Structure of finite dimensional linear bundles morphisms (original) (raw)
Lecture Notes in Physics, 2008
The notion of vector bundle is a basic extension to the geometric domain of the fundamental idea of a vector space. Given a space X, we take a real or complex finite dimensional vector space V and make V the fibre of a bundle over X, where each fibre is isomorphic to this vector space. The simplest way to do this is to form the product X × V and the projection pr X : X × V → X onto the first factor. This is the product vector bundle with base X and fibre V .
On a Lie Algebraic Characterization of Vector Bundles
Symmetry, Integrability and Geometry: Methods and Applications, 2012
We prove that a vector bundle π : E → M is characterized by the Lie algebra generated by all differential operators on E which are eigenvectors of the Lie derivative in the direction of the Euler vector field. Our result is of Pursell-Shanks type but it is remarkable in the sense that it is the whole fibration that is characterized here. The proof relies on a theorem of [Lecomte P., J. Math. Pures Appl. (9) 60 (1981), 229-239] and inherits the same hypotheses. In particular, our characterization holds only for vector bundles of rank greater than 1.
The Topology of Fiber Bundles Lecture Notes
1998
5.4. Applications II: H * (ΩS n) and H * (U (n)) 162 5.5. Applications III: Spin and Spin C structures 165 Bibliography v Corollary 1.9. A vector bundle ζ : p : E → B is trivial if and only if its associated principal GL(n)-bundle p : E GL → B admits a section.
Forum Mathematicum, 2000
A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category F of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theorem.
A G ] 1 0 N ov 2 01 7 An elementary transformation of vector bundles in P n
2018
By considering the equivalence between the category of locally free sheaves and the category of algebraic vector bundles, we show how elementary transformations of vector bundles can be used to prove a case of the maximal rank hypothesis. We in turn show how this can be applied in the study of minimal free resolutions.
Isomorphism classes for higher order tangent bundles
Advances in Geometry, 2017
The tangent bundle T k M of order k, of a smooth Banach manifold M consists of all equivalent classes of curves that agree up to their accelerations of order k. In the previous work of the author he proved that T k M , 1 ≤ k ≤ ∞, admits a vector bundle structure on M if and only if M is endowed with a linear connection or equivalently a connection map on T k M is defined. This bundle structure depends heavily on the choice of the connection. In this paper we ask about the extent to which this vector bundle structure remains isomorphic. To this end we define the notion of the k'th order differential T k g : T k M −→ T k N for a given differentiable map g between manifolds M and N. As we shall see, T k g becomes a vector bundle morphism if the base manifolds are endowed with g-related connections. In particular, replacing a connection with a g-related one, where g : M −→ M is a diffeomorphism, follows invariant vector bundle structures. Finally, using immersions on Hilbert manifolds, convex combination of connection maps and manifold of C r maps we offer three examples to support our theory and reveal its interaction with the known problems such as Sasaki lift of metrics.
Local Properties and Differential Forms of Smooth Map and Tangent Bundle
Mathematical Theory and Modeling, 2013
In this paper, some basic properties of differential forms of smooth map and tangent bundle are developed. The li n ear m ap d e fi n ed b y is t he d eri va tive of at a, where is a smooth map and. If is a smooth bijective ma p and if the maps are all injective, then is a diffeomorphism. Finally, it is shown that if X is a vector field on a manifold M then there is a radial neighbourhood of in and a smooth map such that gh and , where is given by .
On an elementary transformation of vector bundles in P^n
arXiv: Algebraic Geometry, 2017
By considering the equivalence between the category of locally free sheaves and the category of algebraic vector bundles, we show how elementary transformations of vector bundles can be used to prove a case of the maximal rank hypothesis. We in turn show how this can be applied in the study of minimal free resolutions.