The equations of motion of a charged ideal fluid, respectively the superconductivity equation (both in a given magnetic field) are showed to be geodesic equations on a general, respectively central extension of the group of volume preserving diffeomorphisms with right invariant metric. For this, quantization of the magnetic flux is required. We do curvature computations in both cases in order to get some informations about the stability.
A metric Lie algebra g is a Lie algebra equipped with an inner product. A subalgebra h of a metric Lie algebra g is said to be totally geodesic if the Lie subgroup corresponding to h is a totally geodesic submanifold relative to the left-invariant Riemannian metric defined by the inner product, on the simply connected Lie group associated to g. A nonzero element of g is called a geodesic if it spans a one-dimensional totally geodesic subalgebra. We give a new proof of Kaizer's theorem that every metric Lie algebra possesses a geodesic. For nilpotent Lie algebras, we give several results on the…
A naturally parameterised curve in a Lie group with a left invariant metric is a geodesic, if its tangent vector left-translated to the identity satisfies the Euler equation $\dot{Y}=\operatorname{ad}^t_YY$ on the Lie algebra $\mathfrak{g}$ of $G$. Stationary points (equilibria) of the Euler equation are called geodesic vectors: the geodesic starting at the identity in the direction of a geodesic vector is a one-parameter subgroup of $G$. We give a complete classification of Lyapunov stable and unstable geodesic vectors for metric Lie algebras of dimension $3$ and for unimodular metric Lie alg…
We show that every unimodular Lie algebra, of dimension at most 4, equipped with an inner product, possesses an orthonormal basis comprised of geodesic elements. On the other hand, we give an example of a solvable unimodular Lie algebra of dimension 5 that has no orthonormal geodesic basis, for any inner product.
We show that strictly abnormal geodesics arise in graded nilpotent Lie groups. We construct such a group, for which some Carnot geodesics are strictly abnormal; in fact, they are not normal in any subgroup. In the step-2 case we also prove that these geodesics are always smooth. Our main technique is based on the equations for the normal and abnormal curves, that we derive (for any Lie group) explicitly in terms of the structure constants.
It is proved, that if M is a connected, complete submanifold of a complex space form N and each geodesic of M lies in an 1-dimensional totally geodesic complex submanifold of N, then M is totally geodesic in N and is a real space form or a complex space form.
This paper is devoted to geodesic completeness of left-invariant metrics for real and complex Lie groups. We start by establishing the Euler-Arnold formalism in the holomorphic setting. We study the real Lie group $\mathrm{SL}(2, \mathbb{R})$ and reobtain the known characterization of geodesic completeness and, in addition, present a detailed study where we investigate the maximum domain of definition of every single geodesic for every possible metric. We investigate completeness and semicompleteness of the complex geodesic flow for left-invariant holomorphic metrics and, in particular, establ…
On the ground of origins of the theory of Lie groups and Lie algebras, their (co)adjoint representations, and the Pontryagin maximum principle for the time-optimal problem are given an independent foundation for methods of geodesic vector field to search for normal geodesics of left-invariant (sub-)Finsler metrics on Lie groups and to look for the corresponding locally optimal controls in (sub-)Riemannian case, as well as some their applications.
The authors found geodesics, shortest arcs, cut loci, and conjugate sets for left-invariant sub-Riemannian matric on the Lie group $SL(2)$, which is right-invariant relative to the Lie subgroup $SO(2)\subset SL(2)$ (in other words, for invariant sub-Riemannian metric on weakly symmetric space $(SL(2)\times SO(2))/SO(2)$).
We determine, for all three-dimensional non-unimodular Lie groups equipped with a Lorentzian metric, the set of homogeneous geodesics through a point. Together with the results of [C] and [CM2], this leads to the full classification of three-dimensional Lorentzian g.o. spaces and naturally reductive spaces.
In this paper, we investigate left-invariant geodesic orbit metrics on connected simple Lie groups, where the metrics are formed by the structures of generalized flag manifolds. We prove that all these left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.
