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.
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.
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.
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.
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.