Non-positive curvature

Definition

Non-positive curvature refers to a geometric property of a space where, intuitively, triangles formed within the space have angles that sum to less than or equal to 180 degrees. This concept is crucial because it implies that geodesics can diverge, indicating the presence of flat or saddle-like geometry. This property connects to various significant results in geometry and topology, influencing how manifolds behave under certain curvature constraints and shaping the classification of their structures.

5 Must Know Facts For Your Next Test

1. Non-positive curvature implies that a manifold can be locally modeled on Euclidean space or hyperbolic space, affecting its topological properties.
2. In spaces of non-positive curvature, the triangle comparison theorem shows that the sum of angles in a triangle is less than or equal to that in Euclidean triangles.
3. Manifolds with non-positive curvature are often used in geometric group theory because they exhibit properties that relate closely to algebraic structures.
4. The Busemann function provides a way to study geodesics in spaces of non-positive curvature by describing limit points of sequences of geodesics.
5. Berger's classification illustrates how the holonomy groups of Riemannian manifolds relate to their curvature, with non-positive curvature being linked to specific holonomy types.

Review Questions

• How does non-positive curvature affect the behavior of geodesics in a manifold?
  • In manifolds with non-positive curvature, geodesics tend to diverge from one another. This means that if you take two geodesics starting from the same point, they will spread apart as you move away from the starting point. This divergence contrasts with positive curvature, where geodesics tend to converge. This behavior is vital for understanding the global geometry of the manifold and how distances behave in such spaces.
• Discuss the implications of non-positive curvature for triangle geometry within manifolds and how this connects to broader geometric concepts.
  • In manifolds with non-positive curvature, triangles formed by geodesics have angle sums that are less than or equal to 180 degrees. This property leads to significant consequences in geometry, like establishing the triangle comparison theorem. It allows mathematicians to draw parallels between geometric constructions in these manifolds and those in Euclidean or hyperbolic spaces, enriching our understanding of different geometrical frameworks.
• Evaluate the relationship between non-positive curvature and Berger's classification of holonomy groups and its importance in Riemannian geometry.
  • Non-positive curvature is directly linked to specific types of holonomy groups as classified by Berger. For instance, manifolds with non-positive curvature often exhibit holonomy groups that fall into categories like trivial or those related to flat geometries. Understanding this relationship is important because it helps classify Riemannian manifolds based on their geometric properties and underlying structures. This classification not only influences theoretical studies but also has implications for applications in mathematical physics and topology.

Related terms

Riemannian manifold:

A smooth manifold equipped with a Riemannian metric, which allows for the measurement of distances and angles, providing the structure necessary to discuss curvature.

Geodesic:

A curve that represents, locally, the shortest path between points on a manifold, crucial for understanding how non-positive curvature affects distance measurements.

Curvature bounds: Constraints on the curvature of a manifold, which help classify its geometric properties and behavior, particularly when discussing manifolds with non-positive curvature.