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Curvature homogeneity of affine connections on two-dimensional manifolds. (English) Zbl 0942.53019
A pseudo-Riemannian manifold \((M,g)\) (with Levi-Civita connection \(\nabla\) and Riemann curvature tensor \(R\)) is said to be curvature homogeneous up to order \(r\) if, for every pair of points \(p,q \in M\), there exists a linear isometry \(\varphi : T_p M \to T_q M\) of the tangent spaces at \(p\) and \(q\) such that, for all \(i = 0,1,\dots,r\), we have \(\varphi\ast \nabla^i R_q = \nabla^i R_p\). The second author recently generalized this notion of curvature homogeneity to the framework of affine differential geometry. A smooth connection \(\nabla\) on a differentiable manifold \(M\) is said to be curvature homogeneous up to order \(r\) if, for every pair of points \(p,q \in M\), there exists a linear isomorphism \(\varphi : T_pM \to T_qM\) such that \(\varphi\ast \nabla^i R_q = \nabla^i R_p\) for all \(i=0,1,\dots r\).
In the pseudo-Riemannian setting, the case of two-dimensional curvature homogeneous manifolds turns out to be trivial. Indeed, it is easily seen that a two-dimensional pseudo-Riemannian manifold which is curvature homogeneous (up to order zero) has constant sectional curvature. In dimension three, the situation becomes much more interesting. Many examples are known of three-dimensional curvature homogeneous (up to order zero) pseudo-Riemannian manifolds which are not locally homogeneous. In the Riemannian setting, curvature homogeneity up to order one is sufficient to characterize three-dimensional locally homogeneous manifolds. The same result holds in the four-dimensional case. In the framework of Lorentzian geometry, curvature homogeneity up to order two is sufficient to characterize locally homogeneous three-dimensional manifolds.
In the present paper, the authors show that, in the framework of affine geometry, already the case of two-dimensional manifolds is non-trivial. The main theorem of the paper states the following result: Let \(\nabla\) be a torsion-free analytic connection on an analytic two-dimensional manifold \(M\). If the Ricci tensor of \(\nabla\) is skew-symmetric, then curvature homogeneity up to order three implies local homogeneity. This bound cannot be improved. If the Ricci tensor of \(\nabla\) has non-trivial symmetrization, then curvature homogeneity up to order two implies local homogeneity of the manifold. Again, this bound cannot be improved.

MSC:
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53B05 Linear and affine connections
53C30 Differential geometry of homogeneous manifolds
53B20 Local Riemannian geometry
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