nLab submanifold




A submanifold is a manifold inside another manifold.


For a homomorphism of differentiable manifolds

XY X \hookrightarrow Y

to qualify as a submanifold inclusion it is usually required to be an embedding of differentiable manifolds, hence

  1. an embedding of topological spaces;

  2. an immersion of differentiable manifolds.



(submanifolds admit slice charts) For XX a smooth manifold and ι:ΣX\iota \colon \Sigma \hookrightarrow X the embedding of a submanifold, then there exists slice charts:

For each point σΣ\sigma \in \Sigma there is a coordinate chart XUϕ nX \supset U \xrightarrow{\phi} \mathbb{R}^n of XX such that σUM\sigma \in U \subset M and ϕ(ΣU)\phi(\Sigma \cap U) is a rectilinear hyperplane in ϕ(U) n\phi(U) \subset \mathbb{R}^n.

(e.g. Lee 2012, Thm. 5.8)


Historical discussion for submanifolds of Euclidean space:

  • Élie Cartan (translated by Vladislav Goldberg from Cartan’s lectures at the Sorbonne in 1926–27): Part E of: Riemannian Geometry in an Orthogonal Frame, World Scientific (2001) [doi:10.1142/4808, pdf]

Textbook account:

See also

On (isometric) submanifolds of Euclidean space via (the algebraic geometry of) their higher-dimensional coframe fields:

On submanifolds in the generality of supermanifolds:

Last revised on May 18, 2024 at 14:49:14. See the history of this page for a list of all contributions to it.