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I discuss old and new results on fixed points of local actions by Lie groups G on real and complex 2-manifolds, and zero sets of Lie algebras of vector fields. Results of E. Lima, J. Plante and C. Bonatti are reviewed.

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally-compact group that has a compact quotient group modulo its identity component and, thus, in particular, each compact and each connected locally-compact group; it also includes all locally-compact Abelian groups. This paper provides an overview of the structure theory and the Lie theory of pro-Lie groups, including results more recent than those in the authors’ reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly-complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function that links the two. (A topological vector space is weakly complete if it is isomorphic to a power RX of an arbitrary set of copies of R. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected to pro-Lie groups.

This is a survey on harmonic maps and biharmonic maps into (1) Riemannian manifolds of non-positive curvature, (2) compact Lie groups or (3) compact symmetric spaces, based mainly on my recent works on these topics.

In computational anatomy, organ’s shapes are often modeled as deformations of a reference shape, i.e., as elements of a Lie group. To analyze the variability of the human anatomy in this framework, we need to perform statistics on Lie groups. A Lie group is a manifold with a consistent group structure. Statistics on Riemannian manifolds have been well studied, but to use the statistical Riemannian framework on Lie groups, one needs to define a Riemannian metric compatible with the group structure: a bi-invariant metric. However, it is known that Lie groups, which are not a direct product of compact and abelian groups, have no bi-invariant metric. However, what about bi-invariant pseudo-metrics? In other words: could we remove the assumption of the positivity of the metric and obtain consistent statistics on Lie groups through the pseudo-Riemannian framework? Our contribution is two-fold. First, we present an algorithm that constructs bi-invariant pseudo-metrics on a given Lie group, in the case of existence. Then, by running the algorithm on commonly-used Lie groups, we show that most of them do not admit any bi-invariant (pseudo-) metric. We thus conclude that the (pseudo-) Riemannian setting is too limited for the definition of consistent statistics on general Lie groups.

Abstract This paper computes the obstruction to the existence of equivariant extensions of basic gerbes over non-simply connected compact simple Lie groups. By modifying a (finite dimensional) construction of Gawȩdzki and Reis (2004), we exhibit basic equivariant bundle gerbes over non-simply connected compact simple Lie groups. [ABSTRACT FROM AUTHOR]

In this paper, we study the classical semicontinuity-of-automorphism-groups theorem due to Greene-Krantz from the point of view of how sharp the results are. Many examples, in both one and several complex variables, are provided. [ABSTRACT FROM AUTHOR]

Unsteady two-dimensional flows of incompressible viscoelastic Maxwell medium with upper, low and corotational convective derivatives in the rheological constitutive law are considered. A class of partially invariant solutions is analyzed. Using transition to Lagrangian coordinates, an exact solution of the problem of unsteady flow near free-stagnation point was constructed. For the model with Johnson–Segalman convected derivative and special linear dependence of the vertical component of velocity, the general solutions were derived. [ABSTRACT FROM AUTHOR]

We present a new method of investigation of G -structures on orbifolds. This method is founded on the use of appropriate G -structure on an associated model manifold. Applying this method we prove the existence and the uniqueness of finite dimensional Lie group structures on the full automorphism group of an elliptic G -structure on a compact orbifold. This generalizes the well-known theorem for elliptic G -structures on compact manifolds. As an application we get that the group of all automorphisms of an almost complex structure on a compact orbifold is a finite dimensional Lie group that carries a uniquely defined Lie group structure. [ABSTRACT FROM AUTHOR]

Among many physical and mechanical engineering problems, parallel car parking has drawn the attention of many researchers as it shows how methods of Lie algebra and differential geometry can be applied elegantly to explain the behavior of such a system. In this paper we have reviewed some aspects of this problem in this context and analyse the stability and controllability of such a system by some unconventional integrators like Kahan and Lie trotter integrator. At the same time we have also given the results obtained through conventional Runge Kutta integrator to see a comparison. [ABSTRACT FROM AUTHOR]

We find necessary and sufficient conditions for a finite K-bi-invariant measure on a compact Gelfand pair (G,K) to have a square-integrable density. For convolution semigroups, this is equivalent to having a continuous density in positive time. When (G,K) is a compact Riemannian symmetric pair, we study the induced transition density for G-invariant Feller processes on the symmetric space X = G/K. These are obtained as projections of K-bi-invariant Lévy processes on G, whose laws form a convolution semigroup. We obtain a Fourier series expansion for the density, in terms of spherical functions, where the spectrum is described by Gangolli’s Lévy-Khintchine formula. The density of returns to any given point on X is given by the trace of the transition semigroup, and for subordinated Brownian motion, we can calculate the short time asymptotics of this quantity using recent work of Bañuelos and Baudoin. In the case of the sphere, there is an interesting connection with the Funk-Hecke theorem. [ABSTRACT FROM AUTHOR]

