Horizontal distribution

Definition

Horizontal distribution refers to a choice of the tangent bundle of a Riemannian submersion that is orthogonal to the vertical space, essentially capturing the structure of the horizontal tangent spaces at each point of the base manifold. This distribution plays a crucial role in understanding how the geometry of the total space relates to the geometry of the base space and how curves and surfaces behave under the submersion.

5 Must Know Facts For Your Next Test

1. Horizontal distribution ensures that curves in the total space project to curves in the base space, helping to maintain the geometry of the original manifold.
2. In the context of Riemannian submersions, horizontal distributions are characterized by being orthogonal to vertical distributions at every point.
3. O'Neill's formulas relate horizontal and vertical distributions to curvature, revealing how they interact geometrically within Riemannian submersions.
4. The choice of horizontal distribution is essential for defining horizontal lifts of curves from the base manifold to the total space, facilitating analysis of paths and shapes.
5. This concept is crucial for studying properties like geodesic completeness and curvature behavior in lower-dimensional settings.

Review Questions

• How does horizontal distribution relate to the geometric properties preserved in Riemannian submersions?
  • Horizontal distribution is fundamental in preserving geometric properties in Riemannian submersions because it allows for the projection of curves from the total space onto the base manifold while maintaining their lengths. By defining a tangent space that is orthogonal to vertical spaces, this distribution ensures that important geometric characteristics such as angles and distances are kept intact during the projection process. This relationship is key when analyzing how structures behave under such mappings.
• Discuss how O'Neill's formulas utilize horizontal distribution to describe curvature in Riemannian submersions.
  • O'Neill's formulas leverage horizontal distribution to connect curvature properties between the total space and base manifold in Riemannian submersions. They express how the curvature of the total space can be decomposed into contributions from both vertical and horizontal distributions. This decomposition helps understand how curvature behaves locally and globally, providing insights into geodesic flow and stability within these geometrical frameworks.
• Evaluate the significance of horizontal distribution when analyzing geodesics within Riemannian submersions.
  • The significance of horizontal distribution when analyzing geodesics lies in its role in ensuring that geodesics in the total space project correctly onto geodesics in the base manifold. By allowing for horizontal lifts of curves, it aids in determining how paths behave under different geometrical settings. Evaluating these relationships provides valuable insights into concepts such as completeness and the nature of curvature, ultimately enriching our understanding of Riemannian geometry as a whole.

Related terms

Riemannian submersion: A smooth map between Riemannian manifolds that preserves lengths of horizontal vectors and reduces the dimension of the manifold while maintaining geometric structures.

Vertical distribution: The collection of tangent spaces at each point in a Riemannian submersion that are perpendicular to the horizontal distribution, representing directions that project down to zero on the base manifold.

Geodesics: Curves that represent the shortest path between points in a Riemannian manifold, which can be influenced by horizontal distributions in a Riemannian submersion.