Submanifolds and Immersions — Mathematics Knowledge System

Overview

A submanifold of a smooth manifold M is a subset that is itself a manifold, with the inclusion map being smooth in an appropriate sense. Immersions are smooth maps whose differential is everywhere injective; embeddings are immersions that are additionally homeomorphisms onto their image. Submanifold theory underpins the study of curves and surfaces in ℝⁿ, the leaves of foliations, and the integral manifolds of distributions. The second fundamental form measures how a submanifold bends inside the ambient space.

Intuition

A curve in ℝ³ is a 1-dimensional submanifold, a surface is a 2-dimensional one. The key distinction is between an immersion (locally injective and smooth, like a figure-eight curve) and an embedding (globally injective — the image is a 'nice' subset). The second fundamental form captures extrinsic curvature: how much the submanifold curves inside the ambient manifold, beyond its intrinsic geometry.

Formal Definition

Definition

Let M be a smooth n-manifold and N a smooth k-manifold with k ≤ n. A smooth map f: N → M is an immersion if df_p: T_pN → T_{f(p)}M is injective for all p. It is an embedding if additionally f is a homeomorphism onto f(N) ⊆ M (with the subspace topology).

f: N \to M \text{ immersion} \iff \mathrm{rank}(df_p) = \dim N \quad \forall p \in N

Immersion: injective differential everywhere

f: N \hookrightarrow M \text{ embedding} \iff f \text{ immersion and homeomorphism onto } f(N)

Embedding: global injectivity + topological condition

II(X,Y) = (\nabla^M_X Y)^\perp = \bar{\nabla}_X Y - \nabla^N_X Y

II measures the normal component of the ambient covariant derivative

Second fundamental form

H = \frac{1}{k}\,\mathrm{tr}(II) = \frac{1}{k}\sum_{i=1}^k II(e_i, e_i)

Mean curvature vector

Theorems

Theorem 1: Whitney Embedding Theorem

\text{Every smooth } n\text{-manifold embeds smoothly in } \mathbb{R}^{2n}. \text{ Every smooth } n\text{-manifold immerses in } \mathbb{R}^{2n-1}.

Theorem 2: Gauss Equations

\text{For a submanifold } N \hookrightarrow (M,g), \text{ the intrinsic curvature of } N \text{ relates to the ambient curvature and the second fundamental form:} \\ R^N(X,Y,Z,W) = R^M(X,Y,Z,W) - g(II(X,Z), II(Y,W)) + g(II(X,W), II(Y,Z)).

Theorem 3: Constant Mean Curvature

\text{A compact submanifold } N \subseteq M \text{ with } H = 0 \text{ everywhere (minimal submanifold) is a critical point of the volume functional.}

Worked Examples

1
Compute df/dt = (cos t, cos 2t). This is nonzero for all t (if cos t = 0 then t = π/2 + kπ, and cos(2t) = cos(π + 2kπ) = −1 ≠ 0). So f is an immersion.
$f'(t) = (\cos t,\, \cos 2t)$
2
The image is a figure-eight. Check f(0) = (0,0) and f(π) = (0,0). So f(0) = f(π), meaning f is not injective.
$f(0) = (0, 0) = f(\pi)$
3
An embedding requires global injectivity. Since f(0) = f(π), f is not injective, hence not an embedding.
$f \text{ not injective} \implies f \text{ not an embedding}$

✓ Answer

f(t) = (sin t, sin(2t)/2) is an immersion (nonzero derivative everywhere) but not an embedding since f(0) = f(π) = (0,0) — it has a self-intersection forming a figure-eight.

Practice Problems

Mediumfree response

Explain the difference between an immersion and an embedding, giving one example of each that is not the other.

Mediumproof writing

Prove that the preimage f⁻¹(q) of a regular value q ∈ M of a smooth map f: N → M is a smooth submanifold of N of dimension dim N − dim M.

Quiz

Summary

An immersion f: N → M has injective differential; an embedding is an immersion that is also a homeomorphism onto its image.
Submanifolds are the images of embeddings; they inherit a smooth structure and (in the Riemannian case) an induced metric.
The second fundamental form II(X,Y) = (∇̄_X Y)⊥ measures extrinsic bending; its trace gives mean curvature H.
Minimal submanifolds (H = 0) are critical points of the volume functional.
The Gauss equations relate intrinsic curvature of N to ambient curvature and II: R^N = R^M − II ∗ II.