scholarly journals A Note on Tangent Bundles

1967 ◽  
Vol 29 ◽  
pp. 259-267 ◽  
Author(s):  
Kenichi Shiraiwa
Keyword(s):  
Homotopy Type ◽  
Tangent Bundle ◽  
Topological Structure ◽  
Differentiable Manifold ◽  
Differentiable Structure ◽  
Homotopy Types ◽  
Tangent Bundles

The tangent bundle of a differentiable manifold is an important invariant of a differentiable structure. It is determined neither by the topological structure nor by the homotopy type of a manifold. But in some cases tangent bundles depend only on the homotopy types of manifolds.

Generalized Bianchi identities for horizontal distributions

1983 ◽  
Vol 94 (1) ◽  
pp. 125-132 ◽  
Author(s):  
M. Crampin
Keyword(s):  
Lie Algebra ◽  
Tangent Bundle ◽  
Linear Connection ◽  
Differentiable Manifold ◽  
Horizontal Distribution ◽  
Bianchi Identities ◽  
Graded Lie Algebra ◽  
Image Position ◽  
Tangent Bundles

A linear connection on a differentiable manifold M defines a horizontal distribution on the tangent bundle T(M). Horizontal distributions on tangent bundles are of some interest even when they are not generated by connections. Much of linear connection theory generalizes to arbitrary horizontal distributions. In particular, there are generalized versions of Bianchi's two identities for the torsion T and curvature Rwhere C denotes the cyclic sum with respect to X, Y and Z. These identities are derived below by associating with a horizontal distribution a graded derivation of degree 1 in a graded Lie algebra of vertical forms on M. This approach reveals the fundamentally algebraic origin of the Bianchi identities.

scholarly journals Natural Paracontact Magnetic Trajectories on Unit Tangent Bundles

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 72
Author(s):  
Mohamed Tahar Kadaoui Abbassi ◽  
Noura Amri
Keyword(s):  
Gaussian Curvature ◽  
Tangent Bundle ◽  
Two Dimensional ◽  
Constant Gaussian Curvature ◽  
Metric Structures ◽  
Unit Tangent Bundle ◽  
Tangent Bundles ◽  
Full Classification ◽  
Kaluza Klein

In this paper, we study natural paracontact magnetic trajectories in the unit tangent bundle, i.e., those that are associated to g-natural paracontact metric structures. We characterize slant natural paracontact magnetic trajectories as those satisfying a certain conservation law. Restricting to two-dimensional base manifolds of constant Gaussian curvature and to Kaluza–Klein type metrics on their unit tangent bundles, we give a full classification of natural paracontact slant magnetic trajectories (and geodesics).

Homotopy Types

2020 ◽  
pp. 77-106
Author(s):  
David Corfield
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Definite Descriptions ◽  
Homotopy Types ◽  
Homotopy Type Theory

A further innovation is the introduction of an intensional type theory. Here one need not reduce equivalence to mere identity. How two entities are the same tells us more than whether they are the same. This is explained by the homotopical aspect of HoTT, where types are taken to resemble spaces of points, paths, paths between paths, and so on. This allows us to rethink Russell’s definite descriptions. Mathematicians use a ‘generalized the’ in situations where it appears that they might be talking about a multiplicity of instances, but there is essentially a unique instance. Taken together with the ‘univalence axiom’ there results a language in which anything that can be said of a type can be said of an equivalent type. This allows homotopy type theory to become the language of choice for a structuralist, avoiding the need for any kind of abstraction away from multiple instantiations.

On the Simple ℤ2-homotopy Types of Graph Complexes and Their Simple ℤ2-universality

2008 ◽  
Vol 51 (4) ◽  
pp. 535-544 ◽  
Author(s):  
Péter Csorba
Keyword(s):  
Homotopy Type ◽  
Graph Complexes ◽  
Homotopy Types ◽  
Neighborhood Complex

AbstractWe prove that the neighborhood complex N(G), the box complex B(G), the homomorphism complex Hom(K2, G) and the Lovász complex L(G) have the same simple ℤ2-homotopy type in the sense of Whitehead. We show that these graph complexes are simple ℤ2-universal.

scholarly journals The extension of G-foliations to Tangent bundles of higher order

1975 ◽  
Vol 56 ◽  
pp. 29-44 ◽  
Author(s):  
Luis A. Cordero
Keyword(s):  
Infinite Sequence ◽  
Differentiable Manifold ◽  
Higher Order ◽  
Characteristic Classes ◽  
Tangent Bundles

