A note on the volume of $\nabla$-Einstein manifolds with skew-torsion

TL;DR

This paper investigates the volume properties of compact Einstein manifolds with skew-torsion, extending previous results on volume variation to include manifolds with such special geometric structures.

Contribution

It generalizes Ville's result by analyzing the first variation of volume for Einstein manifolds with parallel skew-torsion, linking it to scalar curvature.

Findings

Sign of the first volume variation depends on scalar curvature.

Provides criteria for volume increase or decrease under metric variations.

Extends classical Einstein volume variation results to skew-torsion settings.

Abstract

We study the volume of compact Riemannian manifolds which are Einstein with respect to a metric connection with (parallel) skew-torsion. We provide a result for the sign of the first variation of the volume in terms of the corresponding scalar curvature. This generalizes a result of M. Ville, related with the first variation of the volume on a compact Einstein manifold.