20 research outputs found
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