Frame bundle

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

Polarisation of Graded Bundles

Abstract: We construct the full linearisation functor which takes a graded bundle of degreek (a particular kind of graded manifold) and produces ak-fold vector bundle. We fully characterise the image of the full linearisation functor and show that we obtain a subcategory ofk-fold vector bundles consisting of symmetric k-fold vector bundles equipped with a family of morphisms indexed by the symmetric group Sk. Interestingly, for the degree 2 case this additional structure gives rise to the notion of a symplectical double vector bundle, which is the skew-symmetric analogue of a metric double vector bundle. We also discuss the related case of fully linearising N-manifolds, and how one can use the full linearisation functor to \superise" a graded bundle.

Classification of extensions of principal bundles and transitive Lie groupoids with prescribed kernel and cokernel

Abstract: The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well-known result. A remarkable generalization of this equivalence, given by Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admit an action of the structure group by automorphisms. In this paper the existence of suitably equivariant transition functions is proved for such groupoids, generalizing consequently the classification of principal bundles by means of their transition functions, to extensions of principal bundles by an equivariant form of Cech cohomology.

Transverse totally geodesic submanifolds of the tangent bundle

Abstract: It is well-known that ifis a smooth vector field on a given Rie- mannian manifold M n thennaturally defines a submanifold �(M n ) transverse to the fibers of the tangent bundle TM n with Sasaki metric. In this paper, we are interested in transverse totally geodesic subman- ifolds of the tangent bundle. We show that a transverse submanifold N l of TM n (1 ≤ l ≤ n) can be realized locally as the image of a sub- manifold F l of M n under some vector fielddefined along F l . For such images �(F l ), the conditions to be totally geodesic are presented. We show that these conditions are not so rigid as in the case of l = n, and we treat several special cases (� of constant length, � normal to F l , M n of constant curvature, M n a Lie group anda left invariant vector field).

General connections on vector bundles

Abstract: In this paper, general connections in the sense of T. Otsuki are dealt with. The general connections were defined by T. Otsuki in [01] as a generalized notion of usual connections. Recently they are called Otsuki connections in Europe. He defined the general connections on the tangent tensor bundles of a manifold and defined associating geometrical objects analogous to those of usual ones, for example, their torsion forms and curvature forms. In his papers [Ol]-[O11], several results about general connections were obtained. The purpose of this paper is to define general connections on vector bundles over a manifold and to study the fundamental properties. In § 1, we will prepare notations used in this paper and fundamental facts on the 1-jet bundle of a vector bundle. In §2, the general connections will be defined and some algebraic properties of the space of covariant derivatives of general connections will be studied. In § 3, we will define two types of induced general connections which are induced by a pair of vector bundle homomorphisms and by a bundle map. In § 4, using given general connections, we will construct general connections on the dual bundle and on the tensor product bundles. In § 5, we will define the curvature form of a general connection. The author would like to express his hearty thanks to Professor T. Otsuki for his helpful advice. He also would like to acknowledge the constant encouragement of Professor S. Yamaguchi.

On the torsion-free connections on higher order frame bundles

Abstract: Using the r-jets of flows of vector fields, we show that every torsion-free linear r-th order connection on the tangent bundle of a manifold M determines a reduction of the (r+1)-st order frame bundle of M to the general linear group We deduce that this reduction coincides with another reduction constructed earlier