Affine geometries defined by fiber preserving diffeomorphisms

Abstract

Clearly a given manifold M may support more than one metric tensor and generally one may select a particular metric on M via a variational procedure defined on the class of all metrics on M. Obviously the class of all metrics on M is a subset of the set of all sections of the vector bundle T 0 2 M and thus one has a rigorous framework for any theory which has as its goal the selection of a metric in this way (in particular, general relativity is such a theory). It is our purpose to develop such a framework for affine geometry. We do not consider specific procedures to select an affine geometry analogous to the selection of a metric via the variation of some Lagrangian, but we establish the arena where such procedures would be meaningful. In the case of Riemannian geometry, this arena would be the set of all sections of the finite dimensional vector bundle T 0 2 M and in this context it is important that covariant derivatives of such sections are again sections of the same bundle. Moreover, the covariant derivative of a given metric is relatively simple as it arises from a linear action of Gℓ( n, R on a typical fiber of T 0 2 M. In the case of affine geometry we find that the appropriate arena is the set of all sections of an infinite dimensional vector bundle ν(E, V, δ). Moreover, since the group Aff (δ) relating åchange of basiså is not compact and acts on the fiber of this infinite dimensional bundle, it turns out that the group action is not generally continuous when one uses the Whitney C ∞ topology on a typical fiber; rather one must use the Schwartz C ∞ -topology widely used in the theory of distributions to obtain a differentiable action. Moreover, the action of the group on a typical fiber is not linear so that the usual formulas for covariant derivatives must be modified. An interesting consequence of our investigation is that the vector bundle ν can be extended and the action of the group also extended so that the nonlinearity is only a «second order nonlinearityå, i.e., formulas for the covariant derivative of a section involve only linear terms and bilinear terms (see Equation 4.2). In addition to this feature, formulas for covariant derivatives of affine geometries are developed which are fully analogous to those for covariant derivatives of Riemannian metrics (see Theorem 4.2). Symmetry-breaking properties associated with special classes of affine structures are obtained and parallels are drawn with the metric case (recall that a metric reduces the linear frame bundle to the bundle of orthonormal frames, symmetry is broken from the general linear group to the orthogonal group). In the last section of the paper we show how our formalism relates to certain already-developed applications of affine geometry to charged particle dynamics worked out in more detail by Norris and his collaborators.