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Keywords:
connections with totally skew-symmetric torsion; scalar curvature; $\nabla $-Einstein manifolds; parallel skew-torsion.
connections with totally skew-symmetric torsion; scalar curvature; $\nabla $-Einstein manifolds; parallel skew-torsion.
Summary:
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.
References:
[1] Agricola, I., Ferreira, A.C.: Einstein manifolds with skew torsion. Oxford Quart. J., 65, 2014, 717-741, DOI 10.1093/qmath/hat050 | MR 3261964
[2] Agricola, I., Friedrich, Th.: On the holonomy of connections with skew-symmetric torsion. Math. Ann., 328, 2004, 711-748, DOI 10.1007/s00208-003-0507-9 | MR 2047649 | Zbl 1055.53031
[3] Agricola, I., Becker-Bender, J., Kim, H.: Twistorial eigenvalue estimates for generalized Dirac operators with torsion. Adv. Math., 243, 2013, 296-329, DOI 10.1016/j.aim.2013.05.001 | MR 3062748
[4] Ammann, B., Bär, C.: The Einstein-Hilbert action as a spectral action. Noncommutative Geometry and the Standard Model of Elementary Particle Physics, 2002, 75-108, MR 1998531
[5] Besse, A.L.: Einstein manifolds. 1987, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Zbl 0613.53001
[6] Chrysikos, I.: Invariant connections with skew-torsion and $\nabla $-Einstein manifolds. J. Lie Theory, 26, 2016, 11-48, MR 3384980
[7] Chrysikos, I.: Killing and twistor spinors with torsion. Ann. Glob. Anal. Geom., 49, 2016, 105-141, DOI 10.1007/s10455-015-9483-z | MR 3464216
[8] Chrysikos, I.: A new $\frac{1}{2}$-Ricci type formula on the spinor bundle and applications. Adv. Appl. Clifford Algebras, 27, 2017, 3097-3127, DOI 10.1007/s00006-017-0810-2 | MR 3718180
[9] Chrysikos, I., Gustad, C. O'Cadiz, Winther, H.: Invariant connections and $\nabla $-Einstein structures on isotropy irreducible spaces. J. Geom. Phys., 138, 2019, 257-284, DOI 10.1016/j.geomphys.2018.10.012 | MR 3945042
[10] Draper, C.A., Garvin, A., Palomo, F.J.: Invariant affine connections on odd dimensional spheres. Ann. Glob. Anal. Geom., 49, 2016, 213-251, DOI 10.1007/s10455-015-9489-6 | MR 3485984
[11] Draper, C.A., Garvin, A., Palomo, F.J.: Einstein connections with skew-torsion on Berger spheres. J. Geom. Phys., 134, 2017, 133-141, DOI 10.1016/j.geomphys.2018.08.006 | MR 3886931
[12] Friedrich, Th., Ivanov, S.: Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math., 6, 1962, 64-94,
[13] Friedrich, Th., Kim, E.C.: The Einstein-Dirac equation on Riemannian spin manifolds. J. Geom. Phys., 33, 2000, 128-172, DOI 10.1016/S0393-0440(99)00043-1
[14] Kühnel, W.: Differential Geometry, Curves--Surfaces--Manifolds. 2002, Amer. Math. Soc. Student Math. Library, MR 1882174
[15] Ville, M.: Sur le volume des variétés riemanniennes pincées. Bulletin de la S. M. F., 115, 1987, 127-139,
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