Reducible holonomy

Definition

Reducible holonomy refers to a situation where the holonomy group of a Riemannian manifold can be represented by a subgroup of the general linear group that is reducible, meaning it can be decomposed into smaller, invariant subspaces. This concept is essential in understanding how curvature and geometric properties can influence the structure of the manifold, linking it to the classification of holonomy groups and their properties.

5 Must Know Facts For Your Next Test

1. In a manifold with reducible holonomy, the holonomy representation can act non-trivially on some vector subspaces while leaving others invariant.
2. Reducible holonomy groups are often related to certain types of geometric structures, such as product structures or direct sums of vector bundles.
3. An example of reducible holonomy is when a manifold exhibits both flat and curved regions, leading to a decomposition of its tangent space into subspaces that reflect this behavior.
4. Understanding reducibility is crucial for studying connections on vector bundles and the impact of curvature on the topology of manifolds.
5. In Berger's classification, certain groups are identified as having reducible representations, which can inform us about possible geometric structures on the manifold.

Review Questions

• How does reducible holonomy relate to the structure of a Riemannian manifold and its geometric properties?
  • Reducible holonomy indicates that the tangent spaces of a Riemannian manifold can be split into smaller invariant subspaces. This splitting suggests that certain geometric properties, like curvature, can vary across different regions of the manifold. By analyzing these regions, one can infer how the manifold behaves under parallel transport and how its overall geometric structure is affected by its curvature.
• Compare and contrast reducible and irreducible holonomy groups in terms of their implications for Riemannian geometry.
  • Reducible holonomy groups allow for decompositions of the tangent space into invariant subspaces, suggesting flexibility in the manifold's curvature and structure. In contrast, irreducible holonomy groups indicate a more rigid structure without such decompositions. This difference has significant implications: reducible cases may exhibit more diverse geometric behaviors, while irreducible cases tend to have uniform curvature properties throughout.
• Evaluate the significance of Berger's classification in understanding reducible holonomy within Riemannian geometry.
  • Berger's classification is crucial for identifying and categorizing various holonomy groups, including those that are reducible. By providing a framework for understanding how different groups relate to the geometric properties of manifolds, Berger's work allows mathematicians to predict how certain structures will behave based on their holonomy characteristics. This understanding can lead to deeper insights into curvature, topology, and the overall geometry of Riemannian manifolds.

Related terms

holonomy group: The holonomy group is a group that describes how parallel transport around closed loops in a Riemannian manifold relates to the curvature of that manifold.

irreducible holonomy: Irreducible holonomy occurs when the holonomy group cannot be decomposed into smaller invariant subspaces, indicating a more rigid structure in the manifold's geometry.

Berger's classification: Berger's classification categorizes different types of holonomy groups associated with Riemannian manifolds, distinguishing between reducible and irreducible cases based on their geometric and topological features.