We complete a uniform construction of canonical absolute parallelism for bracket generating rank 2 distributions with 5-dimensional cube on n-dimensional manifold with n >= 5 by showing that the condition of maximality of class that was assumed previously by Doubrov-Zelenko [J. London Math. Soc., 80(3):545-566, 2009] for such a construction holds automatically at generic points. This also gives analogous constructions in the case when the cube is not 5-dimensional but the distribution is not Goursat through the procedure of iterative Cartan deprolongation. This together with the classical theory of Goursat distributions covers in principle the local geometry of all bracket generating rank 2 distributions in a neighborhood of generic points. As a byproduct, for any n >= 5 we describe the maximally symmetric germs among bracket generating rank 2 distributions with 5-dimensional cube, as well as among those which reduce to such a distribution under a fixed number of Cartan deprolongations. Another consequence of our results on maximality of class is for optimal control problems with constraint given by a rank 2 distribution with 5-dimensional cube: it implies that for a generic point q0 of M, there are plenty abnormal extremal trajectories of corank 1 (which is the minimal possible corank) starting at q0. The set of such points contains all points where the distribution is equiregular.

We study the Doubrov–Zelenko symplectification procedure for rank 2 distributions with 5-dimensional cube—originally motivated by optimal control theory—through the lens of Tanaka–Morimoto theory for normal Cartan connections. In this way, for ambient manifolds of dimension n ≥ 5 , we prove the existence of the normal Cartan connection associated with the symplectified distribution. Furthermore, we show that this symplectification can be interpreted as the (n-4) th iterated Cartan prolongation at a generic point. This interpretation naturally leads to two questions for an arbitrary rank 2 distribution with 5-dimensional cube: (1) Is the (n-4) th iterated Cartan prolongation the minimal iteration where the Tanaka symbols become unified at generic points? (2) Is the (n-4) th iterated Cartan prolongation the minimal iteration admitting a normal Cartan connection via Tanaka–Morimoto theory? Our main results demonstrate that: (a) For n > 5 , the answer to the second question is positive (in contrast to the classical n = 5 case from G_2 -parabolic geometries); (b) For n ≥ 5 , the answer to the first question is negative: unification occurs already at the (n-5) th iterated Cartan prolongation.

We generalize the classical Tanaka result on the finiteness of symmetry algebra for non-degenerate pseudo-product structures to the case when the completely-integrable distributions defining the pseudo-product structure are no longer concentrated in the degree -1. In order to do this, we modify the notion of universal prolongation of graded nilpotent Lie algebras and generalize the original finiteness criterion of Tanaka. Using this result, we demonstrate that in real analytic category, distributions that are controllable by regular abnormal extremal trajectories, also known as singularly transitive, have finite-dimensional symmetries. This result settles Problem V in the affirmative from the 2013 list of open problems by Andrei Agrachev. Additionally, we discuss applications to symmetries and natural equivalence problems for systems of ODEs of mixed order.

In 1910, Élie Cartan famously realized the split real form of the exceptional Lie group G_2 as the symmetry group of the maximally symmetric rank 2 distribution on a 5-dimensional manifold with the small growth vector (2,3,5). In this paper, we discover a new appearance of G_2 in the geometric theory of distributions, arising from a rank 3 distribution on an 8-dimensional manifold with the growth vector (3,6,8). The algebra of infinitesimal symmetries of this distribution at any point is 29-dimensional and isomorphic to (𝔤_2 ⊕ℝ) ⋉ W, where 𝔤_2 is the Lie algebra of G_2 and W is an adjoint module of 𝔤_2. Our model possesses three remarkable properties. First, it is maximally symmetric among all bracket-generating rank 3 distributions with a 6-dimensional square (a family that includes both (3,6,8) and (3,6,7,8) distributions). To the best of our knowledge, this is the first example of a family of distributions defined by a set of prescribed small growth vectors in which maximal symmetry is achieved by a member whose growth vector is not the longest. Second, this model provides the first counterexample to the conjecture that all bracket-generating rank 3 distributions with a 6-dimensional square are of maximal class at a generic point (which is known to hold in dimensions 6 and 7). Third, further analysis yields the control-theoretic consequence that all abnormal extremal trajectories of this model originating at any point of the ambient manifold have a corank of at least 2. To our knowledge, this is the first example with this property among bracket-generating distributions with generic small growth vector for a given rank and ambient dimension. We also give an interpretation of our model in terms of split-octonions, more precisely, in terms of a natural algebraic structure on the tangent bundle to split octonions.

