4-dimensional Space forms as determined by the volumes of small geodesic balls

TL;DR

This paper proves that in 4-dimensional manifolds, the volumes of small geodesic balls can determine if the manifold is a space form, extending Gray-Vanhecke's conjecture with new tensor calculus methods.

Contribution

It provides a new proof for 4-dimensional space forms using tensor calculus, avoiding topological characterizations.

Findings

Volumes of small geodesic balls characterize 4-dimensional space forms

New tensor calculus approach replaces topological methods

Extends Gray-Vanhecke conjecture to broader cases

Abstract

Gray-Vanhecke conjectured that the volumes of small geodesic balls could determine if the manifold is a space form, and provided a proof for the compact 4-dimensional manifold, and some cases. In this paper, similar results for the 4-dimensional case are obtained, based upon tensor calculus and classical theorems rather than the topological characterizations in [6].