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 $P : \text{Emb}(M,\mathbb{R}^N) \rightarrow \mathbb{R}$ on the space of embeddings, we give a condition which guarantees that the gradient $\nabla P$ of the penalty function is normal to $\phi(M)$ at every point.