Geodesic orbit spaces are those Riemannian homogeneous spaces (G/H,g) whose geodesics are orbits of one-parameter subgroups of G. We classify the simply connected geodesic orbit spaces where G is a compact Lie group of rank two. We prove that the only such spaces for which the metric g is not induced from a bi-invariant metric on G are certain spheres and projective spaces, endowed with metrics induced from Hopf fibrations.
For finite dimensional real Lie algebras, we investigate the existence of an inner product having a basis comprised of geodesic elements. We give several existence and non-existence results in certain cases: unimodular solvable Lie algebras having an abelian nilradical, algebras having an abelian derived algebra, algebras having a codimension one ideal of a particular kind, nonunimodular algebras of dimension \$\leq 4\$, and unimodular algebras of dimension 5.
In 86, Ranjan questioned whether a submersion $π: G\to B$ from a compact simple Lie group with bi-invariant metric is a coset foliation or not, provided the submersion is Riemannian with totally geodesic fibers. Here we answer this question affirmatively, even when the submersion is defined only in an open subset of $G$ (assuming suitable hypothesis).
R. Zimmer proved that, on a compact manifold, a foliation with a dense leaf, a suitable leafwise Riemannian symmetric metric and a transverse Lie structure has arithmetic holonomy group. In this work we improve such result for totally geodesic foliations by showing that the manifold itself is arithmetic. This also gives a positive answer, for some special cases, to a conjecture of E. Ghys.
We continue the study of the distribution of closed geodesics on nilmanifolds constructed from a simply connected 2-step nilpotent Lie group with a left invariant metric and a lattice. We consider a Lie group with an associated 2-step nilpotent Lie algebra constructed from an irreducible representation of a compact semisimple Lie algebra on a real finite dimensional vector space. We determine sufficient conditions on the semisimple Lie algebra for the nilmanifold to have the density of closed geodesics property for a lattice arising from a Chevalley rational structure.
The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a geodesic sphere or totally geodesic hypersurface in a Riemannian manifold of constant curvature. Using that geodesic spherical curves are normal curves, i.e., they are the image of an Euclidean spherical curve under the exponential map, we are able to characterize geodesic spher…
We give a complete list of those left invariant unit vector fields on three-dimensional Lie groups with the left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group with the Sasaki metric. As a result, each class of three-dimensional Lie groups admits the totally geodesic unit vector field. From geometrical viewpoint, the field is either parallel or characteristic vector field of a natural almost contact structure on the group.
We study the geodesics problem in Heisenberg group H (case SR and riemannian). The sheaf of infinitesimal automorphisms of the (2n,2n+1) distribution D over H is an infinite, transitive Lie algebra sheaf.
For a Riemannian submersion from a simple compact Lie group with a bi-invariant metric, we prove the action of its holonomy group on the fibers is transitive. As a step towards classifying Riemannian submersions with totally geodesic fibers, we consider the parameterized surface induced by lifting a base geodesic to points along a geodesic in a fiber. Such a surface is "doubly ruled" (it is ruled by horizontal geodesics and also by vertical geodesics). Its characterizing properties allow us to define "doubly ruled parameterized surfaces" in any Riemannian manifold, independent of Riemannian su…
In this paper we investigate geodesic completeness of left-invariant Lorentzian metrics on a simple Lie group $G$ when there exists a left-invariant Killing vector field $Z$ on $G$. Among other results, it is proved that if $Z$ is timelike, or $G$ is strongly causal and $Z$ is lightlike, then the metric is complete. We then consider the special complex Lie group $SL_2(\mathbb{C})$ in more details and show that the existence of a lightlike vector field $Z$ on it, implies geodesic completeness. We also consider the existence of a spacelike vector field $Z$ on $SL_2(\mathbb{C})$ and provide an eq…
Geodesics and shortest arcs of some sub-Riemannian metrics on the Lie groups $SU(2)\times\mathbb{R}$ and $SO(3)\times\mathbb{R}$ with three-dimensional generating distributions
Geodesics and shortest arcs of some sub-Riemannian metrics on the Lie groups $SU(1,1)\times\mathbb{R}$ and $SO_0(2,1)\times\mathbb{R}$ with three-dimensional generating distributions
Sub-Lorentzian geodesics on ${\rm GL}^{+}(2,\mathbb{C})$ with the generating space of Hermitian matrices in the Lie algebra $\mathfrak{gl}^{+}(2,\mathbb{C})$
Motivated by a number of recent investigations, we define and investigate the various properties of the ruled surfaces depend on three dimensional Lie groups with a bi-variant metric. We give useful results involving the characterizations of these ruled surfaces. Some special ruled surfaces such as normal surface, binormal surface, tangent developable surface, rectifying developable surface and Darboux developable surface are worked. From those applications, we make use of such a work to interpret the Gaussian, mean curvatures of these surfaces and geodesic, normal curvature and geodesic torsi…
In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to identify, amongst the compact locally homogeneous Lorentzian 3-manifolds with non compact (local) isotropy group, those that are geodesically complete.
