Gradient Flows of Penalty Functions in the Space of Smooth Embeddings

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Motivated by manifold learning techniques, we give an explicit lower bound for how far a smoothly embedded compact submanifold in Rⁿ can move in a normal direction and remain an embedding. In addition, given a penalty function P: Emb(M,Rⁿ) → R on the space of embeddings, we give a condition which guarantees that the gradient ∇ P of the penalty function is normal to ϕ(M) at every point.