nLab submanifold

Idea

A submanifold is a manifold inside another manifold.

Definition

For a homomorphism of differentiable manifolds

$$X \hookrightarrow Y$$

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.

Properties

Proposition

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

For each point $\sigma \in \Sigma$ there is a coordinate chart $X \supset U \xrightarrow{\phi} \mathbb{R}^n$ of $X$ such that $\sigma \in U \subset M$ and $\phi(\Sigma \cap U)$ is a rectilinear hyperplane in $\phi(U) \subset \mathbb{R}^n$.

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

Related concepts

• embedding of differentiable manifolds

• hypersurface

References

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:

• John Lee, Chapter 5 of: Introduction to Smooth Manifolds, Springer (2012) [doi:10.1007/978-1-4419-9982-5, book webpage, pdf]

See also

• Wikipedia, Submanifold

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

• Phillip Griffiths, Joseph Harris, Algebraic geometry and local differential geometry, Annales scientifiques de l’École Normale Supérieure, Serie 4, Volume 12 no. 3 (1979) 355-452 [numdam:ASENS_1979_4_12_3_355_0]

On submanifolds in the generality of supermanifolds:

• Dimitry A. Leites, Section III.2 in: Introduction to the Theory of Supermanifolds, Russ. Math. Surv. 35 1 (1980) [doi:10.1070/RM1980v035n01ABEH001545, MathNet, iop, pdf]

• Veeravalli Varadarajan, Section 4.7 in: Supersymmetry for mathematicians: An introduction, Courant Lecture Notes in Mathematics 11, American Mathematical Society (2004) [doi:10.1090/cln/011]