Odd Pfaffian forms

TL;DR

This paper introduces the odd Pfaffian, an invariant volume form on odd-dimensional Riemannian manifolds, and establishes new Chern-Gauss-Bonnet formulas for manifolds with singularities and boundaries, extending classical results.

Contribution

It defines the odd Pfaffian form and proves intrinsic Chern-Gauss-Bonnet formulas for manifolds with edge singularities and fibered boundaries, including non-compact cases.

Findings

Established the odd Pfaffian as an invariant volume form.

Derived Chern-Gauss-Bonnet formulas for singular and non-compact manifolds.

Proved rationality of the Pfaffian form on orbifolds and identified metric obstructions.

Abstract

On any odd-dimensional oriented Riemannian manifold we define a volume form, which we call the odd Pfaffian, through a certain invariant polynomial with integral coefficients in the curvature tensor. We prove an intrinsic Chern-Gauss-Bonnet formula for incomplete edge singularities in terms of the odd Pfaffian on the fibers of the boundary fibration. The formula holds for product-type model edge metrics where the degeneration is of conical type in each fiber, but also for general classes of perturbations of the model metrics. The same method produces a Chern- Gauss-Bonnet formula for complete, non-compact manifolds with fibered boundaries in the sense of Mazzeo-Melrose and perturbations thereof, involving the odd Pfaffian of the base of the fibration. We deduce the rationality of the usual Pfaffian form on Riemannian orbifolds, and exhibit obstructions for certain metrics on a fibration to be realized as the model at infinity of a flat metric with conical, edge or fibered boundary singularities.