Metrization of connections with regular curvature. (English) Zbl 1212.53020

Summary: We discuss Riemannian metrics compatible with a linear connection that has regular curvature. Combining (mostly algebraic) methods and results of O. Kowalski [Math. Z. 125, 129–138 (1972; Zbl 0234.53024); Note Mat. 8, No. 1, 1–11 (1988; Zbl 0699.53038)], we give an algorithm which allows to decide effectively the existence of positive definite metrics compatible with a real analytic connection with regular curvature tensor on an analytic connected and simply connected manifold, and to construct the family of compatible metrics (determined up to a scalar multiple) in the affirmative case. We also briefly touch related problems concerning geodesic mappings and projective structures.


53B05 Linear and affine connections
53B20 Local Riemannian geometry
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