The Equality of a Manifold's Rank and Dimension

1969 ◽  
Vol 12 (2) ◽  
pp. 183-184
Author(s):  
R. S. D. Thomas
Keyword(s):  
Vector Fields ◽  
Present Note ◽  
Linearly Independent ◽  
Commuting Vector Fields

T. J. Willmore has shown that if a differentiable manifold's rank (the maximum number of everywhere linearly independent commuting vector fields definable on it) equals the manifold's dimension, then the manifold is a torus of the appropriate dimension [1]. This theorem is proved more simply and without any differentiability hypothesis in the present note.

scholarly journals EXTENDED ABSOLUTE PARALLELISM GEOMETRY

2008 ◽  
Vol 05 (07) ◽  
pp. 1109-1135 ◽  
Author(s):  
NABIL. L. YOUSSEF ◽  
A. M. SID-AHMED
Keyword(s):  
Tangent Bundle ◽  
Special Form ◽  
Vector Fields ◽  
A Priori ◽  
Building Blocks ◽  
Absolute Parallelism ◽  
Linearly Independent ◽  
Geometric Objects ◽  
Directional Argument ◽  
Curvature Tensors

In this paper, we study Absolute Parallelism (AP-) geometry on the tangent bundle TM of a manifold M. Accordingly, all geometric objects defined in this geometry are not only functions of the positional argument x, but also depend on the directional argument y. Moreover, many new geometric objects, which have no counterpart in the classical AP-geometry, emerge in this different framework. We refer to such a geometry as an Extended Absolute Parallelism (EAP-) geometry. The building blocks of the EAP-geometry are a nonlinear connection (assumed given a priori) and 2n linearly independent vector fields (of special form) defined globally on TM defining the parallelization. Four different d-connections are used to explore the properties of this geometry. Simple and compact formulae for the curvature tensors and the W-tensors of the four defined d-connections are obtained, expressed in terms of the torsion and the contortion tensors of the EAP-space. Further conditions are imposed on the canonical d-connection assuming that it is of Cartan type (resp. Berwald type). Important consequences of these assumptions are investigated. Finally, a special form of the canonical d-connection is studied under which the classical AP-geometry is recovered naturally from the EAP-geometry. Physical aspects of some of the geometric objects investigated are pointed out and possible physical implications of the EAP-space are discussed, including an outline of a generalized field theory on the tangent bundle TM of M.

scholarly journals Commuting planar polynomial vector fields for conservative Newton systems

2019 ◽  
Vol 22 (04) ◽  
pp. 1950025 ◽  
Author(s):  
Joel Nagloo ◽  
Alexey Ovchinnikov ◽  
Peter Thompson
Keyword(s):  
Differential Equation ◽  
Vector Field ◽  
Elliptic Equation ◽  
Characteristic Zero ◽  
Classical Result ◽  
Vector Fields ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Linearly Independent ◽  
Planar Vector

We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. Let [Formula: see text], where [Formula: see text] is a field of characteristic zero, and [Formula: see text] the derivation that corresponds to the differential equation [Formula: see text] in a standard way. Let also [Formula: see text] be the Hamiltonian polynomial for [Formula: see text], that is [Formula: see text]. It is known that the set of all polynomial derivations that commute with [Formula: see text] forms a [Formula: see text]-module [Formula: see text]. In this paper, we show that, for every such [Formula: see text], the module [Formula: see text] is of rank [Formula: see text] if and only if [Formula: see text]. For example, the classical elliptic equation [Formula: see text], where [Formula: see text], falls into this category.

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.

Commuting Vector Fields on S 3

1965 ◽  
Vol 81 (1) ◽  
pp. 70 ◽  
Author(s):  
Elon L. Lima
Keyword(s):  
Vector Fields ◽  
Commuting Vector Fields
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