In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogeneous geodesics on compact semi-simple Lie group is established. We introduce the notion of naturally reductive homogeneous Finsler space. As a special case, we study homogeneous geodesics in homogeneous Randers spaces. Finally, we study some curvature properties of homogeneous ge…
In this paper, we study the set of homogeneous geodesics of a leftinvariant Finsler metric on Lie groups. We first give a simple criterion that characterizes geodesic vectors. As an application, we study some geometric properties of bi-invariant Finsler metrics on Lie groups. In particular a necessary and sufficient condition that left-invariant Randers metrics are of Berwald type is given. Finally a correspondence of homogeneous geodesics to critical points of restricted Finsler metrics is given. Then results concerning the existence homogeneous geodesics are obtained.
We prove injectivity and a support theorem for the X-ray transform on $2$-step nilpotent Lie groups with many totally geodesic $2$-dimensional flats. The result follows from a general reduction principle for manifolds with uniformly escaping geodesics.
We give a basic treatment of lattices $Γ$ in these groups. Certain tori $T_F$ and $T_B$ provide the model fiber and the base for a submersion of $Γ\backslash N$. This submersion may not be pseudoriemannian in the usual sense, because the tori may be degenerate. We then begin the study of periodic geodesics in these compact nilmanifolds, obtaining a complete calculation of the period spectrum of certain flat spaces.
Let X be a manifold equipped with a complete Riemannian metric of constant negative curvature and finite volume. We demonstrate the finiteness of the collection of totally geodesic immersed hypersurfaces in X that lie in the zero-level set of some Laplace eigenfunction. For surfaces, we show that the number can be bounded just in terms of the area of the surface. We also provide constructions of geodesics in hyperbolic surfaces that lie in a nodal set but that do not lie in the fixed point set of a reflection symmetry.
The unit sphere $\mathbb S^3$ can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving the corresponding Hamiltonian system.
The goals of this article are twofold : 1) to compute the conjugate locus of a geodesic that lies in the center of a simply connected, 2-step nilpotent Lie group with a left invariant metric 2) compare the isometry types of two such nilpotent Lie groups whose conjugate loci for central geodesics are "the same" in a suitable sense. The first goal is achieved. The second is elusive, but we obtain a partial result.
The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic submanifolds that inherit the property to be geodesic orbit. For a given geodesic orbit Riemannian space, we describe the structure of the nilradical and the radical of the Lie algebra of the isometry group. In the final part, we discuss some new tools to study geodesic orbit Riemannian spaces, related to compact Lie group representations with non-trivial pri…
Riemannian geodesic orbit spaces (G/H,g) are natural generalizations of symmetric spaces, defined by the property that their geodesics are orbits of one-parameter subgroups of G. We study the geodesic orbit spaces of the form (G/S,g), where G is a compact, connected, semisimple Lie group and S is abelian. We give a simple geometric characterization of those spaces, namely that they are naturally reductive. In turn, this yields the classification of the invariant geodesic orbit (and also the naturally reductive) metrics on any space of the form G/S. Our approach involves simplifying the intrica…
We study the geodesic orbit property for nilpotent Lie groups $N$ when endowed with a pseudo-Riemannian left-invariant metric. We consider this property with respect to different groups acting by isometries. When $N$ acts on itself by left-translations we show that it is a geodesic orbit space if and only if the metric is bi-invariant. Assuming $N$ is 2-step nilpotent and with non-degenerate center we give algebraic conditions on the Lie algebra $\mathfrak n$ of $N$ in order to verify that every geodesic is the orbit of a one-parameter subgroup of $N\rtimes\operatorname{Auto}(N)$. In addition …
In the present paper, we try to investigate the Noether symmetries and Lie point symmetries of the Vaidya-Bonner geodesics. Classification of one-dimensional subalgebras of Lie point symmetries are considered. In fact, the collection of pairwise non-conjugate one-dimensional subalgebras that are called the optimal system of subalgebras is determined. Moreover, as illustrative examples, the symmetry analysis is implemented on two special cases of the system.
