Affine connections and distributions
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Authors
Lewis, Andrew D.
Date
1996
Type
Language
en_US
Keyword
Alternative Title
Affine connections and distributions with applications to nonholonomic mechanics
Abstract
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.
Description
Appears as ``Affine connections and distributions with applications
to nonholonomic mechanics'' in Reports on Mathematical Physics
42(1/2), pages 135-164, 1998 (Proceedings for the workshop on
Non-Holonomic Constraints in Dynamics held in Calgary in August
1997)