sub-Riemannian Geometry Research Articles

Riemannian optimization on unit sphere with p-norm and its applications

This study deals with Riemannian optimization on the unit sphere in terms of p-norm with general $$p> 1$$ . As a Riemannian submanifold of the Euclidean space, the geometry of the sphere with p-norm is investigated, and several geometric tools used for Riemannian optimization, such as retractions and vector transports, are proposed and analyzed. Applications to Riemannian optimization on the sphere with nonnegative constraints and $$\textit{L}_{\textit{p}}$$ -regularization-related optimization are also discussed. As practical examples, the former includes nonnegative principal component analysis, and the latter is closely related to the Lasso regression and box-constrained problems. Numerical experiments verify that Riemannian optimization on the sphere with p-norm has substantial potential for such applications, and the proposed framework provides a theoretical basis for such optimization.

Comparison of basic equations of the Kaluza–Klein theory with the nonholonomic model of space–time of the sub-Lorentzian geometry

The field equations of gravity coupled to electromagnetism and equations of motion of a charged particle are part of the Kaluza–Klein theory where general relativity is extended to five dimensions. These equations can also be obtained if nonholonomic constrains are imposed on the 5-vector of particle’s velocity. Hence, further development of the general relativity theory can be sub-Riemannian (or sub-Lorentzian) geometry. Gauge transformations become a special case of coordinate transformations in both the Kaluza–Klein theory and the nonholonomic model. Sub-Riemannian geodesics are proved to be equations of motion of a charged particle. The Dirac operator can be extended for a 5-dimensional manifold as a first-order differential operator. Since the base manifold in physics contains the electromagnetic gauge group [Formula: see text], the eigenvalues of the charge operator are always an integer multiplied by the fundamental electric charge.

О почти квазисасакиевых параконтактных структурах на распределениях субримановых многообразий

The concept of an almost quasi-Sasakian paracontact structure is introduced. In contrast to the quasi-Sasakian paracontact structure, the introduced structure does not have the normality property. The invariants of the intrinsic geometry of an almost quasi-Sasakian paracontact structure are studied. The intrinsic geometry of an almost paracontact metric manifold is understood to mean those of its geometric properties that depend only on the parallel translation determined by the intrinsic connection. In particular, it is proved that the vanishing of the Schouten tensor implies that the structure vector field is a Killing vecto r field. An example of an almost quasi-Sasakian paracontact structure is given. Almost quasi-Sasakian paracontact structures arise naturally on distributions of sub-Riemannian manifolds. On a non-involutive distribution D of a sub-Riemannian manifold M of contact type, an almost paracontact metric structure is defined, which is called an extended structure. It is proved that on a distribution of zero curvature, the extended structure is an almost quasi -Sasakian paracontact structure. A characteristic property of a distribution of zero curvature is the vanishing of the Schouten curvature tensor. The well-known classification of extended structures is analyzed based on the properties of the fundamental tensor F of type (0, 3) associated to the extended structure. In accordance with this classification, there are 212 classes of almost paracontact metric structure, among which there are 12 basic ones. A class containing an extended almost quasi-Sasakian paracontact structure is found.

C-R Immersions and Sub-Riemannian Geometry

On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form θ, we consider natural ϵ-contractions, i.e., contractions gϵM of the Levi form Gθ, such that the norm of the Reeb vector field T of (M, θ) is of order O(ϵ−1). We study isopseudohermitian (i.e., f∗Θ=θ) Cauchy–Riemann immersions f:M→(A,Θ) between strictly pseudoconvex CR manifolds M and A, where Θ is a contact form on A. For every contraction gϵA of the Levi form GΘ, we write the embedding equations for the immersion f:M→A,gϵA. A pseudohermitan version of the Gauss equation for an isopseudohermitian C-R immersion is obtained by an elementary asymptotic analysis as ϵ→0+. For every isopseudohermitian immersion f:M→S2N+1 into a sphere S2N+1⊂CN+1, we show that Webster’s pseudohermitian scalar curvature R of (M, θ) satisfies the inequality R≤2n(f∗gΘ)(T,T)+n+1+12{∥H(f)∥gΘf2+∥traceGθΠH(M)∇⊤−∇∥f∗gΘ2} with equality if and only if B(f)=0 and ∇⊤=∇ on H(M)⊗H(M). This gives a pseudohermitian analog to a classical result by S-S. Chern on minimal isometric immersions into space forms.

