Parallel Translation in Vector Bundles with Abelian Structure Group and the Gauss-Bonnet Formula

1973 ◽  
Vol 25 (4) ◽  
pp. 765-771
Author(s):  
Hansklaus Rummler
Keyword(s):  
Riemannian Manifold ◽  
Simple Proof ◽  
General Result ◽  
Tangent Bundle ◽  
Vector Bundles ◽  
Structure Group ◽  
Parallel Translation ◽  
Result Theorem

Most proofs for the classical Gauss-Bonnet formula use special coordinates, or other non-trivial preparations. Here, a simple proof is given, based on the fact that the structure group SO(2) of the tangent bundle of an oriented 2-dimensional Riemannian manifold is abelian. Since only this hypothesis is used, we prove a slightly more general result (Theorem 1).

Chain maps inducing zero homology maps

1965 ◽  
Vol 61 (4) ◽  
pp. 847-854 ◽  
Author(s):  
G. M. Kelly
Keyword(s):  
Simple Proof ◽  
General Result ◽  
Special Cases ◽  
Result Theorem ◽  
Chain Maps

The purpose of this paper is to prove Theorems 1 and 2 below by exhibiting them as special cases of a more general result, Theorem 3, which admits of a simple proof.

Golden Riemannian structures on the tangent bundle with g-natural metrics

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2543-2554
Author(s):  
E. Peyghan ◽  
F. Firuzi ◽  
U.C. De
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Riemannian Metric

Starting from the g-natural Riemannian metric G on the tangent bundle TM of a Riemannian manifold (M,g), we construct a family of the Golden Riemannian structures ? on the tangent bundle (TM,G). Then we investigate the integrability of such Golden Riemannian structures on the tangent bundle TM and show that there is a direct correlation between the locally decomposable property of (TM,?,G) and the locally flatness of manifold (M,g).

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

2021 ◽  
Vol 62 ◽  
pp. 53-66
Author(s):  
Fethi Latti ◽  
◽  
Hichem Elhendi ◽  
Lakehal Belarbi
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Harmonic Vector ◽  
New Class ◽  
Necessary And Sufficient ◽  
Sasakian Metric ◽  
Harmonic Vector Fields

In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.

Parabolic Higgs Bundles for Real Reductive Lie Groups

2018 ◽  
pp. 653-680 ◽  
Author(s):  
Ignasi Mundet i Riera
Keyword(s):  
Riemann Surface ◽  
Lie Groups ◽  
Lie Group ◽  
Vector Bundles ◽  
Structure Group ◽  
Higgs Bundles ◽  
Local Systems ◽  
Parabolic Structure ◽  
Reductive Lie Groups ◽  
Reductive Lie Group

This chapter explains the correspondence between local systems on a punctured Riemann surface with the structure group being a real reductive Lie group G, and parabolic G-Higgs bundles. The chapter describes the objects involved in this correspondence, taking some time to motivate them by recalling the definitions of G-Higgs bundles without parabolic structure and of parabolic vector bundles. Finally, it explains the relevant polystability condition and the correspondence between local systems and Higgs bundles.

ON FINSLER MANIFOLDS WHOSE TANGENT BUNDLE HAS THE g-NATURAL METRIC

2011 ◽  
Vol 08 (07) ◽  
pp. 1593-1610 ◽  
Author(s):  
ESMAEIL PEYGHAN ◽  
AKBAR TAYEBI
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Positive Constant ◽  
Tangent Bundle ◽  
Contact Structure ◽  
Jacobi Operator ◽  
Riemannian Metric ◽  
Finsler Manifolds ◽  
Almost Contact Structure ◽  
Flat Riemannian Manifold

In this paper, we introduce a Riemannian metric [Formula: see text] and a family of framed f-structures on the slit tangent bundle [Formula: see text] of a Finsler manifold Fn = (M, F). Then we prove that there exists an almost contact structure on the tangent bundle, when this structure is restricted to the Finslerian indicatrix. We show that this structure is Sasakian if and only if Fn is of positive constant curvature 1. Finally, we prove that (i) Fn is a locally flat Riemannian manifold if and only if [Formula: see text], (ii) the Jacobi operator [Formula: see text] is zero or commuting if and only if (M, F) have the zero flag curvature.

scholarly journals NOTES AND PROBLEMS A GENERAL BOUND FOR THE LIMITING DISTRIBUTION OF BREITUNG'S STATISTIC

2008 ◽  
Vol 24 (5) ◽  
pp. 1443-1455 ◽  
Author(s):  
James Davidson ◽  
Jan R. Magnus ◽  
Jan Wiegerinck
Keyword(s):  
General Result ◽  
Random Variables ◽  
Nonparametric Test ◽  
Limiting Distribution ◽  
Residue Theorem ◽  
Prove Theorem ◽  
General Bound ◽  
Result Theorem ◽  
Special Case ◽  
Cauchy’S Residue Theorem

We consider the Breitung (2002, Journal of Econometrics 108, 343–363) statistic ξn, which provides a nonparametric test of the I(1) hypothesis. If ξ denotes the limit in distribution of ξn as n → ∞, we prove (Theorem 1) that 0 ≤ ξ ≤ 1/π2, a result that holds under any assumption on the underlying random variables. The result is a special case of a more general result (Theorem 3), which we prove using the so-called cotangent method associated with Cauchy's residue theorem.

scholarly journals LIE ALGEBROIDS AS SPACES

2018 ◽  
Vol 19 (2) ◽  
pp. 487-535 ◽  
Author(s):  
Ryan Grady ◽  
Owen Gwilliam
Keyword(s):  
Tangent Bundle ◽  
Symplectic Structure ◽  
Vector Bundles ◽  
Natural Generalization ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Faithful Functor ◽  
Derived Geometry

In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique) $L_{\infty }$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_{\infty }$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_{\infty }$ space. Indeed, this functor sends the adjoint complex of $L$ to the tangent bundle of the $L_{\infty }$ space. Finally, we show that a shifted symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated $L_{\infty }$ space.

scholarly journals The super-Sasaki metric on the antitangent bundle

2020 ◽  
Vol 17 (08) ◽  
pp. 2050122
Author(s):  
Andrew James Bruce
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Symplectic Form ◽  
Riemannian Metric ◽  
Riemannian Structure ◽  
Sasaki Metric

We show how to lift a Riemannian metric and almost symplectic form on a manifold to a Riemannian structure on a canonically associated supermanifold known as the antitangent or shifted tangent bundle. We view this construction as a generalization of Sasaki’s construction of a Riemannian metric on the tangent bundle of a Riemannian manifold.

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