🔁Elementary Differential Topology Unit 5 – Submanifolds, Immersions & Submersions

Key Concepts and Definitions

• Manifolds are topological spaces that locally resemble Euclidean space and have a well-defined dimension
• Submanifolds are subsets of a manifold that are themselves manifolds with respect to the subspace topology
  • Submanifolds inherit the smooth structure from the ambient manifold
• Immersions are smooth maps between manifolds that have injective differential at every point
  • Immersions need not be injective globally, but they are locally injective
• Submersions are smooth maps between manifolds that have surjective differential at every point
  • Submersions are locally modeled by projections onto a subspace
• Tangent spaces are vector spaces attached to each point of a manifold, capturing the local linear approximation
• Differential of a smooth map is a linear map between tangent spaces that describes the local behavior of the map

Submanifolds: The Basics

• A subset $S$ of a manifold $M$ is a submanifold if for every point $p \in S$, there exists a chart $(U, \varphi)$ of $M$ such that $\varphi(U \cap S)$ is an open subset of a subspace of $\mathbb{R}^n$
• Submanifolds have the same dimension at every point, which is less than or equal to the dimension of the ambient manifold
• The inclusion map $i: S \hookrightarrow M$ of a submanifold is an immersion
• Regular level sets of smooth functions are submanifolds (regular level set theorem)
  • For a smooth function $f: M \to \mathbb{R}$, if $c \in \mathbb{R}$ is a regular value (i.e., $df_p \neq 0$ for all $p \in f^{-1}(c)$), then $f^{-1}(c)$ is a submanifold of $M$
• Submanifolds can be characterized using the inverse function theorem for immersions

Immersions: Diving Deeper

• An immersion is a smooth map $f: M \to N$ between manifolds such that the differential $df_p: T_pM \to T_{f(p)}N$ is injective for every point $p \in M$
• Immersions are locally injective, meaning that for every point $p \in M$, there exists a neighborhood $U$ of $p$ such that $f|_U: U \to f(U)$ is injective
• The rank of an immersion is equal to the dimension of the domain manifold
• Immersions preserve the dimension of the manifold, i.e., if $f: M \to N$ is an immersion, then $\dim M \leq \dim N$
• Compositions of immersions are immersions
• The inverse function theorem characterizes immersions: a smooth map $f: M \to N$ is an immersion if and only if for every point $p \in M$, there exist charts $(U, \varphi)$ around $p$ and $(V, \psi)$ around $f(p)$ such that $\psi \circ f \circ \varphi^{-1}$ is the inclusion of an open subset of a subspace of $\mathbb{R}^m$ into $\mathbb{R}^n$

Submersions: Flattening Things Out

• A submersion is a smooth map $f: M \to N$ between manifolds such that the differential $df_p: T_pM \to T_{f(p)}N$ is surjective for every point $p \in M$
• Submersions are locally modeled by projections onto a subspace
  • For every point $p \in M$, there exist charts $(U, \varphi)$ around $p$ and $(V, \psi)$ around $f(p)$ such that $\psi \circ f \circ \varphi^{-1}$ is the projection of an open subset of $\mathbb{R}^m$ onto $\mathbb{R}^n$, where $n = \dim N$
• The rank of a submersion is equal to the dimension of the codomain manifold
• Submersions decrease the dimension of the manifold, i.e., if $f: M \to N$ is a submersion, then $\dim M \geq \dim N$
• Compositions of submersions are submersions
• The preimage of a point under a submersion is a submanifold of the domain (submersion level set theorem)
  • If $f: M \to N$ is a submersion and $q \in N$, then $f^{-1}(q)$ is a submanifold of $M$ with dimension $\dim M - \dim N$

