Embedded submanifold

Definition

An embedded submanifold is a subset of a Riemannian manifold that is itself a manifold, with a structure that allows it to fit nicely within the larger manifold. This means it retains its manifold properties while being equipped with an induced Riemannian metric, making it possible to study the geometry of both the submanifold and the ambient space. Understanding embedded submanifolds is crucial for analyzing the geometric relationships between different manifolds and for studying properties such as curvature and distance within the context of Riemannian geometry.

5 Must Know Facts For Your Next Test

1. An embedded submanifold inherits the differentiable structure from the ambient manifold, which allows for smooth transitions between points.
2. The dimension of an embedded submanifold is less than or equal to that of the ambient manifold, ensuring it can be smoothly placed within the larger structure.
3. Embedded submanifolds can have their own intrinsic geometry, characterized by curvature that may differ from that of the ambient manifold.
4. Examples of embedded submanifolds include spheres embedded in Euclidean space or surfaces like tori existing within higher-dimensional spaces.
5. The concept of an embedded submanifold is foundational in studying how local geometries interact within larger spaces, especially in applications such as general relativity.

Review Questions

• How do embedded submanifolds relate to the concepts of Riemannian metrics and curvature in a manifold?
  • Embedded submanifolds have their own Riemannian metrics induced from the ambient manifold, allowing them to retain geometric properties such as distances and angles. This connection helps study the curvature of both the submanifold and its surrounding space. For example, if a sphere is an embedded submanifold in Euclidean space, its curvature can be analyzed in relation to the flat geometry of its ambient environment.
• Discuss the significance of immersion when considering embedded submanifolds in Riemannian geometry.
  • Immersion plays a crucial role in defining how an embedded submanifold fits into an ambient manifold. A proper immersion ensures that the embedding preserves the manifold's structure and differentiability, allowing smooth transitions. Without this condition, one could encounter singularities or discontinuities that disrupt the geometric analysis of both spaces.
• Evaluate how understanding embedded submanifolds can influence applications in fields like physics or engineering, particularly in modeling complex systems.
  • Understanding embedded submanifolds is critical for modeling complex systems in physics and engineering because it provides insights into how different geometries interact. For example, when studying general relativity, one must consider how curved spacetime can embed various physical phenomena. By analyzing properties of embedded submanifolds, scientists can better predict behaviors in systems ranging from fluid dynamics to materials science, ultimately leading to more accurate models and simulations.

Related terms

Riemannian metric: A Riemannian metric is a type of inner product defined on the tangent space of a manifold that allows for measuring lengths and angles, playing a key role in the study of geometric properties.

Immersion:

An immersion is a differentiable map between manifolds that preserves the dimension, allowing for the smooth embedding of one manifold into another without collapsing dimensions.

Tangent space: The tangent space at a point on a manifold is a vector space consisting of all possible directions in which one can tangentially pass through that point, essential for defining derivatives on manifolds.