On submanifolds with tamed second fundamental form

TL;DR

This paper proves that complete submanifolds with tamed second fundamental form in certain Riemannian manifolds are proper, have finite topology if the ambient space is Hadamard, and identifies the fundamental tone as an obstruction for such embeddings.

Contribution

It establishes properness and topological finiteness of submanifolds with tamed second fundamental form in non-positive curvature spaces and links the fundamental tone to embedding obstructions.

Findings

Submanifolds are proper in the specified ambient spaces.

Submanifolds have finite topology in Hadamard manifolds.

Fundamental tone obstructs certain submanifold embeddings.

Abstract

We show that a complete submanifold $M$ with tamed second fundamental form in a complete Riemannian manifold $N$ with sectional curvature $K_{N}\leq \kappa \leq 0$ are proper, (compact if $N$ is compact). In addition, if $N$ is Hadamard then $M$ has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realized as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below.