Mollifications of contact mappings of Engel group

The contact mappings belonging to the metric Sobolev classes are studied on an Engel group with a left-invariant sub-Riemannian metric. In the Euclidean space one of the main methods to handle non-smooth mappings is the mollification, i.e., the convolution with a smooth kernel. An extra difficulty arising with contact mappings of Carnot groups is that the mollification of a contact mapping is usually not contact. Nevertheless, in the case considered it is possible to estimate the magnitude of deviation of contactness sufficiently to obtain useful results. We obtain estimates on convergence (or sometimes divergence) of the components of the differential of the mollified mapping to the corresponding components of the Pansu differential of the contact mapping. As an application to the quasiconformal analysis, we present alternative proofs of the convergence of mollified horizontal exterior forms and the commutativity of the pull-back of the exterior form by the Pansu differential with the exterior differential in the weak sense. These results in turn allow us to obtain such basic properties of mappings with bounded distortion as H\"older continuity, differentiability almost everywhere in the sense of Pansu, Luzin $\mathcal{N}$-property.

Almost-Riemannian manifolds do not satisfy the curvature-dimension condition

The Lott–Sturm–Villani curvature-dimension condition textsf{CD}(K,N) provides a synthetic notion for a metric measure space to have curvature bounded from below by K and dimension bounded from above by N. It was proved by Juillet (Rev Mat Iberoam 37(1), 177–188, 2021) that a large class of sub-Riemannian manifolds do not satisfy the textsf{CD}(K,N) condition, for any Kin {mathbb {R}} and Nin (1,infty ). However, his result does not cover the case of almost-Riemannian manifolds. In this paper, we address the problem of disproving the textsf{CD} condition in this setting, providing a new strategy which allows us to contradict the one-dimensional version of the textsf{CD} condition. In particular, we prove that 2-dimensional almost-Riemannian manifolds and strongly regular almost-Riemannian manifolds do not satisfy the textsf{CD}(K,N) condition for any Kin {mathbb {R}} and Nin (1,infty ).

Model predictive control of non-holonomic systems: Beyond differential-drive vehicles

Non-holonomic systems are of immense practical value and increasingly subject to automation. However, controlling them accurately, e.g., when parking vehicles, is challenging for automatic control methods, including model predictive control (MPC). Combining results from MPC theory and sub-Riemannian geometry in the form of homogeneous nilpotent system approximations, this paper proposes the first comprehensive, ready-to-apply design procedure for MPC controllers to steer controllable, driftless non-holonomic systems into given setpoints. It can be ascertained that the resulting controllers nominally asymptotically stabilize the setpoint for a large-enough prediction horizon. The design procedure is exemplarily applied to four systems, including the kinematic car and a differentially driven mobile robot with up to two trailers. The controllers use a non-quadratic cost function tailored to the non-holonomic kinematics. Novelly, it is proven that a quadratic cost employed in an otherwise similar controller is insufficient to reliably asymptotically stabilize the closed loop for all considered vehicles. Since quadratic costs are the conventional choice in control, this highlights the relevance of the findings. To the knowledge of the authors, it is the first time that MPC controllers of the proposed structure are formulated and applied to non-holonomic systems beyond very simple ones, in particular (partly) on hardware.

Integral formulas for a foliated sub-Riemannian manifold

We apply the notion of foliation to a nonholonomic manifold, which was introduced for the geometric interpretation of constrained systems in mechanics. We prove a series of integral formulas for a foliated sub-Riemannian manifold, that is, a Riemannian manifold equipped with a distribution $${{\mathscr {D}}}$$ and a foliation $${{\mathscr {F}}}$$ whose tangent bundle is a subbundle of $${{\mathscr {D}}}$$ . Our integral formulas generalize some results for a foliated Riemannian manifold and involve the shape operators of $${\mathscr {F}}$$ with respect to normals in $${\mathscr {D}}$$ , the curvature tensor of induced connection on $${\mathscr {D}}$$ and arbitrary functions depending on elementary symmetric functions of eigenvalues of the shape operators. For a special choice of these functions, integral formulas with the Newton transformations of the shape operators of $${\mathscr {F}}$$ are obtained. Application to a foliated sub-Riemannian manifold with restrictions on the curvature and extrinsic geometry of $${\mathscr {F}}$$ and also to codimension-one foliations are given.

