Gradient Flows of Penalty Functions in the Space of Smooth Embeddings

Motivated by manifold learning techniques, we give an explicit lower bound for how far a smoothly embedded compact submanifold in ${\mathbb{R}}^N$ can move in a normal direction and remain an embedding. In addition, given a penalty function $P:\text{Emb}(M,{\mathbb{R}}^N)\to \mathbb{R}$ on the space of embeddings, we give a condition which guarantees that the gradient $\nabla P$ofthepenaltyfunctionisnormalto$\phi(M)$ at every point.