Vector Fields and Other Vector Bundle Morphisms — A Singularity Approach

1981 ◽  
Author(s):  
Ulrich Koschorke
Keyword(s):  
Vector Bundle ◽  
Vector Fields

A GENERALIZATION OF THE ATIYAH–DUPONT VECTOR FIELDS THEORY

2002 ◽  
Vol 04 (04) ◽  
pp. 777-796 ◽  
Author(s):  
ZIZHOU TANG ◽  
WEIPING ZHANG
Keyword(s):  
Vector Bundle ◽  
Cross Sections ◽  
Vector Fields ◽  
Complex Structure ◽  
Index Theory ◽  
Index Theorem ◽  
Obstruction Theory ◽  
Real Vector ◽  
Complete Answer ◽  
Linearly Independent

To generalize the Hopf index theorem and the Atiyah–Dupont vector fields theory, one is interested in the following problem: for a real vector bundle E over a closed manifold M with rank E = dim M, whether there exist two linearly independent cross sections of E? We provide, among others, a complete answer to this problem when both E and M are orientable. It extends the corresponding results for E = TM of Thomas, Atiyah, and Atiyah–Dupont. Moreover we prove a vanishing result of a certain mod 2 index when the bundle E admits a complex structure. This vanishing result implies many known famous results as consequences. Ideas and methods from obstruction theory, K-theory and index theory are used in getting our results.

scholarly journals Iterative calculus on tangent floors

2016 ◽  
Vol 24 (1) ◽  
pp. 121-152
Author(s):  
Vladimir Balan ◽  
Maido Rahula ◽  
Nicoleta Voicu
Keyword(s):  
Gauge Theory ◽  
Vector Bundle ◽  
Lie Groups ◽  
Vector Fields ◽  
Essential Role ◽  
Covariant Differentiation

AbstractTangent fibrations generate a “multi-floored tower”, while raising from one of its floors to the next one, one practically reiterates the previously performed actions. In this way, the "tower" admits a ladder-shaped structure. Raising to the first floors suffices for iteratively performing the subsequent steps. The paper mainly studies the tangent functor. We describe the structure of multiple vector bundle which naturally appears on the floors, tangent maps, sector-forms, the lift of vector fields to upper floors. Further, we show how tangent groups of Lie groups lead to gauge theory, and explain in this context the meaning of covariant differentiation. Finally, we will point out within the floors special subbundles - the osculating bundles, which play an essential role in classical theories.

Positivity of metrics on conic neighborhoods of 1-convex submanifolds

2016 ◽  
Vol 27 (05) ◽  
pp. 1650047 ◽  
Author(s):  
Jasna Prezelj
Keyword(s):  
Vector Bundle ◽  
Holomorphic Function ◽  
Vector Fields ◽  
Holomorphic Vector Bundle ◽  
Holomorphic Section ◽  
Exceptional Set ◽  
Exhaustion Function ◽  
Holomorphic Vector Fields ◽  
Small Times ◽  
Bounded Holomorphic

Let [Formula: see text] be a holomorphic submersion from a complex manifold [Formula: see text] onto a 1-convex manifold [Formula: see text] with exceptional set [Formula: see text] and [Formula: see text] a holomorphic section. Let [Formula: see text] be a plurisubharmonic exhaustion function which is strictly plurisubharmonic on [Formula: see text] with [Formula: see text] For every holomorphic vector bundle [Formula: see text] there exists a neighborhood [Formula: see text] of [Formula: see text] for [Formula: see text] conic along [Formula: see text] such that [Formula: see text] can be endowed with Nakano strictly positive Hermitian metric. Let [Formula: see text] [Formula: see text] be a given holomorphic function. There exist finitely many bounded holomorphic vector fields defined on a Stein neighborhood [Formula: see text] of [Formula: see text] conic along [Formula: see text] with zeroes of arbitrary high order on [Formula: see text] and such that they generate [Formula: see text] Moreover, there exists a smaller neighborhood [Formula: see text] such that their flows remain in [Formula: see text] for sufficiently small times thus generating a local dominating spray.

scholarly journals Infinitesimal Automorphisms of VB-Groupoids and Algebroids

2019 ◽  
Vol 70 (3) ◽  
pp. 1039-1089 ◽  
Author(s):  
Chiara Esposito ◽  
Luca Vitagliano ◽  
Alfonso Giuseppe Tortorella
Keyword(s):  
Vector Bundle ◽  
Special Class ◽  
Vector Fields ◽  
Vector Bundles ◽  
Lie Algebroids ◽  
Lie Groupoids ◽  
Singular Spaces ◽  
Infinitesimal Automorphisms

Abstract VB-groupoids and algebroids are vector bundle objects in the categories of Lie groupoids and Lie algebroids, respectively, and they are related via the Lie functor. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. Additionally, they can be seen as models for vector bundles over singular spaces. In this paper we study their infinitesimal automorphisms, i.e. vector fields on them generating a flow by diffeomorphisms preserving both the linear and the groupoid/algebroid structures. For a special class of VB-groupoids/algebroids coming from representations of Lie groupoids/algebroids, we prove that infinitesimal automorphisms are the same as multiplicative sections of a certain derivation VB-groupoid/algebroid.

A family of reflexive vector bundles of reduction number one

2019 ◽  
Vol 124 (2) ◽  
pp. 188-202
Author(s):  
Cleto B. Miranda-Neto
Keyword(s):  
Vector Bundle ◽  
Commutative Algebra ◽  
Vector Fields ◽  
Vector Bundles ◽  
Rees Algebra ◽  
Reduction Number ◽  
Difficult Issue

A difficult issue in modern commutative algebra asks for examples of modules (more interestingly, reflexive vector bundles) having prescribed reduction number $r\geq 1$. The problem is even subtler if in addition we are interested in good properties for the Rees algebra. In this note we consider the case $r=1$. Precisely, we show that the module of logarithmic vector fields of the Fermat divisor of any degree in projective $3$-space is a reflexive vector bundle of reduction number $1$ and Gorenstein Rees ring.

Normal Forms and Bifurcation of Planar Vector Fields

1994 ◽  
Author(s):  
Shui-Nee Chow ◽  
Chengzhi Li ◽  
Duo Wang
Keyword(s):  
Vector Fields ◽  
Normal Forms ◽  
Planar Vector Fields ◽  
Planar Vector
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