Geodesic
A note on the volume of $\nabla $-Einstein manifolds with skew-torsion
Communications in Mathematics, Tome 29 (2021) no. 3, pp. 385-393

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We study the volume of compact Riemannian manifolds which are Einstein with respect to a metric connection with (parallel) skew\--tor\-sion. 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 \cite {Vil} related with the first variation of the volume on a compact Einstein manifold.
Classification : 53B05, 53C05, 53C25
Keywords: connections with totally skew-symmetric torsion; scalar curvature; $\nabla $-Einstein manifolds; parallel skew-torsion. 
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     author = {Chrysikos, Ioannis},
     title = {A note on the volume of $\nabla ${-Einstein} manifolds with skew-torsion},
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Chrysikos, Ioannis. A note on the volume of $\nabla $-Einstein manifolds with skew-torsion. Communications in Mathematics, Tome 29 (2021) no. 3, pp. 385-393. http://geodesic.mathdoc.fr/item/COMIM_2021__29_3_a4/