The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra o(3, 1) as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane E2. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a previously developed discretization procedure, we present a difference scheme that is invariant under the group O(3, 1) and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under O(3, 1) and is itself invariant under a subgroup of O(3, 1), namely, the O(2) rotations of the Euclidean plane. [ABSTRACT FROM AUTHOR]

In this paper, we study Mabuchi's K-energy on a compactification M of a reductive Lie group G , which is a complexification of its maximal compact subgroup K . We give a criterion for the properness of K-energy on the space of K × K -invariant Kähler potentials. In particular, it turns to give an alternative proof of Delcroix's theorem for the existence of Kähler–Einstein metrics in case of Fano manifolds M . We also study the existence of minimizers of K-energy for general Kähler classes of M . [ABSTRACT FROM AUTHOR]

We give a full classification of Lie algebras of specific type in complexified Clifford algebras. These 16 Lie algebras are direct sums of subspaces of quaternion types. We obtain isomorphisms between these Lie algebras and classical matrix Lie algebras in the cases of arbitrary dimension and signature. We present 16 Lie groups: one Lie group for each Lie algebra associated with this Lie group. We study connection between these groups and spin groups. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new approach based on distribution descriptors for action recognition in depth videos. Our local features are computed from binary patterns which incorporate the shape and motion cues for effective action recognition. Given pixel-level features, our approach estimates video local statistics in a hierarchical manner, where the distribution of pixel-level features and that of frame-level descriptors are modeled using single Gaussians. In this way, our approach constructs video descriptors directly from low-level features without resorting to codebook learning required by Bag-of-features (BoF) based approaches. In order to capture the spatial geometry and temporal order of a video, we use a spatio-temporal pyramid representation for each video. Our approach is validated on six benchmark datasets, i.e. MSRAction3D, MSRGesture3D, DHA, SKIG, UTD-MHAD and CAD-120. The experimental results show that our approach gives good performance on all the datasets. In particular, it achieves state-of-the-art accuracies on DHA, SKIG and UTD-MHAD datasets. [ABSTRACT FROM AUTHOR]

We consider the quotient manifold of the manifold of nondegenerate affinor fields on a compact manifold with respect to the action of the group of nowhere vanishing functions. This manifold is endowed with a structure of infinite-dimensional Lie group. On this Lie group, we construct an object of linear connection with respect to which all left-invariant vector fields are covariantly constant (the Cartan connection). We also find the geodesics of the Cartan connection. [ABSTRACT FROM AUTHOR]

Heavy-duty industrial robots have great advantages in the manufacturing industry. Considering the heavy process load and low stiffness of the robot, an accurate and efficient dynamic model plays an important role in the behavior analysis and performance improvement in the robot. This paper presents a novel methodology for the inverse dynamic analysis of the heavy-duty industrial robot with elastic joints. In particular, high-order kinematics and dynamics are concisely deduced using Lie group to deal with elastic joints for the robot inverse dynamic analysis. Meanwhile, position errors of the end-effector due to elastic joints are evaluated through the inverse dynamic analysis when the robot is in heavy-duty applications. Compared with previous approaches, the advantage of proposed method is that new formulas for inverse dynamic analysis are shown to be more concise and computationally efficient using Lie group. Moreover, the position error evaluation method considering dynamic forces is proved to be more accurate than the traditional method when the robot is in the high-speed application. Because of the high computational efficiency and accurate evaluation results, the proposed approach is applicable to trajectory optimization and position error compensation, especially for the robot in heavy-load and high-speed applications. [ABSTRACT FROM AUTHOR]

We consider unsteady flows of incompressible viscoelastic Maxwell medium with upper, low, and Jaumann convective derivatives in the rheological constitutive law. We give characteristics of a system of equations that describe a three-dimensional motion of such a medium for all three types of convective derivative. In the general case, due to the incompressibility condition, the equations of motion have both real and complex characteristics. We study group properties of this system in the two-dimensional case. On this basis, we choose submodels of the Maxwell model that can be reduced to hyperbolic ones. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. We also present concrete examples of invariant solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special class of solvable Lie groups such that for any x, y in their Lie algebras, [x, y] is a linear combination of x and y. We investigate the harmonicity properties of invariant vector fields of this kind of Lorentzian Lie groups. It is shown that any invariant unit time-like vector field is spatially harmonic. Moreover, we determine all vector fields which are critical points of the energy functional restricted to the space of smooth vector fields. [ABSTRACT FROM AUTHOR]