We construct a sequence of rank 3 distributions on $n$-dimensional manifolds for any $n\geq 7$ such that the dimension of their symmetry group grows exponentially in $n$ (more precisely it is equal to $\operatorname{Fib}_{n-1}+n+2$, where $\operatorname{Fib}_n$ is the $n$-th Fibonacci number, starting with $\operatorname{Fib}_1=\operatorname{Fib}_2=1$) and such that the maximal order of weighted jet needed to determine these symmetries grows quadratically in $n$. These examples are in sharp contrast with the parabolic geometries where the dimension of a symmetry group grows polynomially with respect to the dimension of the ambient manifold and the corresponding maximal order of weighted jet space is equal to the degree of nonholonomy of the underlying distribution plus $1$. Our models are closely related to the geometry of certain curves of symplectic flags and of the rational normal curves.

The classical result of Eisenhart states that, if a Riemannian metric g admits a Riemannian metric that is not constantly proportional to g and has the same (parameterized) geodesics as g in a neighborhood of a given point, then g is a direct product of two Riemannian metrics in this neighborhood. We introduce a new generic class of step 2 graded nilpotent Lie algebras, called ad -surjective, and extend the Eisenhart theorem to sub-Riemannian metrics on step 2 distributions with ad -surjective Tanaka symbols. The class of ad-surjective step 2 nilpotent Lie algebras contains a well-known class of algebras of H-type as a very particular case.

In many applied problems one seeks to identify and count the critical points of a particular eigenvalue of a smooth parametric family of self-adjoint matrices, with the parameter space often being known and simple, such as a torus. Among particular settings where such a question arises are the Floquet–Bloch decomposition of periodic Schrödinger operators, topology of potential energy surfaces in quantum chemistry, spectral optimization problems such as minimal spectral partitions of manifolds, as well as nodal statistics of graph eigenfunctions. In contrast to the classical Morse theory dealing with smooth functions, the eigenvalues of families of self-adjoint matrices are not smooth at the points corresponding to repeated eigenvalues (called, depending on the application and on the dimension of the parameter space, the diabolical/Dirac/Weyl points or the conical intersections). This work develops a procedure for associating a Morse polynomial to a point of eigenvalue multiplicity; it utilizes the assumptions of smoothness and self-adjointness of the family to provide concrete answers. In particular, we define the notions of non-degenerate topologically critical point and generalized Morse family, establish that generalized Morse families are generic in an appropriate sense, establish a differential first-order conditions for criticality, as well as compute the local contribution of a topologically critical point to the Morse polynomial. Remarkably, the non-smooth contribution to the Morse polynomial turns out to depend only on the size of the eigenvalue multiplicity and the relative position of the eigenvalue of interest and not on the particulars of the operator family; it is expressed in terms of the homologies of Grassmannians.

The classical result of Eisenhart states that if a Riemannian metric $g$ admits a Riemannian metric that is not constantly proportional to $g$ and has the same (parameterized) geodesics as $g$ in a neighborhood of a given point, then $g$ is a direct product of two Riemannian metrics in this neighborhood. We introduce a new generic class of step $2$ graded nilpotent Lie algebras, called $\mathrm{ad}$-surjective, and extend the Eisenhart theorem to sub-Riemannian metrics on step 2 distributions with $\mathrm{ad}$-surjective Tanaka symbols. The class of ad-surjective step 2 nilpotent Lie algebras contains a well-known class of algebras of H-type as a very particular case.

