# On embeddings of compact manifolds

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• fresh_42
In summary, an embedding of a compact manifold is a way of representing the manifold in a higher-dimensional space while preserving its geometric structure. Not all compact manifolds can be embedded in a higher-dimensional space, and an embedding differs from an immersion in that it preserves both local and global structure. The embedding of a compact manifold is closely related to its topology and has many important applications in mathematics and science.
fresh_42
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I have found the following entry on his blog by Terence Tao about embeddings of compact manifolds into Euclidean space (Whitney, Nash). It contains the theorems and (sketches of) proofs. Since it is rather short some of you might be interested in.

hehe, I used that post a couple of days ago when I was checking an argument I was trying to use :)
Very useful indeed.

A triangulated manifold with n vertexes naturally embeds in the standard n- simplex in ##R^n##. But this embedding is high dimensional and not smooth.

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## 1. What is an embedding of a compact manifold?

An embedding of a compact manifold is a way of representing the manifold within a higher-dimensional space in a manner that preserves its geometric structure.

## 2. Can any compact manifold be embedded in a higher-dimensional space?

No, not all compact manifolds can be embedded in a higher-dimensional space. For example, the famous non-orientable surface known as the Möbius strip cannot be embedded in three-dimensional space.

## 3. What is the difference between an embedding and an immersion?

An immersion is a mapping that preserves the local geometry of a manifold, but may not preserve its global structure. An embedding, on the other hand, preserves both local and global structure.

## 4. How is the embedding of a compact manifold related to its topology?

The embedding of a compact manifold is closely related to its topology, as it determines the number of dimensions needed to embed the manifold and the properties of the resulting embedding.

## 5. What is the importance of embeddings of compact manifolds in mathematics and science?

Embeddings of compact manifolds have many applications in mathematics and science, including in the study of differential geometry, topology, and physics. They allow for a deeper understanding of the structure of manifolds and can be used to solve various problems in these fields.

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