In this paper we generalize the Haefliger theorem on completly solvable Lie foliations. We prove that every completely solvable Lie foliation on a compact manifold is the inverse image of a homogeneous foliation. Every manifold in this paper is compact and our Lie group G is connected and simply connected. [ABSTRACT FROM AUTHOR]

We consider unsteady flows of incompressible viscoelastic Maxwell medium with upper, low, and Jaumann convective derivatives in the rheological constitutive law. We give characteristics of a system of equations that describe a three-dimensional motion of such a medium for all three types of convective derivative. In the general case, due to the incompressibility condition, the equations of motion have both real and complex characteristics. We study group properties of this system in the two-dimensional case. On this basis, we choose submodels of the Maxwell model that can be reduced to hyperbolic ones. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. We also present concrete examples of invariant solutions. [ABSTRACT FROM AUTHOR]

Let G be a simple, simply-connected, compact Lie group of low rank relative to a fixed prime p . After localization at p , there is a space A which “generates” G in a certain sense. Assuming G satisfies a homotopy nilpotency condition relative to p , we show that the Samelson product 〈 1 G , 1 G 〉 of the identity of G equals the order of the Samelson product 〈 ı , ı 〉 of the inclusion ı : A → G . Applying this result, we calculate the orders of 〈 1 G , 1 G 〉 for all p -regular Lie groups and give bounds of the orders of 〈 1 G , 1 G 〉 for certain quasi- p -regular Lie groups. [ABSTRACT FROM AUTHOR]

In this paper, it has been proved that the universal covering space of a top space (local top space) is a top space (local top space). In addition, it has been shown that the universal covering space of a top space with constant rank is an upper top space. However, there are some upper top spaces which are not an universal covering space. As follows, it has been proved that the fundamental group of a right top space is Abelian, and some properties of generalized left action of a top space on a manifold have been investigated. [ABSTRACT FROM AUTHOR]

We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This implies that all algebraic (not necessarily affine) groups over fields of characteristic zero and some transformation groups of complex spaces and Riemannian manifolds are Jordan. [ABSTRACT FROM AUTHOR]

This work presents a novel geometric framework for self-balancing as well as planar motion control of wheeled vehicles with two fewer control inputs than the configuration variables. For self-balancing control, we shape the kinetic energy in such a way that the upright direction of the robot's body becomes a nonlinearly stable equilibrium for the corresponding controlled Lagrangian which is inherently a saddle point. Then for planar motion control of the robot, we set its position and attitude as an element of the special Euclidean group SE(2) and apply a logarithmic feedback control taking advantage of the Lie group exponential coordinates. For simulation and evaluating the controllers, the unified dynamic model of the self-balancing mobile robot (SMR) is developed using the constrained Euler-Lagrange equations. [ABSTRACT FROM AUTHOR]

The first integrals of n -th order autonomous systems are considered. By getting integrating factors using n − 1 first integrals of the systems, a necessary condition on the existence of global first integrals of n -th order autonomous systems is presented. It is also proved that n − 1 functionally independent first integrals of an n -th order autonomous system can be obtained if the system possesses n − 1 functionally independent integrating factors. Based on one-parameter Lie groups admitted by n -th order autonomous systems, several methods to obtain first integrals of the systems are presented. Simultaneously, several related examples are given to illustrate the feasibility and the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We study schemes for interpolating functions that take values in the special orthogonal group SO(n). Our focus is on interpolation schemes obtained by embedding SO(n) in a linear space, interpolating in the linear space, and mapping the result onto SO(n) via the closest point projection. The resulting interpolants inherit both the order of accuracy and the regularity of the underlying interpolants on the linear space. This enables one to construct interpolants having C1-continuity in a straightforward way, even when the data do not lie in a single coordinate chart. The values and derivatives of the interpolants admit efficient evaluation via either explicit formulas or iterative algorithms, which we detail for two choices of embeddings: the embedding of SO(n) in the space of n×n matrices and, when n = 3, the identification of SO(3) with the set of unit quaternions. Along the way, we point out a connection between these interpolation schemes and geodesic finite elements. We illustrate the utility of these interpolation schemes by numerically computing minimum acceleration curves on SO(n), a task which is handled naturally with SO(n)-valued finite elements having C1-continuity. [ABSTRACT FROM AUTHOR]

We find the groups of motions of eight three-dimensional maximal mobility geometries. These groups are actions of just three Lie groups SL2(R)ΔN, SL2(C)R, and SL2(R)⊗SL2(R) on the space R3, where N is a normal abelian subgroup. We also find explicit expressions for these actions. [ABSTRACT FROM AUTHOR]

