Higher transgressions of the Pfaffian

Sergiu Moroianu

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arXiv · 2011.06538

Higher transgressions of the Pfaffian

Sergiu Moroianu

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Abstract

We define transgressions of arbitrary order, with respect to families of unit-vector fields indexed by a polytope, for the Pfaffian of metric connections for semi-Riemannian metrics on vector bundles. We apply this formula to compute the Euler characteristic of a Riemannian polyhedral manifold in the spirit of Chern's differential-geometric proof of the generalized Gauss-Bonnet formula on closed manifolds and on manifolds-with-boundary. As a consequence, we derive an identity for spherical and hyperbolic polyhedra linking the volumes of faces of even codimension and the measures of outer angles.

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Sergiu Moroianu. 2020-11-12. Higher transgressions of the Pfaffian. https://arxiv.org/abs/2011.06538

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