Riemannian submersion

Definition

A Riemannian submersion is a smooth surjective map between two Riemannian manifolds that preserves the length of tangent vectors from the manifold at the base to the total space, while projecting vertical vectors orthogonally. This concept is crucial as it relates to how geometrical structures can be preserved under mappings, and it connects with the study of curvature and parallel transport in differential geometry.

5 Must Know Facts For Your Next Test

1. Riemannian submersions allow for the reduction of complex geometrical structures by projecting them onto simpler manifolds while preserving local geometrical features.
2. The fibers of a Riemannian submersion can have different geometric properties than the base space, which can lead to interesting phenomena in geometry.
3. Parallel transport along fibers in a Riemannian submersion is affected by the curvature of both the total space and the base manifold.
4. The notion of a Riemannian submersion can be applied to understand relationships between various geometrical objects, such as in general relativity.
5. A classic example of a Riemannian submersion is the projection from a sphere onto its equatorial circle, where each point on the circle corresponds to a line of latitude on the sphere.

Review Questions

• How does a Riemannian submersion preserve geometric structures between two manifolds?
  • A Riemannian submersion preserves geometric structures by ensuring that the lengths of tangent vectors are maintained during projection from the total space to the base manifold. This means that if you take a vector tangent to the total space, its projection onto the base maintains its length. Additionally, this process allows for certain 'vertical' vectors to be projected orthogonally, leading to distinct geometric behaviors between the total space and its base.
• What roles do horizontal and vertical distributions play in understanding Riemannian submersions?
  • Horizontal and vertical distributions are essential in analyzing Riemannian submersions as they dictate how tangent spaces are structured above each point in the base manifold. Horizontal distributions correspond to directions that are isometrically projected onto the base manifold, whereas vertical distributions indicate directions within the fibers. This distinction helps in understanding how curves and geometric features behave under the submersion and reveals important insights about curvature and parallel transport.
• Discuss how understanding Riemannian submersions contributes to broader concepts in differential geometry and their applications.
  • Understanding Riemannian submersions contributes significantly to differential geometry as it provides insights into how complex manifolds relate to simpler structures while preserving important geometrical information. This concept aids in exploring curvature properties, allows for parallel transport discussions between different manifolds, and opens pathways for applications in physics, particularly in general relativity where different spacetime geometries can be analyzed. By studying these connections, we gain a deeper comprehension of how various geometrical frameworks interact and influence one another in mathematical physics.

Related terms

Riemannian manifold:

A Riemannian manifold is a real, smooth manifold equipped with a Riemannian metric, which allows for the measurement of distances and angles on the manifold.

Fiber bundle: A fiber bundle is a structure that consists of a base space, a total space, and fibers that are attached to each point in the base space, providing a way to understand how different spaces relate to each other.

Horizontal and vertical distributions: In the context of a Riemannian submersion, horizontal distributions refer to tangent spaces that are mapped isometrically to the base manifold, while vertical distributions are tangent spaces that lie in the fibers above each point in the base manifold.