Submanifold

Remarks

Example: the unit circle $S^1 \subseteq \R^2$ is an embedded submanifold of dimension $1$. Near a point with $y > 0$, the chart $(x, y) \mapsto (x, y - \sqrt{1 - x^2})$ on $\set{(x, y) : \abs{x} < 1, y > 0}$ sends the circle onto the axis $\R \times \set{0}$; the other points are handled in the same way. Non-example: the cross $C = \set{(x, y) \in \R^2 : xy = 0}$ is not an embedded submanifold. Removing the origin from a small connected neighbourhood of $0$ in $C$ leaves four components. Removing a point from an open interval leaves two, so no slice chart exists at the origin. The embedded submanifolds of $M$ are exactly the images of smooth embeddings into $M$. An immersed submanifold is the image of an injective immersion $S \to M$, with the topology and smooth structure carried over from $S$; its topology may be finer than the subspace topology.