Vertical Space

Definition

Vertical space refers to the fibers of a Riemannian manifold that are orthogonal to the horizontal tangent spaces at a given point. This concept plays a crucial role in understanding Riemannian submersions, as it helps characterize how the geometry of a total space relates to that of its base space through vertical and horizontal distributions.

5 Must Know Facts For Your Next Test

1. In the context of Riemannian submersions, vertical space can be thought of as the 'fiber' direction that is collapsed when moving from the total space to the base space.
2. Vertical spaces are essential for understanding how curvature behaves in Riemannian submersions, as they influence the metric structure and geodesics of the total manifold.
3. The dimensions of vertical space and horizontal space together equal the dimension of the total manifold, maintaining consistency in manifold structure.
4. When analyzing vertical spaces, one often uses projections to simplify computations, especially when examining how properties like distance and angles are affected by the submersion.
5. Vertical spaces can be used to derive certain geometrical properties, such as the existence of harmonic maps and their implications for the topology of manifolds involved.

Review Questions

• How does vertical space relate to horizontal space in a Riemannian submersion?
  • Vertical space and horizontal space together form the tangent space at each point in a Riemannian manifold. The horizontal space represents directions in which distances are preserved during a submersion, while vertical space contains directions that are collapsed during this process. Understanding the relationship between these two types of spaces is key for analyzing how a total manifold projects onto its base manifold.
• Discuss how vertical spaces affect the curvature properties of Riemannian submersions.
  • Vertical spaces play a significant role in determining how curvature behaves within Riemannian submersions. Since these spaces represent directions that do not contribute to the metric structure of the base manifold, they can influence the overall geometric properties. The way vertical fibers are collapsed may lead to varying degrees of curvature in different directions, thus affecting geodesics and other intrinsic geometric features of both total and base spaces.
• Evaluate how understanding vertical space contributes to our knowledge of fiber bundles in differential geometry.
  • Understanding vertical space enhances our grasp of fiber bundles because it helps clarify how fibers behave under projections and transformations. Each fiber corresponds to a vertical space associated with points in the base manifold. By studying these connections, we gain insights into more complex structures like connections and curvature on bundles, revealing deeper relationships between geometry and topology within differential geometry.

Related terms

Riemannian Submersion: A smooth map between Riemannian manifolds that preserves lengths of horizontal vectors while collapsing vertical vectors.

Horizontal Space: The tangent space at a point in a Riemannian manifold that is complementary to the vertical space, representing directions in which the submersion is 'flat'.

Fiber: The preimage of a point in the base manifold under a submersion, consisting of points that share the same projection onto the base.