Affine connections and distributions
Lewis, Andrew D.
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We investigate various aspects of the interplay of an affine connection with a distribution. When the affine connection restricts to the distribution, we discuss torsion, curvature, and holonomy of the affine connection. We also investigate transformations which respect both the affine connection and the distribution. A stronger notion than that of restricting to a distribution is that of geodesic invariance. This is a natural generalisation to a distribution of the idea of a totally geodesic submanifold. We provide a product for vector fields which allows one to test for geodesic invariance in the same way one uses the Lie bracket to test for integrability. If the affine connection does not restrict to the distribution, we are able to define its restriction and in the process generalise the notion of the second fundamental form for submanifolds. We characterise some transformations of these restricted connections and derive conservation laws in the case when the original connection is the Levi-Civita connection associated with a Riemannian metric.