Riemannian submersion

Definition

A Riemannian submersion is a smooth map between Riemannian manifolds that preserves the lengths of vertical vectors, while the geometry of horizontal vectors is distorted. This concept allows for the study of lower-dimensional structures within higher-dimensional spaces, revealing insights into the relationships between different geometrical and topological properties.

5 Must Know Facts For Your Next Test

1. A Riemannian submersion has the property that it sends geodesics in the horizontal distribution to geodesics in the base manifold, making them useful in understanding geometric structures.
2. The fibers of a Riemannian submersion can have varying geometries and can be studied through their connection to the base manifold.
3. Riemannian submersions can provide important examples of how curvature behaves under mappings, as they often relate spaces with different curvature properties.
4. An important theorem related to Riemannian submersions states that if the total space has non-positive curvature, then the base space will also inherit certain curvature conditions.
5. Riemannian submersions are closely related to foliations, where the leaves can be thought of as fibers over points in the base manifold.

Review Questions

• How does a Riemannian submersion relate geodesics in the horizontal distribution to geodesics in the base manifold?
  • A Riemannian submersion maps geodesics that lie entirely within the horizontal distribution to geodesics in the base manifold. This means that if you have a curve that follows the horizontal direction, its image will represent a curve that respects the geometry of the lower-dimensional base space. This relationship is crucial for understanding how geometric properties are transferred between manifolds.
• Discuss the implications of having non-positive curvature in relation to Riemannian submersions and their base spaces.
  • When a Riemannian submersion has its total space with non-positive curvature, it influences the geometry of the base space significantly. The theorem associated with this states that such properties regarding curvature are preserved down to the base manifold. This has practical implications for understanding how certain geometric features like distances and angles behave across different manifolds.
• Evaluate how Riemannian submersions contribute to our understanding of fiber bundles and their geometric structures.
  • Riemannian submersions enhance our comprehension of fiber bundles by providing a geometric framework where each fiber can be analyzed through its relationship with the base space. This connection allows mathematicians to explore how different fibers can exhibit unique geometrical features while still being part of a unified structure. Such insights are valuable for both theoretical advancements and practical applications in differential geometry.

Related terms

Riemannian manifold: A Riemannian manifold is a smooth manifold equipped with a Riemannian metric, which defines the notion of angle, length of curves, and distances on the manifold.

Fiber bundle: A fiber bundle is a structure that consists of a base space and fibers attached to each point in the base space, often used to describe the local triviality of spaces in topology.

Horizontal distribution: In the context of a Riemannian submersion, horizontal distribution refers to the subspace of the tangent bundle that contains all horizontal vectors, orthogonal to the vertical vectors.