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2009•Czech digital mathematics libraryOpen access

Metrization of connections with regular curvature

Alena Vanžurová

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Abstract

summary:We discuss Riemannian metrics compatible with a linear connection that has regular curvature. Combining (mostly algebraic) methods and results of [4] and [5] we give an algorithm which allows to decide effectively 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 breafly touch related problems concerning geodesic mappings and projective structures.

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summary:We discuss Riemannian metrics compatible with a linear connection that has regular curvature. Combining (mostly algebraic) methods and results of [4] and [5] we give an algorithm which allows to decide effectively 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 breafly touch related problems concerning geodesic mappings and projective structures.

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Available abstract

summary:We discuss Riemannian metrics compatible with a linear connection that has regular curvature. Combining (mostly algebraic) methods and results of [4] and [5] we give an algorithm which allows to decide effectively 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 breafly touch related problems concerning geodesic mappings and projective structures.

Key concepts: Mathematics, Connection (principal bundle), Scalar curvature, Geodesic, Riemann curvature tensor, Curvature, Curvature form, Pure mathematics

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