scholarly journals Finsler geometry in the tangent bundle

2019 ◽  
Author(s):  
Lajos Tamássy
Keyword(s):  
Tangent Bundle ◽  
Finsler Geometry

GRAVITY IN CURVED PHASE-SPACES, FINSLER GEOMETRY AND TWO-TIMES PHYSICS

2012 ◽  
Vol 27 (12) ◽  
pp. 1250069 ◽  
Author(s):  
CARLOS CASTRO
Keyword(s):  
Phase Space ◽  
Tangent Bundle ◽  
Finsler Geometry ◽  
Space Time ◽  
Vacuum Field ◽  
Field Equations ◽  
Problem Of Time ◽  
Gravitational Field Equations ◽  
Cotangent Space ◽  
Kaluza Klein

The generalized (vacuum) field equations corresponding to gravity on curved 2d-dimensional (dim) tangent bundle/phase spaces and associated with the geometry of the (co)tangent bundle TMd-1, 1(T*Md-1, 1) of a d-dim space–time Md-1, 1 are investigated following the strict distinguished d-connection formalism of Lagrange–Finsler and Hamilton–Cartan geometry. It is found that there is no mathematical equivalence with Einstein's vacuum field equations in space–times of 2d dimensions, with two times, after a d+d Kaluza–Klein-like decomposition of the 2d-dim scalar curvature R is performed and involving the introduction of a nonlinear connection [Formula: see text]. The physical applications of the 4-dim phase space metric solutions found in this work, corresponding to the cotangent space of a 2-dim space–time, deserve further investigation. The physics of two times may be relevant in the solution to the problem of time in quantum gravity and in the explanation of dark matter. Finding nontrivial solutions of the generalized gravitational field equations corresponding to the 8-dim cotangent bundle (phase space) of the 4-dim space–time remains a challenging task.

Finsler geometry and natural foliations on the tangent bundle

2006 ◽  
Vol 58 (1) ◽  
pp. 131-146 ◽  
Author(s):  
Aurel Bejancu ◽  
Hani Reda Farran
Keyword(s):  
Tangent Bundle ◽  
Finsler Geometry

scholarly journals The tangent bundle exponential map and locally autoparallel coordinates for general connections on the tangent bundle with application to Finsler geometry

2016 ◽  
Vol 13 (03) ◽  
pp. 1650023 ◽  
Author(s):  
Christian Pfeifer
Keyword(s):  
Tangent Bundle ◽  
Order Term ◽  
Finsler Geometry ◽  
Exponential Map ◽  
Metric Geometry ◽  
Normal Coordinates ◽  
Concrete Application ◽  
Finsler Spacetime ◽  
Quadratic Order ◽  
General Connection

We construct a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundle of a manifold. As concrete application we use these new coordinates for Finslerian geometries and obtain Finslerian geodesic coordinates. They generalize normal coordinates known from metric geometry to Finsler geometric manifolds and it turns out that they are identical to the Douglas–Thomas normal coordinates introduced earlier. We expand the Finsler Lagrangian of a Finsler spacetime in these new coordinates and find that it is constant to quadratic order. The quadratic order term comes with the nonlinear curvature of the manifold. From physics these coordinates may be interpreted as the realisation of an Einstein elevator in Finslerian spacetime geometries.

scholarly journals Lagrange geometry on tangent manifolds

2003 ◽  
Vol 2003 (51) ◽  
pp. 3241-3266 ◽  
Author(s):  
Izu Vaisman
Keyword(s):  
Tangent Bundle ◽  
Tensor Field ◽  
Finsler Geometry ◽  
Interesting Case ◽  
Lagrangian Function ◽  
Total Space ◽  
Lagrangian Functions ◽  
Tangent Manifold

Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a nondegenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper, we study a generalization which consists of replacing the tangent bundle by a general tangent manifold, and the Lagrangian by a family of compatible, local, Lagrangian functions. We give several examples and find the cohomological obstructions to globalization. Then, we extend the connections used in Finsler and Lagrange geometry, while giving an index-free presentation of these connections.

scholarly journals Spherical harmonic d-tensors

2019 ◽  
Vol 16 (supp02) ◽  
pp. 1941002 ◽  
Author(s):  
Manuel Hohmann
Keyword(s):  
Differential Equations ◽  
Tangent Bundle ◽  
Spherical Harmonic ◽  
Spherical Symmetry ◽  
Finsler Geometry ◽  
Rotation Group ◽  
Mathematical Tool

Tensor harmonics are a useful mathematical tool for finding solutions to differential equations which transform under a particular representation of the rotation group [Formula: see text]. The aim of this work is to make use of this tool also in the setting of Finsler geometry, or more general geometries on the tangent bundle, where the objects of relevance are d-tensors on the tangent bundle, or tensors in a pullback bundle, instead of ordinary tensors. For this purpose, we construct a set of d-tensor harmonics for spherical symmetry and show how these can be used for calculations in Finsler geometry.

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).

Families over a base with a birationally nef tangent bundle

1997 ◽  
Vol 308 (2) ◽  
pp. 347-359 ◽  
Author(s):  
Sándor J. Kovács
Keyword(s):  
Tangent Bundle

scholarly journals Finsler geometry modeling of complex fluids: reduction in viscous resistance

2021 ◽  
Vol 1730 (1) ◽  
pp. 012036
Author(s):  
Masahiko Okumura ◽  
Ippei Homma ◽  
Shuta Noro ◽  
Hiroshi Koibuchi
Keyword(s):  
Complex Fluids ◽  
Finsler Geometry ◽  
Viscous Resistance ◽  
Geometry Modeling

scholarly journals Vortices over Riemann surfaces and dominated splittings

2021 ◽  
pp. 1-26
Author(s):  
THOMAS METTLER ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Euler Characteristic ◽  
Tangent Bundle ◽  
Riemann Surfaces ◽  
Dominated Splitting ◽  
Vortex Equations ◽  
Special Cases ◽  
Unit Tangent Bundle ◽  
Anosov Flows

Abstract We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$ .

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