We construct canonical absolute parallelisms over real-analytic manifolds equipped with 2-nondegenerate, hypersurface-type CR structures of arbitrary odd dimension not less than 7 whose Levi kernel has constant rank belonging to a broad subclass of CR structures that we label as recoverable. For this we develop a new approach based on a reduction to a special flag structure, called the dynamical Legendrian contact structure, on the leaf space of the CR structure's associated Levi foliation. This extends the results of [23] from the case of regular CR symbols constituting a discrete set in the set of all CR symbols to the case of the arbitrary CR symbols for which the original CR structure can be uniquely recovered from its corresponding dynamical Legendrian contact structure. We find an explicit criterion for this recoverability. In particular, if the rank of the Levi kernel is 1 and the dimension of the CR manifold is not less than 7, then for each given signature of the reduced Levi form in the space of all CR symbols (which depend on continuous parameters) there are no more than 2 symbols for which the aforementioned recoverability fails, and while the present method is applicable for all but those 2 cases, they can be treated separately by the method of [23]. Our method clarifies the relationship between the bigraded Tanaka prolongation of regular symbols developed in [23] and their usual Tanaka prolongation, providing a geometric interpretation of conditions under which they are equal. Motivated by the search for homogeneous models with given nonregular symbols, we also describe a process of reduction from the original natural frame bundle, which is inevitable for structures with nonregular CR symbols. We demonstrate this reduction procedure for examples whose underlying manifolds have dimension 7 and 9.We show that, for any fixed rank r>1, in the set of all CR symbols associated with 2-nondegenerate, hypersurface-type CR manifolds of odd dimension greater than 4r+1 with rank r Levi kernel, the CR symbols not associated with any homogeneous model are generic, and, for r=1, the same result holds if the CR structure is pseudoconvex.

We prove that for every n ≥ 3 the sharp upper bound for the dimension of the symmetry groups of homogeneous, 2-nondegenerate, (2n + 1)-dimensional CR manifolds of hypersurface type with a 1-dimensional Levi kernel is equal to n2 + 7, and simultaneously establish the same result for a more general class of structures characterized by weakening the homogeneity condition. This supports Beloshapka's conjecture stating that hypersurface models with a maximal finite dimensional group of symmetries for a given dimension of the underlying manifold are Levi nondegenerate.

Typically, a linear projection of the Grassmannian in its Plücker embedding is generically injective, unless the image of the Grassmannian is a linear space. A notable exception are self-adjoint linear projections, which have even degree. We consider linear projections of Gr3C6 with low-dimensional centers of projection. When the center has dimension less than five, we show that the projection has degree 1. When the center has dimension five and the projection has degree greater than 1, we show that it is self-adjoint.

In the famous 1910 “cinq variables” paper Cartan showed in particular that for maximally nonholonomic rank 2 distributions in ℝ 5 with non-zero covariant binary biquadratic form the dimension of the pseudo-group of local symmetries does not exceed 7 and among such distributions he described the one-parametric family of distributions for which this pseudo-group is exactly 7-dimensional. Using the novel interpretation of the Cartan covariant binary biquadratic form via the classical Wilczynski invariant of curves in projective spaces associated with abnormal extremals of the distributions [4, 27, 28] one can generalize this Cartan result to rank 2 distributions in ℝ n satisfying certain genericity assumption, called maximality of class, for arbitrary n ≥ 5. In the present paper for any rank 2 distribution of maximal class with at least one nonvanishing generalized Wilczynski invariants we construct the canonical frame on a (2 n − 3)-dimensional bundle and describe explicitly the moduli spaces of the most symmetric models. The relation of our results to the divergence equivalence of Lagrangians of higher order is given as well.