In this paper we describe a canonical procedure for constructing the extension of a G-foliation on a differentiable manifold X to its tangent bundles of higher order and by applying the Bott-Haefliger’s construction of characteristic classes of G-foliations ([2], [3]) we obtain an infinite sequence of characteristic classes for those foliations (Theorem 4.8).

scholarly journals LIFTS OF GOLDEN STRUCTURES ON THE TANGENT BUNDLE

2021 ◽  
pp. 643
Author(s):  
Geeta Verma
Keyword(s):  
Tangent Bundle ◽  
Third Order ◽  
Integrability Conditions ◽  
Complete Lift ◽  
Tangent Bundles ◽  
Order Tangent

The present paper aims to study the complete lift of golden structure on tangent bundles. Integrability conditions for complete lift and third-order tangent bundle are established.

Note on Almost Product Manifolds and their Tangent Bundles

1966 ◽  
Vol 9 (05) ◽  
pp. 621-630
Author(s):  
Chorng Shi Houh
Keyword(s):  
Tangent Bundle ◽  
Tangent Vector ◽  
Product Structure ◽  
Dimensional Manifold ◽  
Almost Product Structure ◽  
Local Coordinates ◽  
Natural Frame ◽  
Class C ◽  
Tangent Bundles ◽  
Differentiability Class

Let Mn be an n-dimensional manifold of differentiability class C∞ with an almost product structure . Let have eigenvalue +1 of multiplicity p and eigenvalue -1 of multiplicity q where p+q = n and p≧1, q≧1. Let T(Mn) be the tangent bundle of M. T(Mn) is a 2n dimensional manifold of class C∞. Let xi be the local coordinates of a point P of Mn. The local coordinates of T(Mn) can be expressed by 2n variables (xi, yi) where xi are coordinates of the point P and yi are components of a tangent vector at P with respect to the natural frame constituted by the vectior ∂/∂xi at P.

Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group (ℤ/a ⋊ ℤ/b) × SL2 (p)

2007 ◽  
Vol 50 (2) ◽  
pp. 206-214 ◽  
Author(s):  
Marek Golasiński ◽  
Daciberg Lima Gonçalves
Keyword(s):  
Automorphism Group ◽  
Homotopy Type ◽  
Spherical Space ◽  
Space Forms ◽  
Homotopy Types ◽  
Spherical Space Forms

AbstractLet G = (ℤ/a ⋊ ℤ/b) × SL2(p), and let X(n) be an n-dimensional CW-complex of the homotopy type of an n-sphere. We study the automorphism group Aut(G) in order to compute the number of distinct homotopy types of spherical space forms with respect to free and cellular G-actions on all CW-complexes X(2dn − 1), where 2d is the period of G. The groups ε(X(2dn − 1)/μ) of self homotopy equivalences of space forms X(2dn − 1)/μ associated with free and cellular G-actions μ on X(2dn − 1) are determined as well.

scholarly journals Complete lift of a structure satisfyingFK−(−)K+1F=0

1992 ◽  
Vol 15 (4) ◽  
pp. 803-808 ◽  
Author(s):  
Lovejoy S. Das
Keyword(s):  
Tangent Bundle ◽  
Differentiable Manifold ◽  
Cotangent Bundles ◽  
Integrability Conditions ◽  
Complete Lift

The idea off-structure manifold on a differentiable manifold was initiated and developed by Yano [1], Ishihara and Yano [2], Goldberg [3] and among others. The horizontal and complete lifts from a differentiable manifoldMnof classC∞to its cotangent bundles have been studied by Yano and Patterson [4,5]. Yano and Ishihara [6] have studied lifts of anf-structure in the tangent and cotangent bundles. The purpose of this paper is to obtain integrability conditions of a structure satisfyingFK−(−)K+1F=0andFW−(−)W+1F≠0for1<W<K, in the tangent bundle.

scholarly journals Focal rigidity of flat tori

2011 ◽  
Vol 83 (4) ◽  
pp. 1149-1158 ◽  
Author(s):  
Ferry Kwakkel ◽  
Marco Martens ◽  
Mauricio Peixoto
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Topological Structure ◽  
Closed Riemannian Manifold ◽  
Flat Tori

Given a closed Riemannian manifold (M, g), i.e. compact and boundaryless, there is a partition of its tangent bundle TM = ∪iΣi called the focal decomposition of TM. The sets Σi are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note, we study the topological structure of the focal decomposition of a closed Riemannian manifold and its relation with the metric structure of the manifold. Our main result is that flat n-tori, n > 2, are focally rigid in the sense that if two flat tori are focally equivalent then the tori are isometric up to rescaling. The case n = 2 was considered before by F. Kwakkel.

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