Riemannian Submersions

Definition

Riemannian submersions are smooth maps between Riemannian manifolds that preserve the geometric structure in a specific way, allowing for the projection of one manifold onto another while maintaining certain properties of the original manifold. This concept plays a crucial role in understanding how different geometries relate to one another, especially in the context of O'Neill's formulas, which describe how curvature behaves under such maps.

5 Must Know Facts For Your Next Test

1. Riemannian submersions allow for a natural way to study how curvature is transmitted from one manifold to another.
2. They can be seen as a generalization of the concept of a submersion in differential geometry, providing insights into how manifolds can be related through projections.
3. The fibers of a Riemannian submersion are generally equipped with their own geometry, often reflecting properties of the total space.
4. O'Neill's formulas help compute how sectional curvature is influenced by the submersion process, linking the curvatures of both manifolds.
5. A classic example of a Riemannian submersion is the projection from a 3-dimensional sphere onto its equatorial circle.

Review Questions

• How do Riemannian submersions relate to the study of curvature in differential geometry?
  • Riemannian submersions create a framework for studying curvature relationships between two manifolds. They allow us to understand how the curvature of the total space translates into the curvature of the base space. O'Neill's formulas specifically provide a method to compute these relationships, showing how sectional curvature changes due to this projection process.
• Discuss the significance of vertical and horizontal spaces in the context of Riemannian submersions.
  • Vertical and horizontal spaces play crucial roles in understanding Riemannian submersions. The vertical space consists of vectors tangent to the fibers, while horizontal spaces are orthogonal complements lying within the base manifold. This decomposition helps clarify how geometry behaves under the map, allowing us to analyze interactions between different dimensionalities in terms of curvature and distance preservation.
• Evaluate the impact of O'Neill's formulas on our understanding of geometric structures in Riemannian submersions.
  • O'Neill's formulas significantly deepen our comprehension of geometric structures in Riemannian submersions by detailing how curvature changes across spaces. They provide explicit calculations that relate sectional curvatures from both the total and base spaces. This relationship not only enhances theoretical insights but also has practical implications in fields like general relativity and differential geometry, showcasing how complex geometric properties are interrelated.

Related terms

Riemannian Manifold: A Riemannian manifold is a smooth manifold equipped with a Riemannian metric, which defines the length of curves, angles, and distances within the manifold.

O'Neill's Formulas: O'Neill's formulas provide relationships between the curvature of the total space and the base space in a Riemannian submersion, capturing how curvature changes under the projection.

Vertical and Horizontal Spaces: In the context of Riemannian submersions, vertical spaces correspond to directions tangent to the fibers of the submersion, while horizontal spaces are orthogonal complements that lie in the base manifold.