Horizontal Geodesics Research Articles

Invariant constant mean curvature tubes around a horizontal geodesic in [formula omitted]-spaces

We consider constant mean curvature surfaces invariant by a continuous group of isometries and lying at bounded distance from a horizontal geodesic on any homogeneous 3-manifold E(κ,τ) with isometry group of dimension 4. These surfaces are called horizontal tubes. We show that they foliate E(κ,τ) minus one or two horizontal geodesics provided that (1−x02)κ+4τ2≤0, where x0≈0.833557. We also describe precisely how horizontal and vertical geodesics get deformed by Daniel's sister correspondence and conclude that the family of horizontal tubes is preserved by the correspondence. These tubes are topologically tori in S2×R and Berger spheres, in which case we compute their conformal type and analyze numerically their isoperimetric profiles.

Killing submersions and magnetic curves

We investigate magnetic curves in Killing submersions and we show that the bundle curvature is constant along magnetic curves with respect to the Killing vector field if and only if all vertical tubes derived from these magnetic curves are of constant mean curvature. Then we examine magnetic Jacobi fields along horizontal Killing magnetic curves in the total space of a Killing submersion. We generalize some important results obtained by O'Neill for horizontal geodesics to horizontal magnetic curves. Finally, we study magnetic Jacobi fields along horizontal Killing magnetic trajectories in 3-dimensional Sasakian space forms.

The Schouten Curvature Tensor and the Jacobi Equation in Sub-Riemannian Geometry

We show that if a distribution does not depend on the vertical coordinates, then the Schouten curvature tensor coincides with the Riemannian curvature. The Schouten curvature tensor and the nonholonomicity tensor are used to write the Jacobi equation for the distribution. This leads to a study of second-order optimality conditions for horizontal geodesics in sub-Riemannian geometry. We study conjugate points for horizontal geodesics on the Heisenberg group as an example.

EQUATIONS OF MOTION FOR SPACE-TIME-MATTER THEORY

The purpose of this paper is to present, in a covariant form and in their full generality, the equations of motion for space-time-matter (STM) theory. The whole study is based on the new approach of STM theory developed in our first paper [1] of this series. We show that the theory of geodesics in a general Kaluza–Klein space is best presented and explained by splitting the set of all geodesics into horizontal and non-horizontal geodesics. It is noteworthy that the horizontal geodesics (respectively, non-horizontal geodesics) project on the base manifold on motions which generalize the motions from general relativity theory (respectively, motions from Lorentz force equations).

Index form for nonholonomic distributions

The purpose of this paper is to derive an invariant Jacobi equation for horizontal geodesics on a nonholonomic distribution. The classical Bliss’s theory is known to be not geometrically invariant; it also produces “extra” conjugate points. “Extra” conjugate points also appear in the accessory problem in the optimal control theory. Hence the restriction of the functional of the index form of a curve to the distribution is considered. It is assumed that the distribution satisfies the cyclicity condition. A sufficient condition for the positive definiteness of the index form and the optimality of a curve is obtained.

On totally geodesic foliations and doubly ruled surfaces in a compact Lie group

For a Riemannian submersion from a simple compact Lie group with a bi-invariant metric, we prove that the action of its holonomy group on the fibers is transitive. As a step towards classifying Riemannian submersions with totally geodesic fibers, we consider the parameterized surface induced by lifting a base geodesic to points along a geodesic in a fiber. Such a surface is “doubly ruled” (it is ruled by horizontal geodesics and also by vertical geodesics). Its characterizing properties allow us to define “doubly ruled parameterized surfaces” in any Riemannian manifold, independent of Riemannian submersions. We initiate a study of the doubly ruled parameterized surfaces in compact Lie groups and in other symmetric spaces by establishing several rigidity theorems and constructing examples.

Jacobi fields for a nonholonomic distribution

The variational problem with nonholonomic constraints was considered in detail by Bliss. A distribution is a special case of constraints. Horizontal geodesics on a manifold with flat metric and constant tensor of nonholonomity are considered. It is proved that, in the classical adjoint problem, conjugate points appear, which does not involve any loss of optimality. The second variation of the length (or energy) functional of admissible (horizontal) geodesics for a distribution on a smooth manifold is expressed in terms of the distribution curvature tensor.

Shape curves and geodesic modelling

A family of shape curves is introduced that is useful for modelling the changes in shape in a series of geometrical objects. The relationship between the preshape sphere and the shape space is used to define a general family of curves based on horizontal geodesics on the preshape sphere. Methods for fitting geodesics and more general curves in the non-Euclidean shape space of point sets are discussed, based on minimizing sums of squares of Procrustes distances. Likelihood-based inference is considered. We illustrate the ideas by carrying out statistical analysis of two-dimensional landmarks on rats' skulls at various times in their development and three-dimensional landmarks on lumbar vertebrae from three primate species.

