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On the extendability of parallel sections of linear connections. (English) Zbl 1356.53053

In the paper the authors study the problem of extendability of parallel sections of linear connections on vector bundles. Given a vector bundle \(\pi:E \rightarrow M\) over a simply connected manifold \(M\), a linear connection \(\nabla\) in \(\pi\) and a \(\nabla\)-parallel section \(\sigma: U \rightarrow E\) on a connected open subset \(U\) of \(M\), they obtain sufficient conditions on \(U\) in order to extend \(\sigma\) as parallel section to the whole of \(M\). In particular, they prove that if the complement of \(U\) in \(M\) is contained in a smooth submanifold of codimension greater or equal to \(2\), then \(\sigma\) can be extended as a parallel section to the whole of \(M\). Moreover, they show other sufficient conditions under the assumption that \(M\) is \(2\)-dimensional.
Reviewer: Anna Fino (Torino)

MSC:

53C29 Issues of holonomy in differential geometry
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli

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References:

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