Embedding

Definition

In differential geometry, an embedding refers to a smooth and injective map from one manifold to another, allowing the first manifold to be treated as a submanifold within the second. This concept is essential when discussing submanifolds since it captures how lower-dimensional spaces can exist and interact within higher-dimensional ones, preserving their geometric structure. When a manifold is embedded, it inherits a metric from the ambient space, which allows us to study properties like lengths, angles, and curvature in the context of its surrounding environment.

5 Must Know Facts For Your Next Test

1. Embeddings allow a manifold to be analyzed in terms of its position and properties within an ambient space, which can simplify many geometric considerations.
2. When a manifold is embedded, the induced metric ensures that distances and angles on the submanifold align with those of the surrounding manifold.
3. Not all maps from one manifold to another are embeddings; they must be smooth, injective, and have a smooth inverse on their image.
4. The concept of an embedding plays a critical role in understanding how different geometrical structures relate to each other across varying dimensions.
5. Embeddings can also help visualize complex manifolds by allowing them to be represented within a more familiar setting, aiding intuition in geometric analysis.

Review Questions

• How does the concept of embedding relate to submanifolds and their metrics?
  • Embedding directly connects to submanifolds by providing a framework through which we can treat these lower-dimensional spaces as integral parts of higher-dimensional manifolds. When a submanifold is embedded in an ambient space, it inherits an induced metric that reflects the geometry of the larger manifold. This relationship allows us to analyze the properties of the submanifold while considering its interaction with the surrounding space.
• What distinguishes an embedding from other types of maps between manifolds, such as immersions?
  • The key difference between an embedding and an immersion lies in injectivity. An embedding is both an immersion and injective, meaning it maps distinct points in the source manifold to distinct points in the target manifold without overlaps. In contrast, immersions may not maintain this injectivity globally; they can locally behave like embeddings but may self-intersect when viewed globally. This distinction is crucial when discussing how spaces can be represented within one another.
• Evaluate the significance of embeddings in understanding geometric properties and relationships between different manifolds.
  • Embeddings are vital for grasping geometric properties because they allow us to study lower-dimensional manifolds within higher-dimensional contexts while retaining essential features such as lengths and angles. By embedding one manifold into another, we gain insights into how these spaces interact and reveal complex relationships that might otherwise be difficult to analyze in isolation. This ability to visualize and relate different dimensions enhances our understanding of curvature and topology across various geometrical constructs.

Related terms

Submanifold:

A submanifold is a subset of a manifold that is itself a manifold with a lower dimension, equipped with the subspace topology.

Induced Metric:

An induced metric is the metric on a submanifold that arises from the ambient manifold’s metric, capturing distances and angles from the higher-dimensional space.

Immersion: An immersion is a smooth map between manifolds that has a derivative that is injective at every point, but not necessarily injective globally.