Higher transgressions of the Pfaffian

TL;DR

This paper introduces higher-order transgressions of the Pfaffian for vector bundles with semi-Riemannian metrics, applying them to compute Euler characteristics and derive geometric identities for polyhedral manifolds.

Contribution

It defines new higher-order transgressions of the Pfaffian and applies them to extend classical geometric formulas to polyhedral manifolds.

Findings

Derived a formula for Euler characteristic of polyhedral manifolds.

Established identities linking face volumes and outer angles of polyhedra.

Extended Chern's proof techniques to polyhedral settings.

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.