scholarly journals The extension of G-foliations to Tangent bundles of higher order

1975 ◽  
Vol 56 ◽  
pp. 29-44 ◽  
Author(s):  
Luis A. Cordero
Keyword(s):  
Infinite Sequence ◽  
Differentiable Manifold ◽  
Higher Order ◽  
Characteristic Classes ◽  
Tangent Bundles

In this paper we describe a canonical procedure for constructing the extension of a G-foliation on a differentiable manifold X to its tangent bundles of higher order and by applying the Bott-Haefliger’s construction of characteristic classes of G-foliations ([2], [3]) we obtain an infinite sequence of characteristic classes for those foliations (Theorem 4.8).

scholarly journals A Note on Tangent Bundles

1967 ◽  
Vol 29 ◽  
pp. 259-267 ◽  
Author(s):  
Kenichi Shiraiwa
Keyword(s):  
Homotopy Type ◽  
Tangent Bundle ◽  
Topological Structure ◽  
Differentiable Manifold ◽  
Differentiable Structure ◽  
Homotopy Types ◽  
Tangent Bundles

The tangent bundle of a differentiable manifold is an important invariant of a differentiable structure. It is determined neither by the topological structure nor by the homotopy type of a manifold. But in some cases tangent bundles depend only on the homotopy types of manifolds.

Generalized Bianchi identities for horizontal distributions

1983 ◽  
Vol 94 (1) ◽  
pp. 125-132 ◽  
Author(s):  
M. Crampin
Keyword(s):  
Lie Algebra ◽  
Tangent Bundle ◽  
Linear Connection ◽  
Differentiable Manifold ◽  
Horizontal Distribution ◽  
Bianchi Identities ◽  
Graded Lie Algebra ◽  
Image Position ◽  
Tangent Bundles

A linear connection on a differentiable manifold M defines a horizontal distribution on the tangent bundle T(M). Horizontal distributions on tangent bundles are of some interest even when they are not generated by connections. Much of linear connection theory generalizes to arbitrary horizontal distributions. In particular, there are generalized versions of Bianchi's two identities for the torsion T and curvature Rwhere C denotes the cyclic sum with respect to X, Y and Z. These identities are derived below by associating with a horizontal distribution a graded derivation of degree 1 in a graded Lie algebra of vertical forms on M. This approach reveals the fundamentally algebraic origin of the Bianchi identities.

Non-embedding theorems for Y-spaces

1967 ◽  
Vol 63 (3) ◽  
pp. 601-612 ◽  
Author(s):  
K. H. Mayer ◽  
R. L. E. Schwarzenberger
Keyword(s):  
Lower Bounds ◽  
Vector Bundles ◽  
Differentiable Manifold ◽  
Characteristic Classes ◽  
Embedding Theorems ◽  
Geometric Properties ◽  
Rank 2 ◽  
Dimension 2

Let X be a compact differentiable manifold of dimension 2m. A differentiable map from X to euclidean (2m + t)-space is an immersion if its Jacobian has rank 2m at each point of X; it is an embedding if it is also one–one. The existence of such an embedding or immersion implies that the characteristic classes of X satisfy certain integrality conditions; these can be used to obtain lower bounds for the integer t. In a similar way many other geometric properties of X can be deduced from a single integrality theorem involving characteristic classes of various vector bundles over X (see for instance (5)).

scholarly journals Liftings of Some Types of Tensor Fields and Connections to Tangent Bundles of pr-Velocities

1970 ◽  
Vol 40 ◽  
pp. 13-31 ◽  
Author(s):  
Akihiko Morimoto
Keyword(s):  
Vector Fields ◽  
Higher Order ◽  
Tensor Fields ◽  
Multi Index ◽  
Differentiable Functions ◽  
Tangent Bundles

In the previous paper [6], we studied the liftings of tensor fields to tangent bundles of higher order. The purpose of the present paper is to generalize the results of [6] to the tangent bundles of pr-velocities in a manifold M— notions due to C. Ehresmann [1] (see also [2]). In §1, we explain the pr-velocities in a manifold and define the (Λ)-lifting of differentiable functions for any multi-index λ -(λ1, λ2,…,λp) of non-negative integers λi satisfying ΣΛi≤r. In § 2, we construct ‹Λ›-lifts of any vector fields and ‹Λ›-lifts of 1-forms. The ‹Λ›-lift is a little bit different from the ‹Λ›-lift of vector fields in [6].

scholarly journals Prolongations of Connections to Tangential Fibre Bundles of Higher Order

1970 ◽  
Vol 40 ◽  
pp. 85-97 ◽  
Author(s):  
Akihiko Morimoto
Keyword(s):  
Tangent Bundle ◽  
Higher Order ◽  
Fibre Bundles ◽  
Affine Connections ◽  
Tangent Bundles

In the previous paper [3] we have studied the prolongations of G-structures to tangent bundles of higher order. The purpose of the present paper is to study the prolongations of connections to tangential fibre bundles of higher order, and to generalize the results due to S. Kobayashi [1] for the case of usual tangent bundle —— in fact, the arguments in [1] will be, in a sense, more or less simplified and clarified by using the notion of tangent bundles of higher order. In addition, as a consequence of our results, we shall obtain the prolongations of linear (affine) connections to tangent bundles of higher order.

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