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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 authors use a form of Whitehead's concept of crossed module, in place of the idea of an abstract kernel, to find an obstruction class in ǫ 2(B,ZG) (G the fibre-type of M) whose vanishing gives a necessary and sufficient condition for the existence of such a Lie group bundle isomorphic to M.
35 citations
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TL;DR: In this article, the authors show that for the tautological bundle tensor L tensor the determinant bundle associated to A vanish and the space of global sections is computed in terms of $H^0(A)$ and$H^ 0(L\otimes A)$ on the moduli space of rank 2 semi-stable sheaves on the projective plane, supporting Le Potier's Strange duality conjecture.
Abstract: We compute the cohomology spaces for the tautological bundle tensor the determinant bundle on the punctual Hilbert scheme H of subschemes of length n of a smooth projective surface X. We show that for L and A invertible vector bundles on X, and w the canonical bundle of X, if $w^{-1}\otimes L$, $w^{-1}\otimes A$ and A are ample vector bundles, then the higher cohomology spaces on H of the tautological bundle associated to L tensor the determinant bundle associated to A vanish, and the space of global sections is computed in terms of $H^0(A)$ and $H^0(L\otimes A)$. This result is motivated by the computation of the space of global sections of the determinant bundle on the moduli space of rank 2 semi-stable sheaves on the projective plane, supporting Le Potier's Strange duality conjecture on the projective plane.
35 citations
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TL;DR: In this article, the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers is presented.
Abstract: A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its external powers, one can define an operator similar to the exterior derivative. We present in this paper the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers. All the results (which, for their most part, are already known) are given with detailed proofs. In the final sections, the results are applied to Poisson manifolds.
35 citations
TL;DR: For every pointwise polynomial function on each fiber of the cotangent bundle of a Riemannian manifold M, a family of diffential operators is given, which acts on the space of smooth sections of a vector bundle on M as discussed by the authors.
Abstract: For every pointwise polynomial function on each fiber of the cotangent bundle of a Riemannian manifold M, a family of diffential operators is given, which acts on the space of smooth sections of a vector bundle on M. Such a correspondence may be considered as a rule to quantize classical systems moving in a Riemannian manifold or in a gauge field. Some applications of our construction are also given in this paper
35 citations
TL;DR: In this article, it was shown that a vector bundle X admits an endomorphism of degree > 1 and commuting with the projection to the base, if and only if X trivializes after a finite covering.
Abstract: Let X be a projective bundle. We prove that X admits an endomorphism of degree >1 and commuting with the projection to the base, if and only if X trivializes after a finite covering. When X is the projectivization of a vector bundle E of rank 2, we prove that it has an endomorphism of degree >1 on a general fiber only if E splits after a finite base change.
35 citations