Nonholonomous geodesics as solutions to Euler-Lagrange integral equations and the differential of the exponential mapping

Equations of horizontal geodesics on a Riemannian (or pseudo-Riemannian) manifold with nonholonomous distribution are obtained using the Euler-Lagrange method in Pontryagin’s formulation. It is shown that if the distribution and the metric tensor of the distribution are Ck-smooth, k ≥ 1, then any regular solution to the variational problem is Ck + 1-smooth. The differential of the exponential mapping is obtained for nonholonomous distribution with the condition of cyclicity with respect to “vertical” coordinates. This differential is nonsingular provided that the distribution is strongly bracket generating.

The Euler-Lagrange method in Pontryagin’s formulation

The classical variational problem with nonholonomic constraints is solvable by the Euler-Lagrange method in Pontryagin’s formulation; however, in this case Lagrange multipliers are merely measurable functions. In this paper, we put forward a modified Euler-Lagrange method, in which the original problem involves a Lagrangian dependent only on the independent components of the velocity vector. Under this approach, the Lagrange multipliers make up an absolutely continuous vector function. Our method is applied to the problem of horizontal geodesics for a nonholonomic distribution on a manifold. These equations are established as having two types of connections: connection on the distribution and connection on the manifold; this was not accounted for by other researchers.

Principal bundles and topological quantization of charges

The space of admissible particle velocities is assumed to be a four-dimensional nonholonomic distribution on a principal or associated bundle. Equations for the horizontal geodesics of this distribution coincide with the equations of motion of charged particles in general relativity theory. It is proved that, if the Lie group of the standard model of elementary particle physics is augmented by the 4-torus, then the wave functions are eigenfunctions of charge operators and the horizontal lift does not depend on the coupling constants. These wave functions satisfy the well-known Dirac equation and its generalizations. For such wave functions, the topological quantization of electric, lepton, and baryon charges takes place.

Equations of motion of a charged particle in a five-dimensional model of the general theory of relativity with a nonholonomic four-dimensional velocity space

The equations of motion of a charged particle in a five-dimensional model of the general theory of relativity with a nonholonomic four-dimensional velocity space are considered. A nonholonomic distribution defined by the differential form ω = A 0 dx 0 + A 1 dx 1 + A 2 dx 2 + A 3 dx 3 + dx 4 on a five-dimensional smooth Lorentzian manifold is studied. By means of the Pontryagin maximum principle, it is proved that the equations of horizontal geodesics for this distribution are the same as the equations of motion of a charged particle in the general theory of relativity. Thus, a model of the Kaluza-Klein theory is built by means of the sub-Lorentzian geometry. Finally, the geodesic sphere, which appears in a constant magnetic field, is studied, as well as its singular points.

Systems of rays in the presence of distribution of hyperplanes

Horizontal systems of rays arise in the study of integral curves of Hamiltonian systems vH on T X, which are tangent to a given distribution V of hyperplanes on X. We investigate the local properties of systems of rays for general pairs (H,V ) as well as for Hamiltonians H such that the corresponding Hamiltonian vector fields vH are horizontal with respect to V . As an example we explicitly calculate the space of horizontal geodesics and the corresponding systems of rays for the canonical distribution on the Heisenberg group. Local stability of systems of horizontal rays based on the standard singularity theory of Lagrangian submanifolds is also considered. Introduction. Let X be a differentiable manifold. Let ω be a differential 1-form on X, and V a distribution of tangent hyperplanes in T X annihilated by ω. We consider the Hamiltonian systems vH , with Hamiltonian H on T X, which are horizontal with respect to V , i.e. the projections of the bicharacteristics of vH onto the base of T X are tangent to V . We introduce the notion of the space of geodesics-rays as the reduced symplectic space M of bicharacteristics on H−1( 12 ) equipped with the reduced symplectic form μ defined by the relation ωX |H−1( 2 ) −π μ = 0, where ωX is the canonical Liouville form on T X and π is the projection onto M along bicharacteristics (cf. [9]). Any Lagrangian submanifold L of (M,μ) is called a system of rays. Its counterimage π−1(L) represents an optical system of rays in the phase space (T X,ωX) of geometric optics. The graph of π is a Lagrangian submanifold in the product symplectic structure (T X ×M,π 2μ − π 1ωX), where πi are the projections of the cartesian product T X×M . Then we have a generating function G for graphπ (cf. [6]), which helps to determine the structure of the counterimage π−1(L). Now the stability notion 1991 Mathematics Subject Classification: Primary 57R45, 53B21; Secondary 58F05.

Geodesics on loop spaces

The space of loops smoothly embedded into a Riemannian manifold, being a principal fibre bundle with structure group Diff S 1, is investigated from a Kaluza-Klein type point of view. In particular, the Levi-Civita connection for the natural Diff S 1-invariant metric on this loop space is calculated and the corresponding horizontal geodesics (the analogue of classical free motion of point particles) are characterized. Finally, an explicit solution is given in the case of loops in R 3.