metric is a metric on the space of symplectic two-forms in a fixed cohomology class. It was introduced in [2]. We compute the associated Levi-Civita connection, describe it's geodesics and compute the formula for the covariant Hessian of an energy functional on the space of symplectic structures in a fixed cohomology class, introduced by S.Donaldson in [1].