共查询到20条相似文献,搜索用时 31 毫秒
1.
This paper is concerned with the study of the Monge optimal transport problem in sub-Riemannian manifolds where the cost is given by the square of the sub-Riemannian distance. Our aim is to extend previous results on existence and uniqueness of optimal transport maps to cases of sub-Riemannian structures which admit many singular minimizing geodesics. We treat here the case of sub-Riemannian structures of rank two in dimension four. 相似文献
2.
We introduce the notion of subcomplex structure on a manifold of arbitrary real dimension and consider some important particular cases of pseudocomplex structures: pseudotwistor, affinor, and sub-Kähler structures. It is shown how subtwistor and affinor structures can give sub-Riemannian and sub-Kähler structures. We also prove that all classical structures (twistor, Kähler, and almost contact metric structures) are particular cases of subcomplex structures. The theory is based on the use of a degenerate 1-form or a 2-form with radical of arbitrary dimension. 相似文献
3.
We show that the classes of Hölder mappings of Carnot–Carathéodory spaces are polynomially differentiable in the sub-Riemannian sense. Moreover, we prove the existence of intrinsic (or adapted) bases, which enable us to match the nonholonomic structures of the images of mappings and target spaces. 相似文献
4.
We compare different notions of curvature on contact sub-Riemannian manifolds. In particular, we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Riemannian distance. We explicitly compute their expressions in terms of the standard tensors of contact geometry. As an application of these results, we obtain a sub-Riemannian version of the Bonnet–Myers theorem that applies to any contact manifold. 相似文献
5.
Weyl (Zur Infinitisimalgeometrie: Einordnung der projektiven und der konformen Auffasung, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Göttinger Akademie der Wissenschaften, Göttingen, 1921) demonstrated that for a connected manifold of dimension greater than 1, if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization, then one metric is a constant scaling of the other one. In the present paper we investigate the analogous property for sub-Riemannian metrics. In particular, we prove that the analogous statement, called the Weyl rigidity, holds either in real analytic category for all sub-Riemannian metrics on distributions with a specific property of their complex abnormal extremals, called minimal order, or in smooth category for all distributions such that all complex abnormal extremals of their nilpotent approximations are of minimal order. This also shows, in real analytic category, the genericity of distributions for which all sub-Riemannian metrics are Weyl rigid and genericity of Weyl rigid sub-Riemannian metrics on a given bracket generating distributions. Finally, this allows us to get analogous genericity results for projective rigidity of sub-Riemannian metrics, i.e., when the only sub-Riemannian metric having the same sub-Riemannian geodesics, up to a reparametrization, with a given one, is a constant scaling of this given one. This is the improvement of our results on the genericity of weaker rigidity properties proved in recent paper (Jean et al. in Geom Dedic 203(1):279–319, 2019).
相似文献
6.
The polynomial sub-Riemannian differentiability is established for the large classes of Hölder mappings in the sub-Riemannian sense, namely, the classes of smooth mappings, their graphs, and the graphs of Lipschitz mappings in the sub-Riemannian sense defined on nilpotent graded groups. We also describe some special bases that carry the sub-Riemannian structure of the preimage to the image. 相似文献
7.
An area formula for graph surfaces of codimension 2 on sub-Lorentzian structures with two “negative” directions is obtained. The main tool of study is the polynomial sub-Riemannian differential. 相似文献
8.
We consider G-invariant affinor metric structures and their particular cases, sub-Kähler structures, on a homogeneous space G/H. The affinor metric structures generalize almost Kähler and almost contact metric structures to manifolds of arbitrary dimension. We consider invariant sub-Riemannian and sub-Kähler structures related to a fixed 1-form with a nontrivial radical. In addition to giving some results for homogeneous spaces of arbitrary dimension, we study these structures separately on the homogeneous spaces of dimension 4 and 5. 相似文献
9.
For constant mean curvature surfaces of class C 2 immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classification results for complete surfaces with empty singular set which are stable, i.e., second order minima of the area under a volume constraint, inside the 3-dimensional sub-Riemannian space forms. In the first Heisenberg group we show that such a surface is a vertical plane. In the sub-Riemannian hyperbolic 3-space we give an upper bound for the mean curvature of such surfaces, and we characterize the horocylinders as the unique ones with squared mean curvature 1. Finally we deduce that any complete surface with empty singular set in the sub-Riemannian 3-sphere is unstable. 相似文献
10.
