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

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.

7 pages, to appear in "Communications in Mathematics"