Abstract This article is devoted to the local geometry of everywhere 2-nondegenerate CR manifolds M of hypersurface type. An absolute parallelism for such structures was recently constructed independently by Isaev and Zaitsev, Medori and Spiro, and Pocchiola in the minimal possible dimension ( dim ⁡ M = 5 {\dim M=5} ), and for dim ⁡ M = 7 {\dim M=7} in certain cases by the first author. In the present paper, we develop a bigraded (i.e., ℤ × ℤ {\mathbb{Z}\times\mathbb{Z}} -graded) analog of Tanaka’s prolongation procedure to construct an absolute parallelism for these CR structures in arbitrary (odd) dimension with Levi kernel of arbitrary admissible dimension. We introduce the notion of a bigraded Tanaka symbol – a complex bigraded vector space – containing all essential information about the CR structure. Under the additional regularity assumption that the symbol is a Lie algebra, we define a bigraded analog of the Tanaka universal algebraic prolongation, endowed with an anti-linear involution, and prove that for any CR structure with a given regular symbol there exists a canonical absolute parallelism on a bundle whose dimension is that of the bigraded universal algebraic prolongation. Moreover, we show that for each regular symbol there is a unique (up to local equivalence) such CR structure whose algebra of infinitesimal symmetries has maximal possible dimension, and the latter algebra is isomorphic to the real part of the bigraded universal algebraic prolongation of the symbol. In the case of 1-dimensional Levi kernel we classify all regular symbols and calculate their bigraded universal algebraic prolongations. In this case, the regular symbols can be subdivided into nilpotent, strongly non-nilpotent, and weakly non-nilpotent. The bigraded universal algebraic prolongation of strongly non-nilpotent regular symbols is isomorphic to the complex orthogonal algebra 𝔰 ⁢ 𝔬 ⁢ ( m , ℂ ) {\mathfrak{so}(m,\mathbb{C})} , where m = 1 2 ⁢ ( dim ⁡ M + 5 ) {m=\tfrac{1}{2}(\dim M+5)} . Any real form of this algebra – except 𝔰 ⁢ 𝔬 ⁢ ( m ) {\mathfrak{so}(m)} and 𝔰 ⁢ 𝔬 ⁢ ( m - 1 , 1 ) {\mathfrak{so}(m-1,1)} – corresponds to the real part of the bigraded universal algebraic prolongation of exactly one strongly non-nilpotent regular CR symbol. However, for a fixed dim ⁡ M ≥ 7 {\dim M\geq 7} the dimension of the bigraded universal algebraic prolongations of all possible regular CR symbols achieves its maximum on one of the nilpotent regular symbols, and this maximal dimension is 1 4 ⁢ ( dim ⁡ M - 1 ) 2 + 7 {\frac{1}{4}(\dim M-1)^{2}+7} .

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).

A typical linear projection of the Grassmannian in its Plucker embedding is injective, unless its image is a projective space. A notable exception are self-adjoint linear projections, which have even degree. We consider linear projections of Gr_3(C^6) with low-dimensional centers of projection. When the center has dimension less than five, we show that the projection has degree 1. When the center has dimension five and the projection has degree greater than 1, we show that it is self-adjoint.

Motivated by a problem in local differential geometry of Cauchy-Riemann (CR) structures of hypersurface type, we find a canonical form for pairs consisting of a nondegenerate Hermitian form and a self-adjoint antilinear operator, or, equivalently, consisting of a nondegenerate Hermitian form and a symmetric bilinear form. This generalizes the only previously known results on simultaneous normalization of such pairs, namely, the results of [2] on simultaneous diagonalization of these pairs in the case where the Hermitian form is positive definite and of [11], where a criterion for simultaneous diagonalization is given. (C) 2019 Elsevier Inc. All rights reserved.

