Connection with torsion, parallel spinors and geometry of Spin(7) manifolds

Details Connection with torsion, parallel spinors and geometry of Spin(7) manifolds Connection with torsion, parallel spinors and geometry of Spin(7) manifolds

2001-11-20

arxiv.org/abs/math/0111216v3

Math. Res. Lett. 11 (2004), no. 2-3, 171--186.

Mathematics

Differential Geometry

LaTeX, 15 pages, references added, some typos corrected, final version to appear in Math. Res. Lett

Scientific paper

We show that on every Spin(7) manifold there always exists a unique linear connection with totally skew-symmetric torsion preserving a nontrivial spinor and the Spin(7) structure. We express its torsion and the Riemannian scalar curvature in terms of the fundamental 4-form. We present an explicit formula for the Riemannian covariant derivative of the fundamental 4-form in terms of its exterior differential. We show the vanishing of the (\hat)-A genus and obtain a linear relation between Betti numbers of a compact Spin(7) manifolds which are locally but not globally conformally equivalent to a space with closed fundamental 4-form. A general solution to the Killing spinor equations is presented

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