Complete Affine K$\ddot{a}$hler Manifolds

TL;DR

This paper proves that complete, connected, oriented Kähler affine manifolds with Ricci flatness or zero scalar curvature (for dimensions up to 5) are isometric to Euclidean space, revealing a rigidity property.

Contribution

It establishes a rigidity result for complete Kähler affine manifolds with Ricci flatness or zero scalar curvature in dimensions up to five.

Findings

Universal cover of such manifolds is Euclidean space

Rigidity holds for dimensions up to 5

Conditions imply flatness and Euclidean structure

Abstract

In this paper we prove that for a complete, connected and oriented K\"{a}ler affine manifold $(M,G)$ of dimension $n,$ if it is K\"ahler affine Ricci flat or the K$\ddot{a}$hler affine scalar curvature $S\equiv0,$ ($n\leq 5$), then the universal covering manifold $\widetilde{M}$ of $M$ is isometric to the Euclidean n-space $E^{n}$.