Submanifold

Definition

A submanifold is a subset of a manifold that is itself a manifold, typically characterized by being defined as the zero set of a smooth map or as an embedded subset. Submanifolds maintain the smooth structure of the ambient manifold, which allows for important geometric and topological properties to be preserved. They play crucial roles in various mathematical concepts, including differential maps and transversality, helping to understand the interactions between different manifolds.

5 Must Know Facts For Your Next Test

1. Submanifolds can be classified into two types: regular submanifolds and embedded submanifolds, with embedded submanifolds being those that are smoothly embedded in the ambient manifold.
2. The dimension of a submanifold is always less than or equal to that of the ambient manifold, and it is determined by the number of constraints imposed by its defining equations.
3. When considering differential maps, the behavior of these maps on submanifolds can be studied using their derivatives, which reflects how smooth structures are preserved.
4. In the context of transversality, submanifolds can interact with other submanifolds in specific ways, such as when they intersect transversely, leading to important properties like dimensionality at intersection points.
5. The existence of submanifolds allows for the application of concepts like integration and differentiation in lower dimensions while still being grounded in the properties of higher-dimensional manifolds.

Review Questions

• How does the concept of embedding relate to the definition and properties of submanifolds?
  • Embedding is fundamental to understanding submanifolds because it allows a submanifold to be smoothly inserted into an ambient manifold without losing its intrinsic manifold structure. An embedded submanifold inherits not only the topology but also the differentiable structure from the ambient space. This relationship is crucial since it ensures that any analysis done within the embedded context remains consistent with broader manifold theory.
• Discuss how tangent spaces are associated with submanifolds and their importance in differential topology.
  • Tangent spaces at points on a submanifold provide vital insights into its local structure and behavior. They help in analyzing smooth functions defined on submanifolds by offering a linear framework to approximate these functions. Understanding how tangent spaces interact with those of the ambient manifold allows mathematicians to study curvature and other geometric properties that arise when dealing with multiple manifolds.
• Evaluate the role of transversality in understanding intersections involving submanifolds and its implications in differential topology.
  • Transversality plays a critical role when analyzing intersections between submanifolds, as it dictates how these intersections occur and their resulting properties. When two submanifolds intersect transversely, they do so in such a way that their intersection is well-defined and behaves nicely, typically having maximal dimensionality based on their individual dimensions. This concept not only simplifies calculations but also leads to profound implications in understanding higher-dimensional geometries and their relationships.

Related terms

Embedding: An embedding is a smooth map from one manifold to another that is both an immersion and a homeomorphism onto its image, allowing the submanifold to retain its manifold structure within the ambient manifold.

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, providing a linear approximation of the manifold.

Immersion: An immersion is a differentiable map between manifolds that has injective differentials at every point, meaning that it captures the local structure of the manifolds involved.