Frame bundle

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

Cotangent bundle approach to noninertial frames

Abstract: The most general possible noninertial acceleration in special relativity is formulated with differential forms in the cotangent bundle. We show that the Lie derivative plays the same role in the cotangent bundle that the covariant derivative plays in the tangent bundle. We also show that a cotangent bundle analog of Fermi–Walker transport can be based upon the, ’’cotangent‐geodesic’’ equation, Luω=0. This gives a generalization of the work by Kiehn on classical Hamiltonian mechanics to special relativity.

Holomorphic Vector Bundle on Hopf Manifolds with Abelian Fundamental Groups

Abstract: Let X be a Hopf manifolds with an Abelian fundamental group. E is a holomorphic vector bundle of rank r with trivial pull-back to W = ℂn–{0}. We prove the existence of a non-vanishing section of L⊗E for some line bundle on X and study the vector bundles filtration structure of E. These generalize the results of D. Mall about structure theorem of such a vector bundle E.

Remarks on the problem of lifting spacetime symmetries

Abstract: The problem of lifting the action of a symmetry group K on spacetime M to automorphisms of a principal bundle P(M, G) is discussed. A classification of bundles P admitting a lift is given for a case more general than that considered by Harnad, Shnider, and Vinet.

A fiber bundle theorem

Abstract: A surjective submersion π: M→B from a pseudoriemannian manifold M is a fibre bundle, if its fibres are connected, totally geodesic and geodesically complete submanifolds of M.

Skew Symmetric Bundle Maps on Space-time

Abstract: We study the \Lie Algebra" of the group of Gauge Transforma- tions of Space-time. We obtain topological invariants arising from this Lie Algebra. Our methods give us fresh mathematical points of view on Lorentz Transformations, orientation conventions, the Doppler shift, Pauli matrices, Electro-Magnetic Duality Rotation, Poynting vectors, and the Energy Mo- mentum Tensor T. LetM be a space-time andT (M) its tangent bundle. ThusM is a 4-dimensional manifold with a nondegenerate inner producth ;i on T (M) of index + ++. We study the space of bundle maps F : T (M)! T (M) which are skew symmetric with respect to the metric, i.e. hFv;vi =0f or allv2 Tx(M )a nd allx2 M. A skew symmetric F has invariant planes and eigenvector lines in each Tx(M). We give necessary and sucient conditions as to when these plane systems and line systems form subbundles in Theorem 7.3. Also we determine the space of those F which give the same underlying structure. This is done by introducing the bundle map TF = FF 1 (tr F 2 )I : T (M)! T (M). Then the space of skew symmetric F which give rise to the same T is homeomorphic to Map(M;S 1 ), the space of maps of M into the circle S 1 . (See Theorem 7.11.) We also show that the space of skew symmetric F has a natural complexica- tion. (see Propositions 2.2 and 2.3) This leads to an equivalence between the F and vector elds on the complexied tangent bundle T (M)C. The complexied study leads to several beautiful relations which link our subject matter to Cliord Algebras and Quaternions. (See Corollaries 4.6 and 4.7 and Theorem 4.8.) We naturally nd many points of contact with Physics, especially classical electromag- netism. These considerations frequently govern our choice of notation. The physical motivations and remarks will be explored in the Scholia; and the mathematical mo- tivations and links will be found in the Remarks.