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Unit tangent bundle
About: Unit tangent bundle is a research topic. Over the lifetime, 1056 publications have been published within this topic receiving 15845 citations.
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TL;DR: The amplitude, phase and state of polarization of an electromagnetic monochromatic plane wave is expressed in terms of a two-component (SU(2)) spinor, which can be represented by a tangent vector to the Poincare sphere as mentioned in this paper.
Abstract: The amplitude, phase and state of polarization of an electromagnetic monochromatic plane wave is expressed in terms of a two-component (SU(2)) spinor, which can be represented by a tangent vector to the Poincare sphere. It is shown that the Hermitian interior product between spinors involves the parallel transport of tangent vectors along the geodesics of the sphere and that two waves are in phase, according to Pancharatnam's definition, when the tangent vectors to the sphere representing the two waves are parallel to each other along the great circle arc joining the corresponding points of the sphere.
2 citations
TL;DR: In this paper, the authors studied the prolongation of semibasic projectable tangent valued k-forms on fibered manifolds with respect to a bundle functor F on local isomorphisms that uses an auxiliary linear r-th order connection on the base manifold.
Abstract: We study the prolongation of semibasic projectable tangent valued k-forms on fibered manifolds with respect to a bundle functor F on local isomorphisms that is based on the flow prolongation of vector fields and uses an auxiliary linear r-th order connection on the base manifold, where r is the base order of F. We find a general condition under which the Frolicher-Nijenhuis bracket is preserved. Special attention is paid to the curvature of connections. The first order jet functor and the tangent functor are discussed in detail. Next we clarify how this prolongation procedure can be extended to arbitrary projectable tangent valued k-forms in the case F is a fiber product preserving bundle functor on the category of fibered manifolds with m-dimensional bases and local diffeomorphisms as base maps.
2 citations
TL;DR: In this article, the authors study the sheaf of differential operators on a flag manifold X in characteristic p>0 and generalize the non-vanishing theorem of Haastert on the associated filtration.
Abstract: In this paper we study the sheaf of differential operators \(\) on a flag manifold X in characteristic p>0. We generalize the non-vanishing theorem of Haastert on the associated filtration of \(\). We make use of the lifting of X to characteristic zero and the complex geometry of X.
2 citations
TL;DR: The Tangent Bundle of a Topological Manifold (TBM) as discussed by the authors is a topological manifold that can be used to represent the topological structure of a manifold and its topology.
Abstract: (1972). The Tangent Bundle of a Topological Manifold. The American Mathematical Monthly: Vol. 79, No. 10, pp. 1090-1096.
2 citations
TL;DR: In this article, the authors generalized the projection results concerning the dimension spectrum of projected measures on R n to parametrized families of transversal mappings between smooth manifolds and measures on them.
Abstract: In this work we first generalize the projection results concerning the dimension spectrum of projected measures on R n to parametrized families of transversal mappings between smooth manifolds and measures on them. The projection theorems for the lower q-dimension were first considered in (FO) and (HK). Theorems for the upper q-dimension were first considered in (FO) and (JJ). After proving the generalized results, we compute for 1 < q • 2 the lower and the upper q-dimensions of the natural projection of a probability measure which is invariant under the geodesic flow on the unit tangent bundle of a two-dimensional Riemann manifold.
2 citations