embedding in a Riemannian manifold Let $M$ be a Riemannian manifold with boundary. Can it isometrically embed into a Riemannian manifold without boundary of the same dimension?
 A: This paper  by Pigolla and Veronelli contains a proof of that, with the additional constraint that the larger manifold is complete (which is not difficult to achieve).
Let me note that the main point of the paper is to consider this problem when we ask the extension to satisfy some curvature bound which hold on the manifold with boundary. There are many entertaining questions of this kind, and I recommend to to take a look at the paper to every intrigued Riemannian geometer.
A: A simple direct proof is as follows: Let $M^o = M\setminus \partial M$ be the interior of $M$. Choose a vector field $X$ on $M$ which along the boundary is non-zero and point into the interior, and which is 0 off some neighborhood of the boundary. Use the flow of $X$ to flow $M$ into $M^o$. It remains to extend the Riemannian metric from the image of $M$ in $M^o$ to the whole of $M^o$.
This proof also works for manifolds with corners.
A: This paper at the annals describes such an embedding on page 1097 for compact Riemannian manifolds, although the main results are embeddings between manifolds with boundaries.