Blowups and blowdowns of geodesics in Carnot groups

This paper provides some partial regularity results for geodesics (i.e., isometric images of intervals) in arbitrary sub-Riemannian and sub-Finsler manifolds. Our strategy is to study infinitesimal and asymptotic properties of geodesics in Carnot groups equipped with arbitrary sub-Finsler metrics. We show that tangents of Carnot geodesics are geodesics in some groups of lower nilpotency step. Namely, every blowup curve of every geodesic in every Carnot group is still a geodesic in the group modulo its last layer. Then as a consequence we get that in every sub-Riemannian manifold any $s$ times iterated tangent of any geodesic is a line, where $s$ is the step of the sub-Riemannian manifold in question. With a similar approach, we also show that blowdown curves of geodesics in sub-Riemannian Carnot groups are contained in subgroups of lower rank. This latter result is also extended to rough geodesics.

Gauss-Bonnet Theorems for Surfaces in Sub-Riemannian Three-dimensional Manifolds

Gauss-Bonnet Theorems for Surfaces in Sub-Riemannian Three-dimensional Manifolds

New second-order optimality conditions in sub-riemannian geometry

We study the geometry of the second-order expansion of the extended end-point map for the sub-Riemannian geodesic problem. Translating the geometric reality into equations we derive new second-order necessary optimality conditions in sub-Riemannian Geometry. In particular, we find an ODE for velocity of an abnormal sub-Riemannian geodesics. It allows to divide abnormal minimizers into two classes, which we propose to call 2-normal and 2-abnormal extremals. In the 2-normal case the above ODE completely determines the velocity of a curve, while in the 2-abnormal case the velocity is undetermined at some, or at all points. With some enhancement of the presented results it should be possible to prove the regularity of all 2-normal extremals (the 2-abnormal case seems to require study of higher-order conditions) thus making a step towards solving the problem of smoothness of sub-Riemannian abnormal geodesics. As a by-product we present a new derivation of Goh conditions. We also prove that the assumptions weaker than these used in [Boarotto, Monti, Palmurella, 2020] to derive third-order Goh conditions, imply piece-wise-C2 regularity of an abnormal extremal. Marty me i Zuzi

К геометрии субримановых многообразий, оснащенных канонической четверть-симметрической связностью

In this article, a sub-Riemannian manifold of contact type is under­stood as a Riemannian manifold equipped with a regular distribution of codimension-one and by a unit structure vector field orthogonal to this distribution. This vector field is called a structural. On a sub-Riemannian manifold of contact type, a quarter-symmetric connection is defined, which is associated with an endomorphism that preserves the distribution of the sub-Riemannian manifold. It is proved that if the connection under study is metric, then the endomorphism associated to it is uniquely de­fined. The structure of the associated endomorphism is found. In the case when the structure vector field is a field of infinitesimal isometries, the quarter-symmetric connection is called the canonical N-connection. An expression is found for the curvature tensor of the canonical N-connection in terms of the Riemann curvature tensor. The properties of the Schouten curvature tensor are investigated, which provide, in particular, the neces­sary symmetries of the curvature tensor of an N-connection for its sec­tional curvature to be well-defined. A relation between the sectional cur­vature of the canonical N-connection and the sectional curvature of the Levi-Civita connection is found. Necessary and sufficient conditions are found under which the sectional curvature of the N-connection and the sectional curvature of the Levi-Civita connection coincide.

Harmonic maps into sub-Riemannian Lie groups

<abstract><p>We define harmonic maps between sub-Riemannian manifolds by generalizing known definitions for Riemannian manifolds. We establish conditions for when a horizontal map into a Lie group with a left-invariant metric structure is a harmonic map. We show that sub-Riemannian harmonic maps can be abnormal or normal, just as sub-Riemannian geodesics. We illustrate our study by presenting the equations for harmonic maps into the Heisenberg group.</p></abstract>

Homogeneous geodesics in sub-Riemannian geometry

We study homogeneous geodesics of sub-Riemannian manifolds, i.e., normal geodesics that are orbits of one-parametric subgroups of isometries. We obtain a criterion for a geodesic to be homogeneous in terms of its initial momentum. We prove that any weakly commutative sub-Riemannian homogeneous space is geodesic orbit, that means all geodesics are homogeneous. We discuss some examples of geodesic orbit sub-Riemannian manifolds. In particular, we show that geodesic orbit Carnot groups are only groups of step 1 and 2. Finally, we get a broad condition for existence of at least one homogeneous geodesic.

