Embedding

Definition

An embedding is a type of function that allows one mathematical object to be treated as if it were contained within another, often preserving certain structures like topology and differentiability. This concept is crucial for understanding how submanifolds can be smoothly included in larger manifolds, impacting the way we analyze geometric and topological properties of spaces.

5 Must Know Facts For Your Next Test

1. An embedding is an injective immersion, meaning it not only preserves differentiability but also ensures that the embedded manifold does not intersect itself.
2. Every embedded submanifold inherits a topology from its ambient manifold, allowing us to study properties like compactness and connectedness in the context of the larger space.
3. There are various types of embeddings, including topological embeddings and smooth embeddings, each focusing on different aspects of preservation such as continuity and differentiability.
4. Embedding spaces are often studied through their dimension; for instance, a 2-dimensional surface can be embedded in a 3-dimensional Euclidean space.
5. The concept of transversality plays an important role in embeddings, particularly when analyzing how different submanifolds intersect within an ambient space.

Review Questions

• How does the definition of an embedding connect to the concepts of submanifolds and immersions?
  • An embedding is a specific type of immersion that ensures both smoothness and injectivity, meaning it allows a submanifold to be included within a larger manifold without any overlaps. This connection helps us understand how submanifolds maintain their structure while existing in a higher-dimensional space. Immersions might allow self-intersections, but embeddings guarantee that these do not occur, emphasizing their role in maintaining distinctness and clarity in geometric relationships.
• Discuss how embeddings influence the study of examples such as spheres or tori within differential topology.
  • Embeddings are essential when considering how various manifolds like spheres or tori fit into higher-dimensional spaces. For example, a circle can be embedded in the plane as a simple curve without self-intersection, while a torus can be embedded in three-dimensional space in a way that highlights its unique topology. Understanding these embeddings allows mathematicians to explore properties like curvature and topology through the lens of higher-dimensional geometry.
• Evaluate the implications of transversality in relation to embeddings and their properties within differential topology.
  • Transversality provides a framework for understanding how different submanifolds intersect when embedded in a common ambient manifold. When two submanifolds intersect transversely, it leads to regular intersection points that help define local structures and properties more clearly. This concept influences not just the behavior of individual embeddings but also the overall topology of the space, enabling mathematicians to uncover rich relationships between various embedded objects and their interactions.

Related terms

Submanifold: A submanifold is a subset of a manifold that is itself a manifold, equipped with the subspace topology and differentiable structure inherited from the larger manifold.

Immersion: An immersion is a smooth map between manifolds that has a derivative which is injective at every point, allowing for the representation of one manifold within another without self-intersections.

Smooth Map: A smooth map is a function between manifolds that is infinitely differentiable, ensuring that the structure of the manifolds is preserved under the mapping.