Embeddings

**Jose27** · August 19th 2009, 08:10 PM

Let $M$ be a non-empty differentiable manifold of dimension $n$ . Prove there exists an embedding $f: \mathbb{R} ^k \longrightarrow M$ for all $1 \leq k \leq n$ .

For the sphere $\mathbb{S} ^n$ it's clear that the stereographic projection is an embedding, but I can't prove the general case. Any ideas would be apreciated.

**algtop** · August 19th 2009, 09:24 PM

Originally Posted by Jose27

Let $M$ be a non-empty differentiable manifold of dimension $n$ . Prove there exists an embedding $f: \mathbb{R} ^k \longrightarrow M$ for all $1 \leq k \leq n$ .

For the sphere $\mathbb{S} ^n$ it's clear that the stereographic projection is an embedding, but I can't prove the general case. Any ideas would be apreciated.

Isn't this applicable to all topological n-manifolds including differentiable n-manifolds?

If that is the case,

By the definition of a topological manifold, every point in n-manifold M has a neighborhood which is homeomorphic to Euclidean space $\mathbb{Re}^n$ . That implies that there is an embedding of $f: \mathbb{Re}^k \longrightarrow M$ when k = n, where M is a topological n-manifold. The remaining steps are established by composing two embeddings of g and h if we define g and h such as

$g:\mathbb{Re}^i \longrightarrow \mathbb{Re}^n$ , $h:\mathbb{Re}^n \longrightarrow M$ , where $i = 1, 2, ..., n$ .

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