Submanifolds

Core Idea

A submanifold is a subset of a manifold that is itself a manifold in a compatible way. Embedded submanifolds arise as level sets of smooth maps at regular values (the preimage theorem), as images of injective immersions, or as solution sets of constraint equations. The tangent space of a submanifold at a point is a subspace of the ambient tangent space, and the codimension determines how many independent constraints define the submanifold locally.

Explainer

The simplest way to construct a submanifold is as a level set. If F : M → N is a smooth map and c ∈ N is a regular value (meaning dF_p is surjective for every p ∈ F⁻¹(c)), then the preimage theorem guarantees that F⁻¹(c) is a smooth submanifold of M with dimension dim(M) - dim(N). This is the manifold version of the implicit function theorem from multivariable calculus. The sphere S² = {x² + y² + z² = 1} ⊂ ℝ³, the orthogonal group O(n) = {A : AᵀA = I} ⊂ GL(n), and every smooth curve in the plane defined by an equation f(x,y) = 0 with nonvanishing gradient are examples.

More generally, a smooth map f : S → M is an immersion if df_p is injective at every point of S, and an embedding if it is additionally a homeomorphism onto its image. An embedded submanifold is the image of an embedding — it inherits a smooth structure from the ambient manifold and sits inside M "without crossing itself." Every compact manifold that immerses in M actually embeds (the Whitney embedding theorem gives quantitative dimension bounds). An immersed submanifold may have self-intersections or may fail to have the subspace topology, as with dense curves on tori.

The tangent space of a submanifold S at a point p is a subspace of the ambient tangent space: TpS ⊆ TpM. For a level set S = F⁻¹(c), the tangent space is the kernel of the derivative: TpS = ker(dFp). The codimension of S in M is dim(M) - dim(S), and it equals the number of independent constraints defining S. When M is equipped with a Riemannian metric, the tangent space splits as TpM = TpS ⊕ (TpS)⊥, where the orthogonal complement is the normal space. This split is fundamental to the geometry of submanifolds — curvature, the second fundamental form, and the Gauss-Codazzi equations all arise from studying how TpS sits inside TpM.

Submanifolds are ubiquitous in mathematics and physics. Configuration spaces of mechanical systems (the set of positions satisfying constraints) are submanifolds of a product of copies of ℝ³. Lie groups are submanifolds of matrix spaces. Phase spaces in Hamiltonian mechanics are submanifolds defined by energy conservation. The theory of submanifolds connects the intrinsic geometry of S to the extrinsic geometry of how S sits in M — a theme that runs from the Gauss-Bonnet theorem through modern geometric analysis.