Consider a smooth connected manifold M equipped with a bracket generating distribution D . Two sub-Riemannian metrics on ( M , D ) are said to be projectively (resp. affinely ) equivalent if they have the same geodesics up to reparameterization (resp. up to affine reparameterization). A sub-Riemannian metric g is called rigid (resp. conformally rigid ) with respect to projective/affine equivalence, if any sub-Riemannian metric which is projectively/affinely equivalent to g is constantly proportional to g (resp. conformal to g ). In the Riemannian case the local classification of projectively (resp. affinely) equivalent metrics was done in the classical work (Levi-Civita in Ann Mat Ser 2a 24:255–300, 1896 ; resp. Eisenhart in Trans Am Math Soc 25(2):297–306, 1923 ). In particular, a Riemannian metric which is not rigid with respect to one of the above equivalences satisfies the following two special properties: its geodesic flow possesses a collection of nontrivial integrals of special type and the metric induces certain canonical product structure on the ambient manifold. The only proper sub-Riemannian cases to which these classification results were extended so far are sub-Riemannian metrics on contact and quasi-contact distributions (Zelenko in J Math Sci (NY) 135(4):3168–3194, 2006 ). The general goal is to extend these results to arbitrary sub-Riemannian manifolds. In this article we establish two types of results toward this goal: if a sub-Riemannian metric is not conformally rigid with respect to the projective equivalence, then, first, its flow of normal extremals has at least one nontrivial integral quadratic on the fibers of the cotangent bundle and, second, the nilpotent approximation of the underlying distribution at any point admits a product structure. As a consequence we obtain two types of genericity results for rigidity: first, we show that a generic sub-Riemannian metric on a fixed pair ( M , D ) is conformally rigid with respect to projective equivalence. Second, we prove that, except for special pairs ( m , n ), for a generic distribution D of rank m on an n -dimensional manifold, every sub-Riemannian metric on D is conformally rigid with respect to the projective equivalence. For the affine equivalence in both genericity results conformal rigidity can be replaced by usual rigidity.

The aim of these notes is to describe how to construct canonical bundles of moving frames and differential invariants for parametrized curves in Lagrangian Grassmannians, at least in the monotonic case. Such curves appear as Jacobi curves of sub-Riemannian extremals [1,2] . Originally this construction was done in [6,7], where it uses the specifics of Lagrangian Grassmannian. In later works [3,4] a much more general theory for construction of canonical bundles of moving frames for parametrized or unparametrized curves in the so-called generalized flag varieties was developed, so that the problem which is discussed here can be considered as a particular case of this general theory. Although this was briefly discussed at the very end of [3], the application of the theory of [3,4] to obtain the results of [6,7] were never written in detail and this is our goal here. We believe that this exposition gives a more conceptual point of view on the original results of [6,7] and especially clarifies the origin of the normalization conditions of the canonical bundles of moving frames there, which in fact boil down to a choice of a complement to a certain subspace of the symplectic Lie algebra. The notes in almost the same form are included under my authorship as the Appendix in the book "A Comprehensive Introduction to sub-Riemannian Geometry from Hamiltonian view point" by A. Agrachev D. Barilari, U. Boscain. Following the style of this book, some not difficult statements are given as exercises for pedagogical purposes.

We previously obtained a congruence modulo four for the number of real solutions to many Schubert problems on a square Grassmannian given by osculating flags. Here we consider Schubert problems given by more general isotropic flags, and prove this congruence modulo four for the largest class of Schubert problems that could be expected to exhibit this congruence.

The positive impacts of platooning on travel time reliability, congestion, emissions, and energy consumption have been shown for homogeneous roadway segments. However, speed limit changes frequently throughout the transportation network, due to either safety-related considerations (e.g., workzone operations) or congestion management schemes (e.g., speed harmonization systems). These abrupt changes in speed limit can result in shock- wave formation and cause travel time unreliability. Therefore, designing a platooning strategy for tracking a reference velocity profile is critical to enabling end-to-end platooning. Accordingly, this study introduces a generalized control model to track a desired velocity profile, while ensuring safety in the platoon of autonomous vehicles. We define appropriate natural error terms and the target curve in the state space of the control system, which is the set of points where all error terms vanish and corresponds to the case when all vehicles move with the desired velocities and in the minimum safe distance between them. In this way, we change the tracking velocity profile problem into a state- feedback stabilization problem with respect to the target curve. Under certain mild assumptions on the Lipschitz constant of the speed drop profile, we show that the stabilizing feedback can be obtained via introducing a natural dynamics for the maximum of the error terms for each vehicle. Moreover, we show that with this stabilizing feedback collisions will not occur if the initial state of the system of vehicles is sufficiently close to the target curve. We also show that the error terms remain bounded throughout the time and space. Two scenarios were simulated, with and without initial perturbations, and results confirmed the effectiveness of the proposed control model in tracking the speed drop while ensuring safety and string stability.