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Geodesic completeness for type \(\mathcal{A}\) surfaces. (English) Zbl 1378.53048
An affine surface is a smooth surface endowed with a torsion-free connection on the tangent bundle. A theorem of Opozda classifies the locally-homogeneous affine surfaces in three types, the first one being type \({\mathcal{A}}\) which corresponds to constant Christoffel symbols. Let \(\rho(\xi, \eta) := \operatorname{Tr}(\sigma \to R(\sigma, \xi)\eta)\) be the Ricci tensor. It is symmetric for type-\({\mathcal{A}}\) surfaces which insures that we can associate to these surfaces models of the form: \({\mathcal{M}}_C:=(\mathbb{R}^2, \nabla^C)\) where \(C=\{C_{ij}^k\}\) is a collection of constant symbols, with \(C_{ij}^k=C_{ji}^k\), representing the Christoffel symbols. A model \(\mathcal{M}_C\) is said to be essentially geodesically complete if there exists a type-\({\mathcal{A}}\) surface which is modelled on \(\mathcal{M}_C\) and is geodesically complete; otherwise \(\mathcal{M}_C\) is said to be essentially geodesically incomplete.
The authors show that some models for type-\({\mathcal{A}}\) surfaces are geodesically complete, that some others admit an incomplete geodesic, but model geodesically complete surfaces, and that there are also others which do not model any geodesically complete surface. This gives rise to a simple algorithm to test whether a given model \({\mathcal{M}}_C\) is essentially geodesically incomplete. If the rank of the Ricci tensor is \(1\), then the model of the type-\({\mathcal{A}}\) surface is essentially geodesically incomplete if and only if \(\nabla \rho \neq 0\). If the rank of the Ricci tensor is 2, the surface is essentially geodesically incomplete if and only if there exists a non-trivial geodesic of the form \(\sigma_{a,b}(t) = (a, b) \cdot log(t)\).

MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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