Affine connection

Definition

An affine connection is a mathematical structure that allows for the comparison of vectors in tangent spaces of a manifold, enabling the definition of parallel transport and covariant derivatives. It provides a way to differentiate vector fields along curves on the manifold and plays a crucial role in understanding the geometric properties of Riemannian manifolds.

5 Must Know Facts For Your Next Test

1. An affine connection allows for the definition of parallel transport, which describes how vectors are moved along curves in a consistent manner without twisting or turning.
2. The Levi-Civita connection is a specific type of affine connection that is compatible with the metric tensor and has zero torsion, making it unique among all affine connections.
3. Affine connections can be used to define covariant derivatives, which extend the notion of differentiation to include changes in direction on curved manifolds.
4. Properties of affine connections include linearity, compatibility with the tangent bundle, and the ability to define curvature through the Riemann curvature tensor.
5. Berger's classification of Riemannian holonomy relates to affine connections by characterizing how different types of curvature affect the parallel transport of vectors around closed loops.

Review Questions

• How does an affine connection facilitate the process of parallel transport along curves in a manifold?
  • An affine connection provides a systematic way to transport vectors along curves while maintaining their geometric relationships. By defining how vectors change as they move from one point to another, an affine connection ensures that these vectors do not twist or rotate unexpectedly. This concept is essential for understanding how vectors behave in curved spaces and underpins the notion of parallelism in differential geometry.
• Discuss the significance of the Levi-Civita connection and its unique properties among all affine connections.
  • The Levi-Civita connection is significant because it is the only affine connection that is both compatible with the metric tensor and has zero torsion. This means it preserves lengths and angles during parallel transport, making it particularly useful for Riemannian geometry. Its uniqueness ensures that for any given Riemannian manifold, there is a well-defined way to differentiate vector fields that respects the geometric structure of the manifold.
• Evaluate how Berger's classification of Riemannian holonomy relates to affine connections and their impact on geometric structures.
  • Berger's classification categorizes Riemannian holonomy groups based on how they affect the parallel transport of vectors around loops in a manifold. This classification ties closely to affine connections since different types of curvature associated with these connections influence how vectors behave during parallel transport. By understanding holonomy, we gain insights into the global geometric properties of manifolds and can classify them according to their inherent symmetries and shapes.

Related terms

Tangent space: The tangent space at a point on a manifold is a vector space that consists of all possible tangent vectors at that point, representing directions in which one can move away from the point.

Curvature: Curvature measures how a manifold deviates from being flat; it can be described in terms of an affine connection, influencing the behavior of geodesics and the overall geometry.

Geodesic: A geodesic is a curve that locally minimizes distance between points on a manifold and is characterized by having zero acceleration with respect to an affine connection.