Suppose $(M,g)$ is an open Riemannian manifold with bounded geometry, i.e., the injectivity radius is $\ge \epsilon>0$ and each iterated covariant derivative of curvature is bounded with respect to $g$.

Question: Does there exist an embedding into some high dimensional $\mathbb R^N$ with the following properties:

- isometric
- proper (i.e., compact sets have compact inverse images)
- The normal tubular neighborhood contains a uniformly thick disk in each fiber?

The Gauss equations give a relationship between the (normal bundle valued) second fundamental form (or shape operator); but by embedding arc-length parameterized curves one can show that this is not enough to force focal points uniformly away.

There is a related question Existence of an isometric embedding into Euclidean space with bounded second fundamental form, but it does not answer this question.

By taking a product with another Euclidean space one can then have embedding where the normal bundle is trivial. Then one can use such embedding to carry over results valid on $\mathbb R^n$ to Riemannian manifolds with bounded geometry.

Edit: Many thanks for the comments and Anton for a very succinct answer.

I end with a wild guess: Maybe, one can characterize the Riemannian manifolds of bounded geometry admitting such embeddings as those whose ends are asymptotically flat times something compact (by which I mean an bundle)?