We prove the coarea formula for sufficiently smooth contact mappings of Carnot manifolds to Carnot–Carathéodory spaces. In particular, we investigate level surfaces of these mappings, and compare Riemannian and sub-Riemannian measures on them. Our main tool is the sharp asymptotic behavior of the Riemannian measure of the intersection of a tangent plane to a level surface and a sub-Riemannian ball. This calculation in particular implies that the sub-Riemannian measure of the set of characteristic points (i.e., the points at which the sub-Riemannian differential is degenerate) equals zero on almost every level set. 相似文献
11.
The polynomial sub-Riemannian differentiability of classes of mappings of Carnot groups and graphs is proved. Examples of polynomial sub-Riemannian differentials preserving Hausdorff dimension are given. 相似文献
12.
We prove an analogue of the Cheeger–Gromoll splitting theorem for sub-Riemannian manifolds with the measure contraction property instead of the nonnegativity of the Ricci curvature. If such a sub-Riemannian manifold contains a straight line, then the manifold splits diffeomorphically, where the splitting is not necessarily isometric. We prove that such a sub-Riemannian manifold containing a straight line cannot split isometrically under some typical condition in sub-Riemannian geometry. Heisenberg groups are such examples. 相似文献
13.
We discuss rank 2 sub-Riemannian structures on low-dimensional manifolds and prove that some of these structures in dimensions 6, 7 and 8 have a maximal amount of symmetry but no integrals polynomial in momenta of low degrees, except for those coming from the Killing vector fields and the Hamiltonian, thus indicating nonintegrability of the corresponding geodesic flows. 相似文献
14.
For the classes of the mappings Lipschitz in the sub-Riemannian sense and taking values in the Heisenberg group we introduce some suitable notions of variation of an argument and the corresponding increment of the area functional and derive several basic properties of maximal surfaces on the five-dimensional sub-Lorentzian structures. 相似文献
15.
The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics
on generic corank 1 distributions. Using the Pontryagin maximum principle, we treat Riemannian and sub-Riemannian cases in
a unified way and obtain some algebraic necessary conditions for the geodesic equivalence of (sub-)Riemannian metrics. In
this way, first we obtain a new elementary proof of the classical Levi-Civita theorem on the classification of all Riemannian
geodesically equivalent metrics in a neighborhood of the so-called regular (stable) point w.r.t. these metrics. Second, we
prove that sub-Riemannian metrics on contact distributions are geodesically equivalent iff they are constantly proportional.
Then we describe all geodesically equivalent sub-Riemannian metrics on quasi-contact distributions. Finally, we give a classification
of all pairs of geodesically equivalent Riemannian metrics on a surface that are proportional at an isolated point. This is
the simplest case, which was not covered by Levi-Civita’s theorem.
__________
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric
Problems in Control Theory, 2004. 相似文献
16.
The curvature and the reduced curvature are basic differential invariants of the pair (Hamiltonian system, Lagrangian distribution) on a symplectic manifold. We
consider the Hamiltonian flows of the curve of least action of natural mechanical systems in sub-Riemannian structures with
symmetries. We give sufficient conditions for the reduced flows (after reduction of the first integrals induced from the symmetries)
to be hyperbolic in terms of the reduced curvature and show new examples of Anosov flows. 相似文献
17.
Differential Equations - Homogeneous sub-Riemannian geodesics are described for the standard sub-Riemannian structure on the group $${\mathrm {SE}}(2)$$ of proper motions of the plane. It is shown... 相似文献
18.
We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group \(\mathbb H^n\). Our results include a natural sub-Riemannian version of the celebrated curvature-dimension condition of Lott–Villani and Sturm and also a geodesic version of the Borell–Brascamp–Lieb inequality akin to the one obtained by Cordero-Erausquin, McCann and Schmuckenschläger. The latter statement implies sub-Riemannian versions of the geodesic Prékopa–Leindler and Brunn–Minkowski inequalities. The proofs are based on optimal mass transportation and Riemannian approximation of \(\mathbb H^n\) developed by Ambrosio and Rigot. These results refute a general point of view, according to which no geometric inequalities can be derived by optimal mass transportation on singular spaces. 相似文献
20.
We show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This result allows us to describe the sub-Riemannian geodesic flow on totally geodesic Riemannian foliations in terms of the Riemannian geodesic flow. Also, given a submersion \(\pi :M \rightarrow B\), we describe when the projections of a Riemannian and a sub-Riemannian geodesic flow in M coincide. 相似文献
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