Constant curvature models in sub-Riemannian geometry (original) (raw)

Jacobi equations and Comparison Theorems for corank 1 sub-Riemannian structures with symmetries

Journal of Geometry and Physics, 2011

The Jacobi curve of an extremal of optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. In our previous works we constructed the canonical bundle of moving frames and the complete system of symplectic invariants, called curvature maps, for parametrized curves in Lagrange Grassmannians satisfying very general assumptions. The structural equation for a canonical moving frame of the Jacobi curve of an extremal can be interpreted as the normal form for the Jacobi equation along this extremal and the curvature maps can be seen as the "coefficients"of this normal form. In the case of a Riemannian metric there is only one curvature map and it is naturally related to the Riemannian sectional curvature. In the present paper we study the curvature maps for a sub-Riemannian structure on a corank 1 distribution having an additional transversal infinitesimal symmetry. After the factorization by the integral foliation of this symmetry, such sub-Riemannian structure can be reduced to a Riemannian manifold equipped with a closed 2-form (a magnetic field). We obtain explicit expressions for the curvature maps of the original sub-Riemannian structure in terms of the curvature tensor of this Riemannian manifold and the magnetic field. We also estimate the number of conjugate points along the sub-Riemannian extremals in terms of the bounds for the curvature tensor of this Riemannian manifold and the magnetic field in the case of an uniform magnetic field. The language developed for the calculation of the curvature maps can be applied to more general sub-Riemannian structures with symmetries, including sub-Riemmannian structures appearing naturally in Yang-Mills fields.

Symmetries of sub-Riemannian surfaces

2009

We obtain some results on symmetries of sub-Riemannian surfaces. In case of contact sub-Riemannian surface we base on invariants found by Hughen . Using these invariants, we find conditions under which a sub-Riemannian surface does not admit symmetries. If a surface admits symmetries, we show how invariants help to find them. It is worth noting, that the obtained conditions can be explicitly checked for a given contact sub-Riemannian surface. Also, we consider sub-Riemannian surfaces which are not contact and find their invariants along the surface where the distribution fails to be contact.

SOME RESULTS ON SUB-RIEMANNIAN GEOMETRY

Sub-Riemannian structures naturally occur in different branches of Mathematics in the study of constrained systems in classical mechanics, in optimal control, geometric measure theory and differential geometry. In this paper, we show that Sub-Riemannian structures on three manifolds locally depend on two functions 1 and K of three variables and we investigate how these differential invariants influence the geometry.

The Geometry of Sub-Riemannian Three-Manifolds

2003

The local equivalence problem for sub-Riemannian structures on threemanifolds is solved. In the course of the solution, it is shown how to attach a canonical Riemannian metric and connection to the given sub-Riemannian metric and it is shown how all of the differential invariants of the sub-Riemannian structure can be calculated. The relation between the completeness of the sub-Riemannian metric, the associated Riemannian metric, and geodesic completeness is investigated, and an example is given of a manifold that is complete in the sub-Riemannian metric but not complete in the canonical associated Riemannian metric. It is shown that the Jacobi equations for subRiemannian geodesics can be interpreted as a scalar, fourth-order, self-adjoint linear operator along each geodesic. The influence of the differential invariants of the subRiemannian structure on the conjugate points is investigated, and the results are used to prove a Bonnet-Myers-type theorem for complete sub-Riemannian 3-m...

Connections on Sub-Riemannian Manifolds

Cambridge University Press eBooks, 2013

We provide invariant formulas for the Euler-Lagrange equation associated to sub-Riemannian geodesics. They use the concept of curvature and horizontal connection introduced and studied in the paper.

Constant mean curvature surfaces in sub-Riemannian geometry

2008

We investigate the minimal and isoperimetric surface problems in a large class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of Bryant, Griffiths and Grossman, derive succinct forms of the Euler-Lagrange equations for critical points for the associated variational problems. Using the Euler-Lagrange equations, we show that minimal and isoperimetric surfaces satisfy a constant horizontal mean curvature conditions away from characteristic points. Moreover, we use the formalism to construct a horizontal second fundamental form, II 0 , for vertically rigid spaces and, as a first application, use II 0 to show that minimal surfaces cannot have points of horizontal positive curvature and, that minimal surfaces in Carnot groups cannot be locally horizontally geometrically convex. We note that the convexity condition is distinct from others currently in the literature.

Riemannian geometry of

Journal of Functional Analysis, 2006

The main result of the paper is a computation of the Ricci curvature of Diff(S 1 )/S 1 . Unlike earlier results on the subject, we do not use the Kähler structure symmetries to compute the Ricci curvature, but rather rely on classical finite-dimensional results of Nomizu et al. on Riemannian geometry of homogeneous spaces.