Quotient Manifold

Definition

A quotient manifold is a type of manifold that is formed by taking a differentiable manifold and partitioning it into disjoint subsets, where each subset corresponds to an equivalence class under a certain equivalence relation. This process allows for the construction of new manifolds that retain some properties of the original manifold while capturing the geometric structure associated with the equivalence relation. Quotient manifolds often arise in examples involving symmetries or group actions on manifolds, showcasing how these structures can simplify complex geometrical relationships.

5 Must Know Facts For Your Next Test

1. Quotient manifolds can be understood as the set of equivalence classes formed by identifying points based on an equivalence relation defined by a group action.
2. The process of forming a quotient manifold involves using local charts from the original manifold and creating new charts that reflect the equivalence classes.
3. An important example of a quotient manifold is the real projective space, which is obtained by taking spheres and identifying antipodal points.
4. The topology and differentiable structure of a quotient manifold depend heavily on the nature of the equivalence relation used to define it.
5. Quotient manifolds can simplify problems in Riemannian geometry by allowing one to study the geometry of orbits under group actions, rather than working with more complex original manifolds.

Review Questions

• How does an equivalence relation contribute to the formation of a quotient manifold?
  • An equivalence relation is essential for forming a quotient manifold as it defines how points in a manifold are grouped into equivalence classes. Each class represents a set of points that are considered identical under the relation, allowing us to collapse these points into single entities in the quotient space. This grouping not only simplifies our study of the manifold's geometry but also retains key structural properties related to the original manifold.
• Discuss how group actions lead to examples of quotient manifolds and provide a specific example.
  • Group actions play a crucial role in producing quotient manifolds by acting on a manifold's points and creating equivalence classes based on this action. A specific example is when considering the action of the circle group $S^1$ on the unit circle $S^1$ itself, where points are identified based on their angular position. This results in the quotient manifold being homeomorphic to a single point, illustrating how symmetry can drastically change our view of geometric structures.
• Evaluate the significance of quotient manifolds in understanding complex geometric structures within Riemannian geometry.
  • Quotient manifolds hold significant importance in Riemannian geometry as they allow for simplification and analysis of complex geometric structures by focusing on symmetry and group actions. They provide insight into how geometric properties behave under identification of points, which can reveal invariants and assist in solving problems related to curvature and geodesics. By studying these quotients, mathematicians can better understand the interplay between local and global geometric properties, leading to deeper insights into the nature of spaces we investigate.

Related terms

Equivalence Relation: A relationship that groups elements into sets such that any two elements in the same set are equivalent under some criteria, allowing for the formation of quotient spaces.

Action of a Group: A way in which a group operates on a manifold, transforming its points while preserving the manifold's structure and allowing for the creation of quotient manifolds based on these actions.

Topological Space: A fundamental concept in mathematics that generalizes the notion of 'closeness' or continuity, serving as a basis for defining manifolds and understanding their properties.