Embedding

Definition

An embedding is a mathematical concept where one manifold is placed into another manifold in such a way that the structure of the first manifold is preserved. This means that the embedding allows for a smooth and continuous inclusion, maintaining the manifold's topological and geometric properties. Understanding embeddings is crucial when discussing smooth maps and their differentials, as it helps illustrate how one space can relate to another while preserving essential characteristics.

5 Must Know Facts For Your Next Test

1. An embedding must be an injective (one-to-one) map, meaning no two points in the original manifold can map to the same point in the target manifold.
2. Embeddings preserve the differentiable structure, which means that smooth functions defined on the embedded manifold correspond to smooth functions in the larger manifold.
3. The concept of an embedding can be used to study the local and global properties of manifolds, providing insights into their shape and behavior.
4. Every smooth manifold can be embedded into some Euclidean space, thanks to results from differential topology like Whitney's embedding theorem.
5. Embeddings are essential for understanding how manifolds interact with each other and are often used to simplify complex problems by considering lower-dimensional representations.

Review Questions

• How does an embedding differ from an immersion, and why is this distinction important in understanding smooth maps?
  • An embedding differs from an immersion in that an embedding is both a smooth map and a homeomorphism onto its image, ensuring it is injective, whereas an immersion may not maintain this injectivity globally. This distinction is important because it affects how we interpret the relationship between manifolds; embeddings allow for a clear representation of one manifold within another while preserving structure, while immersions may lead to overlapping points. Understanding this difference helps clarify how smooth maps operate between spaces.
• Discuss how embeddings relate to submanifolds and provide an example illustrating this relationship.
  • Embeddings are closely related to submanifolds since any embedded manifold can be viewed as a submanifold of a larger manifold. For example, consider the circle $S^1$ embedded in the plane $ extbf{R}^2$. The circle retains its geometric properties as it sits within the plane while also forming a submanifold. This relationship highlights how embeddings allow us to understand lower-dimensional structures within higher-dimensional contexts, facilitating deeper exploration of their properties.
• Evaluate the significance of Whitney's embedding theorem in the context of embeddings and smooth maps between manifolds.
  • Whitney's embedding theorem is significant because it establishes that any smooth manifold can be embedded in some Euclidean space, which broadens our understanding of how manifolds can be represented geometrically. This result shows that we can always find a suitable 'home' for any manifold within a familiar setting like $ extbf{R}^n$, which simplifies complex interactions between different manifolds. The theorem lays groundwork for further studies in differential geometry and topology by confirming that embeddings exist universally across smooth manifolds.

Related terms

Immersion: An immersion is a smooth map between manifolds that is locally injective, meaning that small neighborhoods in the domain map to distinct neighborhoods in the codomain.

Submanifold: A submanifold is a subset of a manifold that itself has the structure of a manifold, often arising as the image of an embedding.

Tangent Space: The tangent space at a point on a manifold consists of all possible tangent vectors at that point, representing directions in which one can move from that point.