Extrinsic curvature and conformal Gauss-Bonnet for four-manifolds with corner

TL;DR

This paper introduces new extrinsic curvature invariants for four-manifolds with corners, including a conformal invariant and a related PDE operator, leading to a reformulation of the Gauss-Bonnet theorem in this setting.

Contribution

It defines novel extrinsic curvature quantities on corners of four-manifolds and establishes a conformal Gauss-Bonnet theorem involving these invariants.

Findings

One of the invariants is a pointwise conformal invariant.

The other invariant's transformation is governed by a new PDE operator.

The Gauss-Bonnet theorem is expressed using these quantities.

Abstract

This paper defines two new extrinsic curvature quantities on the corner of a four-dimensional Riemannian manifold with corner. One of these is a pointwise conformal invariant, and the conformal transformation of the other is governed by a new linear second-order pointwise conformally invariant partial differential operator. The Gauss-Bonnet theorem is then stated in terms of these quantities.