[Paper Review] When is a Connection a Levi-Civita Connection?

Richard Atkins|ArXiv.org|Apr 16, 2008

Fixed Point Theorems Analysis4 references3 citations

TL;DR

This paper provides a comprehensive criterion for determining when a symmetric connection on a manifold is a Levi-Civita connection by analyzing the existence of local and global parallel metrics. It introduces a derived flag construction to detect local parallel sections in the bundle of symmetric 2-tensors, and uses sheaf cohomology—particularly de Rham cohomology in the rank-one case—to characterize global obstructions. The key result is that while analytic connections on simply-connected manifolds are always metric if locally metric, smooth connections may fail to admit a global parallel metric even on contractible spaces, as demonstrated by a constructed counterexample with varying fiber dimensions in the derived flag.

ABSTRACT

We consider the more general question as to when a connection is a metric connection. There are two aspects to this investigation: first, the determination of the integrability conditions that ensure the existence of a local parallel metric in the neighbourhood of a given point and second, the characterization of the topological obstruction to a globally defined parallel metric.

Motivation & Objective

• To determine the necessary and sufficient conditions for a symmetric connection on a manifold to be a Levi-Civita connection.
• To characterize the local existence of parallel metrics using integrability conditions on the derived flag of a vector bundle.
• To identify topological obstructions to the global existence of a parallel metric via sheaf cohomology of the general linear group.
• To contrast analytic and smooth connections, showing that analyticity ensures global metric extension where smoothness does not.
• To construct a counterexample of a smooth, locally metric connection on R² that is not globally metric, highlighting a pathology in the smooth category.

Proposed method

• Construct a derived flag of subsets of the bundle of symmetric covariant 2-tensors to detect local non-trivial parallel sections.
• Define a map αW′: W′ → (W/W′) ⊗ T*M using the connection and projection, and analyze its kernel to determine integrability conditions.
• Use the terminal subset W̃ to determine whether a local parallel metric exists; if W̃ is a non-zero vector bundle, such metrics exist locally.
• For regular connections, formulate global metric existence in terms of the first cohomology group H¹(M, Gₙ) of the constant sheaf of GL(n, ℝ) sections.
• Apply de Rham cohomology when the derived bundle has rank one, linking the obstruction to a closed 1-form in H¹dR(M).
• Use analytic continuation to prove that analytic connections on analytic manifolds yield a globally well-defined derived bundle W̃, ensuring global metric extension.

Experimental results

Research questions

• RQ1 Under what conditions does a symmetric connection on a manifold admit a local parallel metric in a neighborhood of each point?
• RQ2 What topological or geometric obstructions prevent a locally metric connection from admitting a globally defined parallel metric?
• RQ3 How does the derived flag of the bundle of symmetric 2-tensors determine the existence of local parallel sections?
• RQ4 Why do smooth locally metric connections fail to be globally metric even on contractible manifolds, unlike analytic ones?
• RQ5 In what sense does the failure of the derived bundle W̃ to be a vector bundle (e.g., varying fiber dimension) obstruct global metric existence?

Key findings

• A connection is locally metric if and only if the derived flag of the symmetric 2-tensor bundle yields a non-trivial subbundle W̃ that is locally a vector bundle.
• For regular connections, the existence of a global parallel metric is equivalent to the vanishing of a cohomology class in H¹(M, GL(n, ℝ)), generalizing the classification of flat vector bundles.
• When the derived bundle W̃ has rank one, the obstruction to a global metric is encoded in a de Rham cohomology class in H¹dR(M).
• Analytic connections on analytic, simply-connected manifolds are metric if locally metric, due to analytic continuation ensuring global consistency of parallel sections.
• A smooth, locally metric connection on R² exists that is not globally metric, due to discontinuous fiber dimension in W̃ (1 → 3 → 1), preventing sheaf cohomology formulation.
• The example demonstrates that smooth connections can have local parallel metrics that do not extend globally, even on contractible manifolds, unlike the analytic case.