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.

Finsler-spacetime tangent bundle

1992 ◽  
Vol 5 (3) ◽  
pp. 221-248 ◽  
Author(s):  
Howard E. Brandt
Keyword(s):  
Tangent Bundle ◽  
Spacetime Tangent Bundle ◽  
Finsler Spacetime

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.

Composition of Local Normal Coordinates and Polyhedral Geometry in Riemannian Manifold Learning

2015 ◽  
Vol 5 (2) ◽  
pp. 37-68 ◽  
Author(s):  
Gastão F. Miranda Jr. ◽  
Gilson Giraldi ◽  
Carlos E. Thomaz ◽  
Daniel Millàn
Keyword(s):  
Riemannian Manifold ◽  
Manifold Learning ◽  
Barycentric Coordinates ◽  
Synthesis Process ◽  
Exponential Map ◽  
Normal Coordinate ◽  
Normal Coordinates ◽  
Polyhedral Geometry ◽  
Representational Space ◽  
Dimensional Domain

The Local Riemannian Manifold Learning (LRML) recovers the manifold topology and geometry behind database samples through normal coordinate neighborhoods computed by the exponential map. Besides, LRML uses barycentric coordinates to go from the parameter space to the Riemannian manifold in order to perform the manifold synthesis. Despite of the advantages of LRML, the obtained parameterization cannot be used as a representational space without ambiguities. Besides, the synthesis process needs a simplicial decomposition of the lower dimensional domain to be efficiently performed, which is not considered in the LRML proposal. In this paper, the authors address these drawbacks of LRML by using a composition procedure to combine the normal coordinate neighborhoods for building a suitable representational space. Moreover, they incorporate a polyhedral geometry framework to the LRML method to give an efficient background for the synthesis process and data analysis. In the computational experiments, the authors verify the efficiency of the LRML combined with the composition and discrete geometry frameworks for dimensionality reduction, synthesis and data exploration.

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 Finsler spacetime geometry in physics

2019 ◽  
Vol 16 (supp02) ◽  
pp. 1941004 ◽  
Author(s):  
Christian Pfeifer
Keyword(s):  
Gravitational Field ◽  
Riemannian Geometry ◽  
Laboratory Experiments ◽  
Finsler Geometry ◽  
Gravitational Field Equation ◽  
Physical Systems ◽  
Spacetime Geometry ◽  
Finsler Spacetime ◽  
Generalized Geometry ◽  
Straightforward Generalization

Finsler geometry naturally appears in the description of various physical systems. In this review, I divide the emergence of Finsler geometry in physics into three categories: dual description of dispersion relations, most general geometric clock and geometry being compatible with the relevant Ehlers–Pirani–Schild axioms. As Finsler geometry is a straightforward generalization of Riemannian geometry there are many attempts to use it as generalized geometry of spacetime in physics. However, this generalization is subtle due to the existence of non-trivial null directions. I review how a pseudo-Finsler spacetime geometry can be defined such that it provides a precise notion of causal curves, observers and their measurements as well as a gravitational field equation determining the Finslerian spacetime geometry dynamically. The construction of such Finsler spacetimes lays the foundation for comparing their predictions with observations, in astrophysics as well as in laboratory experiments.

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.

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