Axial curvatures for corank 1 singular $n$-manifolds in $\mathbb R^{n+k}$

TL;DR

This paper introduces a new set of axial curvatures for corank 1 singular manifolds in Euclidean space, generalizing existing second order curvatures and linking them to principal curvatures and geometric properties of singular sets.

Contribution

It defines multiple axial curvatures for singular manifolds, relates them to principal curvatures, and interprets their geometric significance, extending curvature concepts for singular geometries.

Findings

Axial curvatures generalize second order curvatures for singular manifolds.

Relations established between axial curvatures and principal curvatures.

Geometric interpretations of axial curvatures for specific cases like $n=2,3$.

Abstract

For singular $n$-manifolds in $\mathbb R^{n+k}$ with a corank 1 singular point at $p\in M^n_{\mbox{sing}}$ we define up to $l(n-1)$ different axial curvatures at $p$, where $l=\min\{n,k+1\}$. These curvatures are obtained using the curvature locus (the image by the second fundamental form of the unitary tangent vectors) and are therefore second order invariants. In fact, in the case $n=2$ they generalise all second order curvatures which have been defined for frontal type surfaces. We relate these curvatures with the principal curvatures in certain normal directions of an associated regular $(n-1)$-manifold contained in $M^n_{\mbox{sing}}$. We obtain many interesting geometrical interpretations in the cases $n=2,3$. For instance, for frontal type 3-manifolds with 2-dimensional singular set, the Gaussian curvature of the singular set can be expressed in terms of the axial curvatures. Similarly for the curvature of the singular set when it is 1-dimensional. Finally, we show that all the umbilic curvatures which have been defined for singular manifolds up to now can be seen as the absolute value of one of our axial curvatures.