Mathematics – Differential Geometry
Scientific paper
2009-05-20
Mathematics
Differential Geometry
minor changes
Scientific paper
A general framework for different averaging procedures is introduced. We motivate the existence of this framework through three examples: 1. The Averaging Principle that appears in Classical Mechanics, which is on the basis of Perturbation Theory, 2. The integration along the fiber leading to the Thom isomorphism theorem in Algebraic Topology and 3. The averaging of some linear connections in some pull-back bundles. The resulting {\it averaged connections} are affine connections on the tangent bundle of the manifold {\bf M}. The second motivation comes from the problem of, given vector bundle automorphism, to define a {\it push-forward} vector bundle automorphism. We will see that a definition of averaging exists such that it solves this problem and contains the three examples of averaging procedure described before. After this, we explain the notion of {\it convex invariance} [3] for the case of orientable Riemannian vector bundles. As a consequence, we consider a new approach to Finsler Geometry.
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