A remark on the reach and upper bounds on some extrinsic geometry invariants of submanifolds
Reza Mirzaie
TL;DR
This work investigates how the reach $\tau$ of a compact submanifold $M$ inside a Riemannian manifold $N$ governs its extrinsic geometry, extending Euclidean results to curved ambient spaces. Using the second variation formula, the authors derive sharp inequalities that bound the exterior acceleration of unit-speed geodesics on $M$ in terms of $\tau$, ambient curvature, and the second fundamental form, and they establish existence results for reach-assigning points. They show that reach imposes concrete geometric relations (e.g., a precise second-variation bound and an intrinsic-extrinsic comparison) and they derive explicit bounds when $N$ has curvature bounded below, including special constant-curvature scenarios. The results provide a framework to compare intrinsic and extrinsic geometries in a curved ambient space and generalize key Euclidean phenomena, with potential implications for geometric analysis and topological considerations like bottleneck points.
Abstract
We consider a compact submanifold $M$ of a Riemannian manifold $N$ and we use the second variation formula as a tool to drive some geometric results on reach$(M, N)$ the reach of $M$ in $N$, including some useful relations between the extrinsic geometry of $M$ in $N$ and reach$(M, N)$. Our results generalize some theorems previously proved for the special case where $N$ is Euclidean space.