[Paper Review] On the extendability of parallel sections of linear connections

Antonio J. Di Scala, Gianni Manno|arXiv (Cornell University)|May 29, 2014

Geometric Analysis and Curvature Flows10 references4 citations

TL;DR

This paper establishes sufficient conditions for extending parallel sections of linear connections on vector bundles over simply connected manifolds, particularly in 2D. It proves that if the complement of the domain has codimension ≥2 or satisfies specific geometric conditions (e.g., compact zero-measure sets), then a parallel section defined on a dense open subset extends globally, with applications to Killing vector fields on Riemann surfaces.

ABSTRACT

Let $π:E o M$ be a vector bundle over a simply connected manifold and $ abla$ a linear connection in $π$. Let $σ: U ightarrow E$ be a $ abla$-parallel section of $π$ defined on a connected open subset $U$ of $M$. We give sufficient conditions on $U$ in order to extend $σ$ to the whole $M$. We mainly concentrate to the case when $M$ is a $2$-dimensional simply connected manifold.

Motivation & Objective

• To determine sufficient conditions under which a parallel section defined on a dense open subset of a simply connected manifold extends globally.
• To address the non-uniqueness and non-extendability of parallel sections in geometric contexts such as Killing vector fields and tractor connections.
• To provide a general framework for extending sections when the base manifold is 2-dimensional and the domain complement has low codimension.
• To resolve a MathOverflow problem on extending Killing vector fields on compact Riemann surfaces minus finitely many points.
• To introduce and analyze geometric conditions (R and R⁺) on domains in ℝ² ensuring extension of parallel sections for metric and general connections.

Proposed method

• Uses the fact that if a parallel section exists on a dense open subset with complement of codimension ≥2, then the curvature of the connection must vanish identically, implying flatness and global extendability.
• Applies the theory of flat connections and holonomy to extend sections over simply connected domains, leveraging the topological triviality of the holonomy group.
• Introduces two geometric conditions, R and R⁺, on open subsets of ℝ², where R requires the complement to be compact and of zero Lebesgue measure, and R⁺ is a weaker topological condition.
• Employs the Kostant connection on the bundle TM ⊕ so(TM) to relate Killing vector fields to parallel sections, enabling extension via the kernel of curvature derivatives.
• Uses local coordinate analysis and curvature computations to show that the kernel of certain curvature derivatives defines a flat parallel line subbundle, allowing section extension.
• Applies the density of the domain U in M to extend pointwise identities (e.g., ξ ∧ ∇Yξ ≡ 0) from U to the whole manifold, ensuring the subbundle is parallel and flat.

Experimental results

Research questions

• RQ1 Under what conditions can a parallel section defined on a dense open subset of a simply connected manifold be extended to the entire manifold?
• RQ2 Can the extendability of parallel sections be guaranteed when the complement of the domain has codimension at least 2?
• RQ3 What geometric or topological conditions on the domain U ⊂ ℝ² ensure that a parallel section of a metric or general connection extends globally?
• RQ4 Does the existence of a Killing vector field on a Riemann surface minus a segment imply its global extension?
• RQ5 Can the extension of parallel sections be reduced to the flatness of the connection when the domain is dense and the complement has low dimension?

Key findings

• If the complement of the domain U ⊂ M has codimension ≥2, then any parallel section on U extends uniquely to a global parallel section on M, even without assuming M is simply connected.
• For rank-one bundles, the existence of a non-zero parallel section on a dense open subset implies the connection is flat, hence the section extends globally.
• For rank-two bundles with a compatible metric, the existence of a parallel section on a dense open subset forces the curvature to vanish, enabling global extension.
• When U ⊂ ℝ² has complement of zero Lebesgue measure and is compact, condition R holds, ensuring extension of parallel sections for metric connections.
• Condition R⁺ on U ⊂ ℝ² ensures extension of parallel sections even for general (non-metric) connections, generalizing previous results.
• A Killing vector field defined on a Riemann surface minus a segment (in local coordinates) can always be extended to the whole surface, resolving a problem posed by R. Bryant on MathOverflow.