Embedding与diffeomorphism的差异解析：请求举例及极端情况说明

Great question—these two concepts are easy to mix up, but they’re fundamentally different in what they guarantee. Let’s break this down clearly, with examples you can wrap your head around and edge cases that highlight their boundaries.

Before diving into differences, let’s set the baseline to avoid confusion:

• Embedding: A smooth map ( f: M \to N ) between smooth manifolds is an embedding if:

  1. It’s injective (one-to-one, no overlapping points in the target),
  2. The induced map ( f: M \to f(M) ) (where ( f(M) ) uses the subspace topology from ( N )) is a homeomorphism (the source manifold’s topology matches the image’s subspace topology),
  3. The differential ( df_p ) is injective at every point ( p \in M ) (preserves local smooth structure without collapsing dimensions).
     In short: An embedding tucks one manifold into another as a "well-behaved" submanifold.

• Diffeomorphism: A smooth map ( f: M \to N ) is a diffeomorphism if:

  1. It’s a bijection (every target point is hit exactly once),
  2. Its inverse map ( f^{-1}: N \to M ) is also smooth.
     This means ( M ) and ( N ) are essentially identical manifolds—just relabeled points with smooth structure fully preserved.

Let’s distill the critical distinctions:

1. Bijectivity is non-negotiable for diffeomorphisms

   • Diffeomorphisms must cover the entire target manifold and never repeat points. No exceptions—bijection is a hard requirement.
   • Embeddings only need to be injective (no repeats), but they don’t have to cover the entire target. Most embeddings are "partial" maps into a higher-dimensional space.

2. Topological compatibility has different rules

   • Embeddings require the image ( f(M) ) to be a submanifold of ( N ), with no weird topological mismatches (e.g., no infinitely crinkled regions that break the homeomorphism requirement).
   • Diffeomorphisms don’t care about submanifolds—they’re equivalence relations between full manifolds. The entire source and target are smoothly interchangeable.

3. Dimension constraints are strict for diffeomorphisms

   • Diffeomorphisms can only exist between manifolds of the same dimension. You can’t have a diffeomorphism between a line (1D) and a plane (2D)—bijection across different dimensions is impossible while preserving smooth structure.
   • Embeddings can go from lower to higher dimensions (the most common case), or even same dimension (e.g., embedding ( \mathbb{R}^2 ) into ( \mathbb{R}^2 ) as a subset, which is just an injective diffeomorphism onto its image).

Let’s make this tangible with relatable (for topologists, at least) examples:

Embedding Examples #

• Line into Plane: The map ( f(x) = (x, 0) ) embeds ( \mathbb{R} ) into ( \mathbb{R}^2 ) as the x-axis. It’s injective, smooth, and the x-axis’s subspace topology matches ( \mathbb{R} )’s topology perfectly—this is the simplest embedding you can get.
• Circle into 3D Space: ( f(\theta) = (\cos\theta, \sin\theta, 0) ) embeds ( S^1 ) (the unit circle) into ( \mathbb{R}^3 ) as the equatorial circle. The image is a clean 1-dimensional submanifold with no overlaps.
• Infinite Spiral: ( f(t) = (\cos t, \sin t, t) ) embeds ( \mathbb{R} ) into ( \mathbb{R}^3 ) as a spiral. It’s injective, smooth, and every open interval in ( \mathbb{R} ) maps to an open segment of the spiral (preserving topology).

Diffeomorphism Examples #

• Plane Translation: ( f(x,y) = (x+2, y-5) ) is a diffeomorphism from ( \mathbb{R}^2 ) to itself. It’s a bijection, smooth, and its inverse ( f^{-1}(x,y) = (x-2, y+5) ) is also smooth—just shifting the plane around.
• Open Disk to Plane: The map ( f(x,y) = \left( \frac{x}{1 - \sqrt{x^2 + y^2}}, \frac{y}{1 - \sqrt{x^2 + y^2}} \right) ) is a diffeomorphism from the open unit disk ( D^2 ) to all of ( \mathbb{R}^2 ). It stretches the disk out to cover the entire plane smoothly, with a smooth inverse that "shrinks" the plane back into the disk.
• Line to Open Interval: ( f(x) = \frac{1}{1 + e^{-x}} ) maps ( \mathbb{R} ) bijectively to ( (0,1) ). Both ( f ) and its inverse ( f^{-1}(y) = \ln\left( \frac{y}{1-y} \right) ) are smooth, so this is a diffeomorphism—proving an infinite line and a finite open interval are smoothly equivalent.

These cases really clarify the boundaries of each concept:

• Embedding that can’t be a diffeomorphism: Any low-to-high dimensional embedding (like ( \mathbb{R} \to \mathbb{R}^2 )) is automatically not a diffeomorphism—you can’t have a bijection between a 1D and 2D manifold, so diffeomorphism is impossible by definition.
• Smooth injective map that’s NOT an embedding: Here’s a classic gotcha: define ( f: \mathbb{R} \to \mathbb{R}^2 ) as:
  [
  f(t) =
  \begin{cases}
  (t, 0) & t \leq 0 \
  (t, t^2 \sin(1/t)) & t > 0
  \end{cases}
  ]
  This map is smooth and injective, but it’s not an embedding. Near ( t=0 ), the image oscillates infinitely often between ( y = t^2 ) and ( y = -t^2 )—the subspace topology on ( f(\mathbb{R}) ) doesn’t match ( \mathbb{R} )’s topology (open sets around 0 in ( \mathbb{R} ) map to sets that aren’t open in ( f(\mathbb{R}) )’s subspace topology). This violates the embedding’s topological requirement.
• Local diffeomorphism that’s neither embedding nor diffeomorphism: The map ( f: \mathbb{R} \to S^1 ) given by ( f(t) = (\cos t, \sin t) ) is a local diffeomorphism (every point has a neighborhood where it’s a smooth bijection), but it’s not injective (it repeats every ( 2\pi )) so it’s not an embedding, and not a bijection—hence not a diffeomorphism. This shows local smoothness doesn’t guarantee global embedding or diffeomorphism.
• Unexpected diffeomorphism: The 2-sphere minus a single point ( S^2 \setminus {p} ) is diffeomorphic to ( \mathbb{R}^2 ) via stereographic projection. Even though one looks like a "punctured sphere" and the other like a flat plane, they’re smoothly equivalent—this is a great example of how diffeomorphisms can hide geometric intuition.

内容的提问来源于stack exchange，提问作者EEEB