Loop space of a K\"ahler manifold

TL;DR

This paper demonstrates that the loop space of a Kähler manifold naturally inherits a Kähler structure, is complete and unbounded, and allows geodesics to be constructed from those of individual leaves.

Contribution

It establishes the Kähler structure on the loop space and analyzes its geometric properties, including completeness, unboundedness, and geodesic construction methods.

Findings

Loop space inherits a Kähler structure

The loop space is complete and unbounded

Geodesics can be constructed from leaf geodesics

Abstract

We prove that the loop space of a K\"ahler manifold inherits a K\"ahler structure. Then we prove that equipped with this natural metric the loop space is complete and unbounded. Additionally, we show that a geodesic on the loop space can be constructed by piecing together geodesics from each individual leaf.