Pseudo-Riemannian metric singularities and the extendability of parallel transport

Abstract:

We are given a $C\infty$ immersion $i:N \to (M,\langle \;\rangle )$, and $p \in N$ is a point where $N_p^ \bot \cap {T_p}N$ is one-dimensional. We have shown that there is a tensor ${\text {I}}{{\text {I}}_p}:{T_p}N \times {T_p}N \times \operatorname {Rad}_p \to {\mathbf {R}}$ intrinsic to $(N,{i^*}\langle \;\rangle )$ which determines an extrinsic feature of the immersion. The purpose of this paper is to show that II controls the following two intrinsic properties. First, II determines which pairs of vector fields $X$, $Y$ on $N$ have the property that intrinsic covariant derivative ${\nabla _x}Y$ extends smoothly to all of $N$. Second, given a curve in $N$ containing $p,{\text {I}}{{\text {I}}_p}$ determines which parallel vector fields along the curve extend smoothly through $p$. As an application we locally characterize product and flat metric singularities.