Topic
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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17 citations
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TL;DR: In this paper, it was shown that the cotangent bundle T ∗ T M of the tangent bundle of any differentiable manifold M carries an integrable almost tangent structure which is generated by a natural lifting procedure from the canonical almost-tent structure (vertical endomorphism) of T M.
17 citations
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TL;DR: In this paper, a necessary and sufficient criterion for there to exist a Fourier type decomposition of such a bundle was given, which is a reduction of the structure group to the finite rank unitary group U(n), viewed as the subgroup of LU(n) consisting of constant loops.
Abstract: In this paper we investigate bundles whose structure group is the loop group LU(n). These bundles are classified by maps to the loop space of the classifying space, LBU(n). Our main result is to give a necessary and sufficient criterion for there to exist a Fourier type decomposition of such a bundle �. This is essentially a decomposition ofasLC, whereis a finite dimensional subbundle ofand LC is the space of functions, C 1 (S 1 , C). The criterion is a reduction of the structure group to the finite rank unitary group U(n) viewed as the subgroup of LU(n) consisting of constant loops. Next we study the case where one starts with an n dimensional bundle � ! M classified by a map f : M ! BU(n) from which one constructs a loop bundle L� ! LM classified by Lf : LM ! LBU(n). The tangent bundle of LM is such a bundle. We then show how to twist such a bundle by elements of the automorphism group of the pull back ofover LM via the map LM ! M that evaluates a loop at a basepoint. Given a connection on �, we view the associated parallel transport operator as an element of this gauge group and show that twisting the loop bundle by such an operator satisfies the criterion and admits a Fourier decomposition.
16 citations
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TL;DR: In this article, the authors studied the von Neumann algebra V ⁎ (T ) consisting of operators commuting with both T and T from a geometric viewpoint and identified operators with connection-preserving bundle maps on E ( T ), the holomorphic Hermitian vector bundle associated to T.
16 citations
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01 Jan 1988
TL;DR: In this article, it was shown that the Dolbeault cohomology groups vanish if E is positive in the sense of Griffiths and p+q≥n+1, l≥r+C (n, p, q) vanish when E is a holmorphic vector bundle of rank r over a compact complex manifold X of dimension n.
Abstract: Let E be a holmorphic vector bundle of rank r over a compact complex manifold X of dimension n It is shown that the Dolbeault cohomology groups H p,q (X, E ⊗k ⊗(det E)l) vanish if E is positive in the sense of Griffiths and p+q≥n+1, l≥r+C (n, p, q) The proof rests on the wellknown fact that every tensor power E ⊗k splits into irreducible representations of Gl(E), each component being canonically isomorphic to the direct image on X of a positive homogeneous line bundle over a flag manifold of E The vanishing property is then obtained by a suitable generalization of Le Potier's isomorphism theorem, combined with a new curvature estimate for the bundle of X-relative differential forms on the flag manifold of E
16 citations