Stability of compact smooth embeddings

A smooth embedding of a compact manifold remains an embedding under sufficiently small C1 metric on a smooth mapping space perturbations. A positive uniform lower bound on its differential preserves immersion. Finite coordinate neighborhoods give uniform local injectivity for nearby maps, and compactness separates the remaining pairs of points. A uniform second-derivative bound gives an explicit local Taylor estimate, although it is not necessary for the C1 openness statement.