Smooth embedding

Definition

A smooth embedding is a type of function that maps one manifold into another in a way that preserves the smooth structure. This means that the function is not only continuous but also has continuous derivatives up to any desired order, making it a very 'nice' way to embed one space into another. Smooth embeddings are important because they allow us to study the properties of submanifolds within larger manifolds, maintaining essential geometric and topological features.

5 Must Know Facts For Your Next Test

1. For a function to be a smooth embedding, it must be an injective (one-to-one) function that is also a smooth map, meaning all partial derivatives exist and are continuous.
2. The image of a smooth embedding is always a submanifold of the target manifold, capturing the local structure of the original manifold within the larger context.
3. Smooth embeddings allow us to relate different geometrical objects, helping us understand how one shape can fit within another smoothly.
4. Every smooth embedding has an associated differential that must be injective at each point, which guarantees that locally around each point, the embedding behaves like an open set in Euclidean space.
5. The concept of smooth embeddings is fundamental in differential topology, where it is used to classify and study various geometric shapes and their properties.

Review Questions

• How does the concept of a smooth embedding relate to the notion of differentiable structures on manifolds?
  • A smooth embedding relies heavily on differentiable structures because it requires that the function preserves smoothness when mapping one manifold into another. Differentiable structures define how we can differentiate functions on manifolds, and for an embedding to be considered smooth, all derivatives must exist and be continuous. This connection ensures that we can study the embedded manifold's geometry and topology in terms of its original differentiable structure.
• Discuss the significance of injectivity in the definition of a smooth embedding and its implications for the image of such embeddings.
  • Injectivity in a smooth embedding ensures that every point in the source manifold maps to a unique point in the target manifold. This property is crucial because it guarantees that the image of the embedding does not overlap itself, allowing it to form a proper submanifold. If an embedding were not injective, we could lose essential information about the original manifold's structure, making it impossible to study or analyze its properties effectively.
• Evaluate how smooth embeddings contribute to our understanding of topology and geometry in higher dimensions and their applications in modern mathematics.
  • Smooth embeddings play a pivotal role in bridging topology and geometry, particularly in higher dimensions. By allowing manifolds to be embedded within other manifolds smoothly, mathematicians can explore intricate relationships between different geometric objects. This understanding has far-reaching implications in fields like algebraic geometry and theoretical physics, where complex shapes need to be analyzed. Moreover, they aid in classifying manifolds by understanding their embeddings and studying their invariants under different transformations.

Related terms

manifold: A manifold is a topological space that locally resembles Euclidean space and can be described by coordinates, allowing for calculus to be done on it.

differentiable structure: A differentiable structure on a manifold defines how functions can be differentiated, ensuring that transition maps between local coordinate charts are smooth.

submanifold: A submanifold is a subset of a manifold that itself has the structure of a manifold, allowing it to be studied within the context of the larger manifold.