Examples and Visualizations

• The projection of a cylinder onto its axis is a submersion
  • Consider the cylinder $S^1 \times \mathbb{R}$ and the projection $\pi: S^1 \times \mathbb{R} \to \mathbb{R}$ given by $\pi(x, t) = t$
• The inclusion of a sphere into Euclidean space is an immersion
  • The inclusion map $i: S^n \hookrightarrow \mathbb{R}^{n+1}$ is an immersion for any $n \geq 0$
• The Möbius strip is a submanifold of $\mathbb{R}^3$
  • The Möbius strip can be parametrized by $f: [0, 1] \times [0, 1] \to \mathbb{R}^3$ given by $f(u, v) = ((1 + v/2 \cos(\pi u/2))\cos(2\pi u), (1 + v/2 \cos(\pi u/2))\sin(2\pi u), v/2 \sin(\pi u/2))$
• The torus is a submanifold of $\mathbb{R}^3$
  • The torus can be parametrized by $f: [0, 2\pi] \times [0, 2\pi] \to \mathbb{R}^3$ given by $f(u, v) = ((R + r\cos(v))\cos(u), (R + r\cos(v))\sin(u), r\sin(v))$, where $R > r > 0$
• The graph of a smooth function is a submanifold of the product manifold
  • If $f: M \to N$ is a smooth function, then the graph $\Gamma_f = \{(p, f(p)) \mid p \in M\}$ is a submanifold of $M \times N$

Theorems and Proofs

• Inverse function theorem for immersions: A smooth map $f: M \to N$ is an immersion if and only if for every point $p \in M$, there exist charts $(U, \varphi)$ around $p$ and $(V, \psi)$ around $f(p)$ such that $\psi \circ f \circ \varphi^{-1}$ is the inclusion of an open subset of a subspace of $\mathbb{R}^m$ into $\mathbb{R}^n$
  • Proof sketch: Use the local form of immersions and the inverse function theorem for smooth maps between Euclidean spaces
• Submersion level set theorem: If $f: M \to N$ is a submersion and $q \in N$, then $f^{-1}(q)$ is a submanifold of $M$ with dimension $\dim M - \dim N$
  • Proof sketch: Use the local form of submersions and the preimage theorem for smooth maps between Euclidean spaces
• Constant rank theorem: If $f: M \to N$ is a smooth map of constant rank $r$, then for every point $p \in M$, there exist charts $(U, \varphi)$ around $p$ and $(V, \psi)$ around $f(p)$ such that $\psi \circ f \circ \varphi^{-1}$ is the projection of an open subset of $\mathbb{R}^m$ onto the first $r$ coordinates of $\mathbb{R}^n$
  • Proof sketch: Use the local form of immersions and submersions, and the fact that locally, a constant rank map can be decomposed into an immersion followed by a submersion

Applications in Topology

• Submanifolds, immersions, and submersions are fundamental tools in the study of the topology of manifolds
• The regular level set theorem allows the construction of submanifolds as level sets of smooth functions
  • This is useful in the study of cobordisms and Morse theory
• Immersions and submersions are used to define and study fiber bundles, which are important objects in algebraic and differential topology
  • A fiber bundle is a submersion with a local product structure, i.e., locally, it looks like a product of the base space and the fiber
• The constant rank theorem is used to prove the Frobenius theorem, which characterizes integrable distributions on manifolds
  • Integrable distributions are important in the study of foliations and geometric structures on manifolds
• Immersions and submersions are used to define and study Lie group actions on manifolds
  • A Lie group action is a smooth map $G \times M \to M$ that satisfies certain properties, where $G$ is a Lie group and $M$ is a manifold

Common Pitfalls and Tips

• Remember that immersions and submersions are local properties, while injectivity and surjectivity are global properties
  • An immersion need not be globally injective, and a submersion need not be globally surjective
• Be careful when composing immersions and submersions
  • The composition of two immersions is an immersion, and the composition of two submersions is a submersion, but the composition of an immersion and a submersion (in either order) may not have any special properties
• When working with submanifolds, always check that the subset is locally modeled by a subspace of Euclidean space
  • Not every subset of a manifold is a submanifold, even if it looks like one globally
• Use the inverse function theorem and the constant rank theorem to identify immersions and submersions
  • These theorems provide a powerful way to characterize immersions and submersions using local charts
• Visualize examples of submanifolds, immersions, and submersions in low dimensions to build intuition
  • Examples in dimensions 1, 2, and 3 can often be easily visualized and can help in understanding the general concepts