In this paper, a stabilizer dedicated for a unicycle-like robot is considered. The proposed smooth control law ensures the global boundedness of position and orientation trajectories to a neighbourhood of the desired point with an arbitrarily selected radius and it is robust to bounded additive measurement noises. The controller consists of a smooth hybrid navigation algorithm and a smooth feedback based on the transverse function approach. The stability proof, simulation and experimental results illustrating properties of the algorithm are discussed. [ABSTRACT FROM AUTHOR]

For a complete Cartan foliation (M,F) we introduce two algebraic invariants g0(M,F) and g1(M,F) which we call structure Lie algebras. If the transverse Cartan geometry of (M,F) is effective then g0(M,F) = g1(M,F). Weprove that if g0(M,F) is zero then in the category of Cartan foliations the group of all basic automorphisms of the foliation (M,F) admits a unique structure of a finite-dimensional Lie group. In particular, we obtain sufficient conditions for this group to be discrete. We give some exact (i.e. best possible) estimates of the dimension of this group depending on the transverse geometry and topology of leaves. We construct several examples of groups of all basic automorphisms of complete Cartan foliations. [ABSTRACT FROM AUTHOR]

Let be an almost coKähler -manifold such that the vertical Ricci curvatures are invariant along the Reeb vector field. In this paper, we prove that is semi-symmetric if and only if it is coKähler. [ABSTRACT FROM AUTHOR]

Abstract: In the setting of a Lie group of polynomial volume growth, we derive inequalities of Caffarelli–Kohn–Nirenberg type, where the weights involved are powers of the Carnot–Caratheodory distance associated with a fixed system of vector fields which satisfy the Hörmander condition. The use of weak L p spaces is crucial in our proofs and we formulate these inequalities within the framework of L p , q Lorentz spaces (a scale of (quasi)‐Banach spaces which extend the more classical L p Lebesgue spaces) thereby obtaining a refinement of, for instance, Sobolev and Hardy–Sobolev inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we suppose that M is a Riemannian manifold of constant negative curvature under the action of a Lie subgroup G of IsoðMÞ such that the maximum of the dimension of the orbits is dim M - 2. Then, we study topological properties of M under some conditions. [ABSTRACT FROM AUTHOR]

Let G be a semisimple Lie group with a maximal torus T. We present an explicit formula for the Borel transgression τ: H¹(T) → H²(G=T) of the fibration G → G=T. This formula corrects an error in the paper [9], and has been applied to construct the integral cohomology rings of compact Lie groups in the sequel works [4, 6]. [ABSTRACT FROM AUTHOR]

Here, the global and coordinate‐free approach has been applied to study the leaderless attitude synchronisation problem of multiple rigid bodies with kinematic evolving on Lie group SO (3). A decentralised consensus law based on the gradient of a smooth cost function which depends on the relative attitude of neighbouring agents is proposed. The advantage of using this cost function lies in the fact that there is a relation between the Hessian matrix of this function in a consensus configuration and the Laplacian matrix of a graph. In addition, a new approach for analysing the attitude synchronisation problem was developed and proved that convergence to synchronisation can be achieved for any connected and undirected graph if the initial attitudes of neighbouring agents are sufficiently close. Using the critical points of the cost function, the effects of graph topology on synchronisation problem were investigated. The initial attitudes sets ensuring the convergence to synchronisation are obtained for the undirected tree, complete, and ring graphs. The initial set is more restrictive for ring graph owing to the existence of other critical points induced by graph topology. Eventually, using the appropriate Lyapunov function, sufficient condition that guarantees convergence to synchronisation for general connected graph was obtained, and numerical simulation carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Let G be a full connected semisimple isometry Lie group of a connected Riemannian symmetric space M = G/K with the stabilizer K; p : G → G/K = M the canonical projection which is a Riemannian submersion for some G-left invariant and K-right invariant Riemannian metric on G, and d is a (unique) sub-Riemannian metric on G defined by this metric and the horizontal distribution of the Riemannian submersion p. It is proved that each geodesic in (G, d) is normal and presents an orbit of some one-parameter isometry group. By the Solov'ev method, using the Cartan decomposition for M = G/K, the author found the curvatures of the homogeneous sub-Riemannian manifold (G, d). In the case G = Sp(1) × Sp(1) with the Riemannian symmetric space S3 = Sp(1) = G/ diag(Sp(1) × Sp(1)) the curvatures and torsions are calculated of images in S3 of all geodesics on (G, d) with respect to p. [ABSTRACT FROM AUTHOR]

We define a diffeology on the Milnor classifying space of a diffeological group G , constructed in a similar fashion to the topological version using an infinite join. Besides obtaining the expected classification theorem for smooth principal bundles, we prove the existence of a diffeological connection on any principal bundle (with mild conditions on the bundles and groups), and apply the theory to some examples, including some infinite-dimensional groups, as well as irrational tori. [ABSTRACT FROM AUTHOR]