Riemannian Foliation

Definition

A Riemannian foliation is a geometric structure on a Riemannian manifold where the manifold can be decomposed into a collection of disjoint submanifolds called leaves, such that the Riemannian metric is compatible with this decomposition. This means that at every point, one can find orthogonal projections from the ambient manifold to these leaves, preserving the distances within the leaves. The concept is crucial for understanding how curvature and geometry behave within the framework of foliated spaces.

5 Must Know Facts For Your Next Test

1. Riemannian foliations allow for a systematic study of the geometry and topology of manifolds by breaking them down into simpler components, making analysis more tractable.
2. The existence of Riemannian foliations is often tied to specific curvature conditions on the ambient manifold, which can impact the behavior of the leaves.
3. In O'Neill's framework, the relationships among curvature tensors for the ambient manifold and its leaves are captured in formulas that illustrate how the geometry interacts across different dimensions.
4. The transverse structure associated with a Riemannian foliation gives rise to interesting phenomena such as how distances are measured between points in different leaves.
5. Applications of Riemannian foliations extend beyond pure mathematics; they are also significant in physics, particularly in theories that explore gauge fields and general relativity.

Review Questions

• How does a Riemannian foliation decompose a manifold, and what are its implications for understanding geometry?
  • A Riemannian foliation decomposes a manifold into disjoint submanifolds called leaves, which simplifies the analysis of geometric properties by focusing on these lower-dimensional spaces. This decomposition helps in studying curvature, as the distances within each leaf can be preserved while examining relationships across different leaves. Understanding this structure allows mathematicians to develop insights into global geometric behavior based on local properties.
• Discuss O'Neill's formulas and their significance in studying Riemannian foliations.
  • O'Neill's formulas provide critical relationships between various curvature tensors associated with both the ambient Riemannian manifold and its leaves. These formulas help quantify how curvature behaves across different dimensions within the foliation. By analyzing these relationships, one gains deeper insights into how geometric properties like distance and curvature interact between leaves and their surrounding space, thereby enriching our understanding of the overall manifold structure.
• Evaluate the role of holonomy in understanding Riemannian foliations and its impact on global geometric properties.
  • Holonomy plays a vital role in examining Riemannian foliations by revealing how paths within the leaves influence the geometric structure of the entire manifold. It assesses how much curvature is retained or altered when transitioning between different leaves, thus providing insights into global geometric properties like connectivity and potential singularities. The interplay between holonomy and foliation offers a comprehensive view of how local behaviors contribute to broader characteristics of the manifold as a whole.

Related terms

Leaves: The submanifolds that form the decomposition in a foliation; each leaf is a smooth submanifold of the Riemannian manifold.

Holonomy: A measure of how much the geometry of a manifold can change when one moves along paths in the leaves of a foliation, reflecting the way different leaves are connected.

O'Neill's Formulas: Formulas that relate various geometric properties of the ambient manifold and the leaves in a Riemannian foliation, providing important insights into curvature and distance.