ODD Metrics

TL;DR

This paper introduces ODD metrics, a new class of semi-positive symmetric tensors that can degenerate on submanifolds but still retain key Riemannian properties, enabling a generalized geometric framework.

Contribution

It defines ODD metrics, explores their fundamental properties, and demonstrates their ability to induce a metric space structure and support well-defined geodesics despite degeneracies.

Findings

ODD metrics induce a metric space structure on manifolds.

ODD vector fields are integrable at general points of degeneracy.

ODD geodesics exist and are unique at general points of the degeneracy locus.

Abstract

We introduce the concept of ODD ('$\mathbf{O}$rthogonally $\mathbf{D}$egenerating on a $\mathbf{D}$ivisor') Riemannian metrics on real analytic manifolds $M$. These semipositive symmetric $2$-tensors may degenerate on a finite collection of submanifolds, while their restrictions to these submanifolds satisfy the inductive compatibility criterion to be an ODD metric again. In this first in a series of articles on these metrics, we show that they satisfy basic properties that hold for Riemannian metrics. For example, we introduce orthonormal frames, the lowering and raising of indices, ODD volume forms and the Levi-Civita connection. We finally show that an ODD metric induces a metric space structure on $M$ and that at least at general points of the degeneracy locus $\mathcal{D}$, ODD vector fields are integrable and ODD geodesics exist and are unique.