Higher transgressions of the Pfaffian

  • Sergiu Moroianu

    Academiei Române, Bucharest, Romania
Higher transgressions of the Pfaffian cover
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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, very much 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, Higher transgressions of the Pfaffian. Rev. Mat. Iberoam. 38 (2022), no. 5, pp. 1425–1452

DOI 10.4171/RMI/1358