Immersed submanifold

Definition

An immersed submanifold is a subset of a manifold that, while it may not be embedded, locally resembles the manifold around it in terms of its structure and differentiability. This concept is crucial because it allows for the study of lower-dimensional structures within higher-dimensional manifolds while still maintaining a degree of geometric and analytical properties relevant to the larger manifold.

5 Must Know Facts For Your Next Test

1. An immersed submanifold allows for local charts that map points to Euclidean space, preserving differentiability but not necessarily global properties like topology.
2. In an immersed submanifold, the inclusion map can fail to be an embedding; hence, self-intersections may occur even if locally it looks like a lower-dimensional manifold.
3. Immersed submanifolds can be used to study phenomena such as curves and surfaces within higher-dimensional spaces without requiring them to be embedded.
4. The dimension of an immersed submanifold is strictly less than that of the ambient manifold, allowing it to locally resemble an open subset of Euclidean space.
5. In the context of Riemannian geometry, immersed submanifolds can inherit geometric properties from the larger manifold, such as metrics and curvature.

Review Questions

• How does an immersed submanifold differ from an embedded submanifold, and what are the implications of these differences?
  • An immersed submanifold allows for local differentiable structures similar to those in the ambient manifold but can have self-intersections and is not necessarily globally injective. In contrast, an embedded submanifold maintains a homeomorphic relationship with its Euclidean representation without any self-intersections. This difference implies that while both types can retain local geometrical features, only embedded submanifolds preserve topological properties globally.
• Discuss the significance of tangent spaces in relation to immersed submanifolds and how they facilitate understanding the geometry involved.
  • Tangent spaces are crucial when analyzing immersed submanifolds as they provide linear approximations at each point. For an immersed submanifold, its tangent space at a given point will correspond to directions within the larger manifold that are tangent to the submanifold itself. This relationship helps in exploring the curvature and differential structure of both the immersed submanifold and its surrounding ambient space.
• Evaluate how immersed submanifolds contribute to Riemannian geometry and their role in understanding geometric phenomena in higher dimensions.
  • Immersed submanifolds play a significant role in Riemannian geometry by allowing mathematicians to analyze lower-dimensional structures within higher-dimensional manifolds while inheriting geometric properties like metrics and curvature. This ability enables deeper insights into complex geometric phenomena such as geodesics and curvature behavior in multi-dimensional spaces. By studying immersed submanifolds, one can gain perspectives on deformation, limits, and other properties that emerge when lower-dimensional entities exist in more complex ambient settings.

Related terms

embedded submanifold: An embedded submanifold is a submanifold that is also a topological embedding, meaning it is homeomorphic to an open subset of Euclidean space, allowing for smooth inclusion within the parent manifold.

tangent space: The tangent space at a point on a manifold consists of all tangent vectors at that point, providing a linear approximation of the manifold in the vicinity of that point.

differentiable structure: A differentiable structure on a manifold refers to the ways in which functions can be differentiated, enabling the application of calculus to manifold theory.