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Frame bundle
About: Frame bundle is a research topic. Over the lifetime, 1600 publications have been published within this topic receiving 23049 citations.
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TL;DR: In this paper, the tangent bundle of a closed, connected, non-orientable smooth manifold is embedded as a sub-bundle of a 2-plane bundle over a CW complex of dimension m or less.
Abstract: Let ζ be a nonorientable m-plane bundle over a CW complex X of dimension m or less Given a 2-plane bundle η over X, we wish to know whether η can be embedded as a sub-bundle of ζ The bundle η need not be orientable When ζ is even-dimensional there is the added complication of twisted coefficients In that case, we use Postnikov decomposition of certain nonsimple fibrations in order to describe the obstructions for the embedding problem Emery Thomas [11] and [12] treated this problem for ζ and η both orientable The results found here are applied to the tangent bundle of a closed, connected, nonorientable smooth manifold, as a special case
6 citations
TL;DR: In this article, the authors give an explicit computation of the spectral density function, by constructing certain quasimodes on the associated principle bundle, for a manifold with quantizing line bundle.
Abstract: For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin–Uribe showed that there is a well-defined cluster of low-lying eigenvalues, whose distribution is described by a spectral density function. We give an explicit computation of the spectral density function, by constructing certain quasimodes on the associated principle bundle.
6 citations
TL;DR: In this article, the authors studied the Riemannian metrics that arise when the manifold is R or S 1 with a constant connection and showed that the frame bundle admits a Riemanian metric which Schmidt introduced to construct the b-boundary.
Abstract: On a manifold with connection the frame bundle admits a Riemannian metric which Schmidt introduced to construct the b-boundary of the underlying manifold. Here we study the metrics that arise when the manifold is R or S1 with a constant connection.
6 citations
TL;DR: In this article, a generalization of Lagrange-Poincar\'e reduction scheme to these types of variational problems allows us to relate it with the Einstein-Hilbert variational problem.
Abstract: The present article deals with a formulation of the so called (vacuum) Palatini gravity as a general variational principle. In order to accomplish this goal, some geometrical tools related to the geometry of the bundle of connections of the frame bundle $LM$ are used. A generalization of Lagrange-Poincar\'e reduction scheme to these types of variational problems allows us to relate it with the Einstein-Hilbert variational problem. Relations with some other variational problems for gravity found in the literature are discussed.
6 citations
TL;DR: The most relevant geometrical aspects of the gauge theory of gravitation are considered in this paper, where a global definition of tetrad fields is given and emphasis is placed on their role in defining an isomorphism between the tangent bundle of space-time and an appropriate vector bundle B associated to the gauge bundle.
Abstract: The most relevant geometrical aspects of the gauge theory of gravitation are considered. A global definition of the tetrad fields is given and emphasis is placed on their role in defining an isomorphism between the tangent bundle of space-time and an appropriate vector bundle B associated to the gauge bundle. It is finally shown how to construct the fundamental geometrical objects on space-time, starting from B.
6 citations