Transfinite thin plate spline interpolation

Duchon's method of thin plate splines defines a polyharmonic interpolant to scattered data values as the minimizer of a certain integral functional. For transfinite interpolation, i.e. interpolation of continuous data prescribed on curves or hypersurfaces, Kounchev has developed the method of polysplines, which are piecewise polyharmonic functions of fixed smoothness across the given hypersurfaces and satisfy some boundary conditions. Recently, Bejancu has introduced boundary conditions of Beppo Levi type to construct a semi-cardinal model for polyspline interpolation to data on an infinite set of parallel hyperplanes. The present paper proves that, for periodic data on a finite set of parallel hyperplanes, the polyspline interpolant satisfying Beppo Levi boundary conditions is in fact a thin plate spline, i.e. it minimizes a Duchon type functional

A Finslerian version of 't Hooft Deterministic Quantum Models

Using the Finsler structure living in the phase space associated to the tangent bundle of the configuration manifold, deterministic models at the Planck scale are obtained. The Hamiltonian function are constructed directly from the geometric data and some assumptions concerning time inversion symmetry. The existence of a maximal acceleration and speed is proved for Finslerian deterministic models. We investigate the spontaneous symmetry breaking of the orthogonal symmetry SO(6N) of the Hamiltonian of a deterministic system. This symmetry break implies the non-validity of the argument used to obtain Bell's inequalities for spin states. It is introduced and motivated in the context of Randers spaces an example of simple 't Hooft model with interactions.Comment: 25 pages; no figures. String discussion deleted. Some minor change

Nonholonomic Ricci Flows and Running Cosmological Constant: I. 4D Taub-NUT Metrics

In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some conceptual issues on spacetimes provided with generic off-diagonal metrics and associated nonlinear connection structures are analyzed. The limit from gravity/Ricci flow models with nontrivial torsion to configurations with the Levi-Civita connection is allowed in some specific physical circumstances by constraining the class of integral varieties for the Einstein and Ricci flow equations.Comment: latex2e, final variant to be published in IJMP

A Geometric characterization of Finsler manifolds of constant curvature K

We prove that a Finsler manifold 𝔽m is of constant curvature K=1 if and only if the unit horizontal Liouville vector field is a Killing vector field on the indicatrix bundle IM of 𝔽m

Some constructions of almost para-hyperhermitian structures on manifolds and tangent bundles

In this paper we give some examples of almost para-hyperhermitian structures on the tangent bundle of an almost product manifold, on the product manifold $M\times\mathbb{R}$, where $M$ is a manifold endowed with a mixed 3-structure and on the circle bundle over a manifold with a mixed 3-structure.Comment: 10 pages; This paper has been presented in the "4th German-Romanian Seminar on Geometry" Dortmund, Germany, 15-18 July 200

Finsler and Lagrange Geometries in Einstein and String Gravity

We review the current status of Finsler-Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kahler geometry and geometric mechanics how they could perform their results in order to be accepted by the community of ''orthodox'' physicists. Although the bulk of former models of Finsler-Lagrange spaces where elaborated on tangent bundles, the surprising result advocated in our works is that such locally anisotropic structures can be modelled equivalently on Riemann-Cartan spaces, even as exact solutions in Einstein and/or string gravity, if nonholonomic distributions and moving frames of references are introduced into consideration. We also propose a canonical scheme when geometrical objects on a (pseudo) Riemannian space are nonholonomically deformed into generalized Lagrange, or Finsler, configurations on the same manifold. Such canonical transforms are defined by the coefficients of a prime metric and generate target spaces as Lagrange structures, their models of almost Hermitian/ Kahler, or nonholonomic Riemann spaces. Finally, we consider some classes of exact solutions in string and Einstein gravity modelling Lagrange-Finsler structures with solitonic pp-waves and speculate on their physical meaning.Comment: latex 2e, 11pt, 44 pages; accepted to IJGMMP (2008) as a short variant of arXiv:0707.1524v3, on 86 page

Two-Connection Renormalization and Nonholonomic Gauge Models of Einstein Gravity

A new framework to perturbative quantum gravity is proposed following the geometry of nonholonomic distributions on (pseudo) Riemannian manifolds. There are considered such distributions and adapted connections, also completely defined by a metric structure, when gravitational models with infinite many couplings reduce to two--loop renormalizable effective actions. We use a key result from our partner work arXiv:0902.0911 that the classical Einstein gravity theory can be reformulated equivalently as a nonholonomic gauge model in the bundle of affine/de Sitter frames on pseudo-Riemannian spacetime. It is proven that (for a class of nonholonomic constraints and splitting of the Levi-Civita connection into a "renormalizable" distinguished connection, on a base background manifold, and a gauge like distortion tensor, in total space) a nonholonomic differential renormalization procedure for quantum gravitational fields can be elaborated. Calculation labor is reduced to one- and two-loop levels and renormalization group equations for nonholonomic configurations.Comment: latex2e, 40 pages, v4, accepted for Int. J. Geom. Meth. Mod. Phys. 7 (2010

Fedosov Quantization of Lagrange-Finsler and Hamilton-Cartan Spaces and Einstein Gravity Lifts on (Co) Tangent Bundles

We provide a method of converting Lagrange and Finsler spaces and their Legendre transforms to Hamilton and Cartan spaces into almost Kaehler structures on tangent and cotangent bundles. In particular cases, the Hamilton spaces contain nonholonomic lifts of (pseudo) Riemannian / Einstein metrics on effective phase spaces. This allows us to define the corresponding Fedosov operators and develop deformation quantization schemes for nonlinear mechanical and gravity models on Lagrange- and Hamilton-Fedosov manifolds.Comment: latex2e, 11pt, 35 pages, v3, accepted to J. Math. Phys. (2009

Weak Gravitational Field in Finsler-Randers Space and Raychaudhuri Equation

The linearized form of the metric of a Finsler - Randers space is studied in relation to the equations of motion, the deviation of geodesics and the generalized Raychaudhuri equation are given for a weak gravitational field. This equation is also derived in the framework of a tangent bundle. By using Cartan or Berwald-like connections we get some types "gravito - electromagnetic" curvature. In addition we investigate the conditions under which a definite Lagrangian in a Randers space leads to Einstein field equations under the presence of electromagnetic field. Finally, some applications of the weak field in a generalized Finsler spacetime for gravitational waves are given.Comment: 22 pages, matches version published in GER

Lagrange-Fedosov Nonholonomic Manifolds

We outline an unified approach to geometrization of Lagrange mechanics, Finsler geometry and geometric methods of constructing exact solutions with generic off-diagonal terms and nonholonomic variables in gravity theories. Such geometries with induced almost symplectic structure are modelled on nonholonomic manifolds provided with nonintegrable distributions defining nonlinear connections. We introduce the concept of Lagrange-Fedosov spaces and Fedosov nonholonomic manifolds provided with almost symplectic connection adapted to the nonlinear connection structure. We investigate the main properties of generalized Fedosov nonholonomic manifolds and analyze exact solutions defining almost symplectic Einstein spaces.Comment: latex2e, v3, published variant, with new S.V. affiliatio