On the structure of Finsler and areal spaces

We study underlying geometric structures for integral variational functionals, depending on submanifolds of a given manifold. Applications include (first order) variational functionals of Finsler and areal geometries with integrand the Hilbert 1-form, and admit immediate extensions to higher-order functionals.Comment: 8 pages, Proceedings for AGMP-8, to be published in the special issue of Miskolc Mathematical Note

Parameter invariant Lagrangian formulation of Kawaguchi geometry

This Ph.D. thesis is devoted to the constructions of Lagrangian formulation on Finsler and Kawaguchi manifolds. While Finsler geometry is a natural extension of Riemannian geometry, Kawaguchi geometry is the extension of Finsler geometry to higher order derivatives and to k-dimensional parameter space. The latter extension is also called areal geometry in some references. On Finsler (Kawaguchi) manifold, we can define a reparameterisation invariant 1 (k)-dimensional area by the Hilbert (Kawaguchi) form, which we take as an action. The equation of motion obtained from such action also has the property of reparameterisation invariance. In this framework, the solution manifold of the Euler-Lagrange equation is realised as a submanifold of Finsler/Kawaguchi manifold, and no fibered structure over the parameter space is needed. We also show that for the case of first order k-dimensional parameter space and second order 1-dimensional parameter space, a global Lagrangian could be constructed. For second order k-dimensional parameter space, Lagrangian is not global but it still has the reparameterisation invariant property. Such theory is expected to provide the ideal stage for formulating fundamental theories of physics, especially for cases such as when one needs to consider the mixing of spacetime and field variables. It is shown that locally, any conventional Lagrangian could be reformulated by the parameter independent Lagrangian. Furthermore, the parameter independent property will gives us the freedom of choosing a parameter, which in some cases turns out to be useful in finding solutions and symmetries.Comment: 138 pages, Ph.D. thesi

On Metrizability of Invariant Affine Connections

The metrizability problem for a symmetric affine connection on a manifold, invariant with respect to a group of diffeomorphisms G, is considered. We say that the connection is G-metrizable, if it is expressible as the Levi-Civita connection of a G-invariant metric field. In this paper we analyze the G-metrizability equations for the rotation group G = SO(3), acting canonically on three- and four-dimensional Euclidean spaces. We show that the property of the connection to be SO(3)-invariant allows us to find complete explicit description of all solutions of the SO(3)-metrizability equations.Comment: 17 pages, To appear in IJGMMP vol.9 No.1 (2012

Chiral gravity in higher dimensions

We construct a chiral theory of gravity in 7 and 8 dimensions, which are equivalent to Einstein-Cartan theory using less variables. In these dimensions, we can construct such higher dimensional chiral gravity because of the existence of gravitational instanton. The octonionic-valued variables in the theory represent the deviation from the gravitational instanton, and from their non-associativity, prevents the theory to be SO(n) gauge invariant. Still the chiral gravity holds G_2 (7-D), and Spin(7) (8-D) gauge symmetry.Comment: 18 pages, no figures. Minor typos corrected. Updated reference

General Relativity by Kawaguchi geometry

We construct a parameterisation invariant Lagrange theory of fields up to second order by using multivector bundles and Kawaguchi geometry. In this setup, the spacetime is an dynamical object which is a submanifold of the greater manifold, and the actual spacetime is the solution of Euler-Lagrange equations. Such theory is a reasonable mathematical foundation to describe an extended theory of Einstein’s general relativity, and is capable of being a stage for unification with other physical fields