submanifold

A submanifold is a subset S of a smooth manifold M that itself carries a smooth manifold structure in such a way that the inclusion map i: S → M is a smooth embedding. In general terminology, one also speaks of immersed submanifolds, which are the images of smooth immersions f: N → M; such submanifolds need not be embedded (they may have self-intersections). In many common texts, the word submanifold is used to mean embedded submanifold.

Dimension is fundamental: if M has dimension n and S has dimension k, then S is a k-dimensional submanifold of M. Locally, every point p ∈ S has a neighborhood in M and a chart in which S appears as a k-dimensional plane inside the chart domain. Concretely, there exists a chart (U, φ) of M around p with φ(U ∩ S) = φ(U) ∩ (R^k × {0}) in R^n. Equivalently, S may be described locally as the common zero set of n − k smooth functions with linearly independent differentials at p (a regular level set).

Tangent and normal spaces are intrinsic to the submanifold structure. For p ∈ S, T_p S is a k-dimensional subspace of T_p M. If M is equipped with a metric, the normal space is the quotient T_p M / T_p S, or, in a Riemannian setting, the orthogonal complement of T_p S.

Regular level sets provide a practical construction: if F: M → N is smooth and a ∈ N is a regular value, then F^{-1}(a) is an embedded submanifold of M of codimension dim N.

Examples include circles in the plane, spheres in Euclidean space, and more generally any smooth submanifold formed by level sets or images of regular immersions.