А)нормальные экстремали левоинвариантных субфинслеровых квазиметрик на группе 𝐺2,1 × 𝐺2,1

In this paper, we find abnormal extremals of leftinvariant subFinsler quasimetrics on the direct Cartesian square of the connected twodimensional noncommutative Lie group, where the quasimetric is defined by a seminorm on the twodimensional or threedimensional generating subspace of the Lie algebra of this fourdimensional Lie group. A criterion for the nonstrict abnormality of these extremals is established. The solution to these problems is based on the previous general author's results on generating subspaces of fourdimensional Lie algebras and abnormal extremals of leftinvariant subFinsler quasimetrics on connected Lie groups with these Lie algebras. In conclusion, we consider the search problem for geodesics of a leftinvariant subRiemannian metric on the given Lie group, defined by the scalar product on the twodimensional generating subspace of its Lie algebra.

Riemannian and sub-Riemannian structures on a cotangent bundle of Heisenberg group

In this paper we give a classification of left invariant sub-Riemannian structures on cotangent bundle of 2n+1 dimensional Heisenberg group T*H2n+1. We show that the sub-Riemannian metric is tamed by the corresponding Riemannian metric on T*H2n+1. We also describe Riemannian and sub-Riemannian geodesics on T*H2n+1.

On isometric immersions of sub-Riemannian manifolds

We study curvature invariants of a sub-Riemannian manifold (i.e., a manifold with a Riemannian metric on a non-holonomic distribution) related to mutual curvature of several pairwise orthogonal subspaces of the distribution, and prove geometrical inequalities for a sub-Riemannian submanifold. As applications, inequalities are proved for submanifolds with mutually orthogonal distributions that include scalar and mutual curvature. For compact submanifolds, inequalities are obtained that are supported by known integral formulas for almost-product manifolds.

Surfaces of genus g ≥ 1 in 3D contact sub-Riemannian manifolds

We consider smooth embedded surfaces in a 3D contact sub-Riemannian manifold and the problem of the finiteness of the induced distance (i.e., the infimum of the length of horizontal curves that belong to the surface). Recently it has been proved that for a surface having the topology of a sphere embedded in a tight co-orientable structure, the distance is always finite. In this paper we study closed surfaces of genus larger than 1, proving that such surfaces can be embedded in such a way that the induced distance is finite or infinite. We then study the structural stability of the fmiteness/not-finiteness of the distance.

Geometric Inequalities for a Submanifold Equipped with Distributions

The article introduces invariants of a Riemannian manifold related to the mutual curvature of several pairwise orthogonal subspaces of a tangent bundle. In the case of one-dimensional subspaces, this curvature is equal to half the scalar curvature of the subspace spanned by them, and in the case of complementary subspaces, this is the mixed scalar curvature. We compared our invariants with Chen invariants and proved geometric inequalities with intermediate mean curvature squared for a Riemannian submanifold. This gives sufficient conditions for the absence of minimal isometric immersions of Riemannian manifolds in a Euclidean space. As applications, geometric inequalities were obtained for isometric immersions of sub-Riemannian manifolds and Riemannian manifolds equipped with mutually orthogonal distributions.

Riemannian Stochastic Variance-Reduced Cubic Regularized Newton Method for Submanifold Optimization

We propose a stochastic variance-reduced cubic regularized Newton algorithm to optimize the finite-sum problem over a Riemannian submanifold of the Euclidean space. The proposed algorithm requires a full gradient and Hessian update at the beginning of each epoch while it performs stochastic variance-reduced updates in the iterations within each epoch. The iteration complexity of $$O(\epsilon ^{-3/2})$$ to obtain an $$(\epsilon ,\sqrt{\epsilon })$$ -second-order stationary point, i.e., a point with the Riemannian gradient norm upper bounded by $$\epsilon $$ and minimum eigenvalue of Riemannian Hessian lower bounded by $$-\sqrt{\epsilon }$$ , is established when the manifold is embedded in the Euclidean space. Furthermore, the paper proposes a computationally more appealing modification of the algorithm which only requires an inexact solution of the cubic regularized Newton subproblem with the same iteration complexity. The proposed algorithm is evaluated and compared with three other Riemannian second-order methods over two numerical studies on estimating the inverse scale matrix of the multivariate t-distribution on the manifold of symmetric positive definite matrices and estimating the parameter of a linear classifier on the sphere manifold.