Submanifold

Summary

A submanifold of a smooth manifold $M$ is a subset that is itself a smooth manifold in a way compatible with $M$. The most common kind, an embedded submanifold, looks near each of its points like a flat coordinate subspace $\R^k \subseteq \R^n$. Level sets of submersions, images of embeddings and graphs of smooth maps are submanifolds. A wider notion, the immersed submanifold, is the image of an injective immersion with its own topology.