Frame bundle

About: Frame bundle is a research topic. Over the lifetime, 1600 publications have been published within this topic receiving 23049 citations.

Note on generalized connections and affine bundles

Abstract: We develop an alternative view on the concept of connections over a vector bundle map, which consists of a horizontal lift procedure to a prolonged bundle. We further focus on prolongations to an affine bundle and introduce the concept of affineness of a generalized connection.

On canonical Cartan connections associated to filtered G-structures

Abstract: A filtered manifold is a smooth manifold $M$ together with a filtration of the tangent bundle by smooth subbundles which is compatible with the Lie bracket of vector fields in a certain sense. The Lie bracket of vector fields then induces a bilinear operation on the associated graded of each tangent space of $M$ making it into a nilpotent graded Lie algebra. Assuming that these symbol algebras are the same for all points, one obtains a natural frame bundle for the associated graded to the tangent bundle, and filtered G--structures are defined as reductions of structure group of this bundle. Generalizing the case of parabolic geometries, this article is devoted to the question of whether a filtered G-structure of given type determines a canonical Cartan connection on an extended bundle. As for existence, the result are roughly as general as Morimoto's theorem from 1993, but it has several specific features. First, we allow for general candidates for a homogeneous model and a general version of normalization conditions. Second, the construction is entirely phrased in terms of Lie algebra valued forms and leads to an explicit characterization of the canonical Cartan connection. To verify that the procedure can be applied to a given type of filtered G-structures, only finite dimensional algebraic verifications have to be carried out.

Tensor fields and connections on a cross-section in the tangent bundle of order r

Abstract: Let M be an ^-dimensional differentiable manifold and Tr(M) the tangent bundle of order r over M, ri^l being an integer [1], [3], [4]. The prolongations of tensor fields and connections given in the differentiable manifold M to its tangent bundle of order r have been studied in [1J, [2], [3] [4], [7], [8] and [9]. If V is a vector field given in M, V determines a cross-section in Tr(M). For the cases r=l and r=2, Yano [7] and Tani [5] have studied, on the cross-section determined by a vector field F, the behavior of the prolongations of tensor fields and connections in M to T(M) (i.e., Ά(M)) and Γa(M), respectively. The purpose of this paper is to study, on the cross-section determined by a vector field V, the behavior of the prolongations of these geometric objects in M to Tr(M) ( r^ l ) . In §1 we summarize the results and properties we need concerning the prolongations of tensor fields and connections in M to Tr(M). Proofs of the statements in §1 can be found in [1], [2], [3], [4] and [8]. In § 2 we study the cross-section determined in Tr(M) by a given vector field V in M In § 3 we study the behavior of prolongations of tensor fields on the cross-section. In §4 we study the prolongations of connections given in M to Tr{M) along the cross-section and some of their properties. We assume in the squel that the manifolds, functions, tensor fields and connections under consideration are all of differentiability of class C°°. Several kinds of indices are used as follows: The indices λ, μ, v, • ••, s, t, u, ••• run through the range 0,1, 2, ••• r; the indices h, i, j , k, m, ••• run through the range 1, 2, ••• n. Double indices like {v)h are used, where O^i^r, l^h^n. The indices Λ, B, C, ••• run through the range (1)1, (1)2, •••, (l)n, (2)1, •••, (2)n, •••, (r)l, •••, (r)n. For a given function / on M, the notation / ( 0 ) is sometimes substituted by f° for simplicity. Summation notation Σ J β l with respect to h> i, j , k, m, ••• (=1, 2, ••• n) is omitted while summation notation with respect to λ, μ, v, •••, s, t, u •••, from 0 to r, will be kept. For example,