We solve explicitly the geodesic equation for a wide class of (pseudo)-Riemannian homogeneous manifolds (G/H,m), including those with G compact, as well as non-compact semisimple Lie groups, under a simple algebraic condition for the metric m. We prove that these manifolds are geodesically complete and their geodesics are orbits in G/H of a product of N one-parameter subgroups of G, where N is a natural number. This is intimately related to the fact that the metrics m can be regarded as N-parameter deformations of a Riemannian metric in G/H with special symmetries. We show that there exists a …
Normal geodesic flows flows of Carnot-Caratheodory are discussed from the point of view of the theory of Hamiltonian systems. The geodesic flows corresponding to left-invariant metrics and left- and -right-invariant rank 2 distributions on the three-dimensional Heisenberg group are analysed as integrable systems. The flows corresponding to left-invariant metrics and left-invariant distributions on Lie groups are reduced to Euler equations on Lie groups. Relation of these constructions to problems of analytical mechanics is discussed.
In this paper we study the geodesic flow on nilmanifolds equipped with a left-invariant metric. We write the underlying definitions and find general formulas for the Poisson involution. As an example we develop the Heisenberg Lie group equipped with its canonical metric. We prove that a family of first integrals giving the complete integrability can be read off at the Lie algebra of the isometry group. We also explain the complete integrability on compact quotients and for any invariant metric.
Some nilpotent Lie groups possess a transformation group analogous to the similarity group acting on the Euclidean space. We call such a pair a nilpotent similarity structure. It is notably the case for all Carnot groups and their dilatations. We generalize a theorem of Fried: closed manifolds with a nilpotent similarity structure are either complete or radiant and, in the latter case, complete for the structure of the space deprived of a point. The proof relies on a generalization of convexity arguments in a setting where, in the coordinates given by the Lie algebra, we study geodesic segment…
The intimate relationship between coherent states and geodesics is pointed out. For homogenous manifolds on which the exponential from the Lie algebra to the Lie group equals the geodesic exponential, and in particular for symmetric spaces, it is proved that the cut locus of the point $0$ is equal to the set of coherent vectors orthogonal to $\vert 0>$. A simple method to calculate the conjugate locus in Hermitian symmetric spaces with significance in the coherent state approach is presented. The results are illustrated on the complex Grassmann manifold.
Let $G/K$ be an orbit of the adjoint representation of a compact connected Lie group $G$, $σ$ be an involutive automorphism of $G$ and $\tilde G$ be the Lie group of fixed points of $σ$. We find a sufficient condition for the complete integrability of the geodesic flow of the Riemannian metric on $\tilde G/(\tilde G\cap K)$, which is induced by the bi-invariant Riemannian metric on $\tilde G$. The integrals constructed here are real analytic functions, polynomial in momenta. It is checked that this sufficient condition holds when $G$ is the unitary group $U(n)$ and $σ$ is its automorphism defi…
The goal of this paper is the study of algebraic relations on the Lie algebra of first integrals of the geodesic flow on nilpotent Lie groups equipped with a left-invariant metric. It is proved that the isometry algebra of the $k$-step nilpotent Lie group, $k=2,3$, gives rise to a isomorphic family of first integrals for the geodesic flow. Also invariant first integrals are analyzed and new involution conditions are shown. Finally it is proved that in low dimensions complete families of first integrals can be constructed with Killing vector fields and symmetric Killing 2-tensor fields. This ho…
In this paper we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left-invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of thes…
We prove the integrability of geodesic flows on the Riemannian g.o. spaces of compact Lie groups, as well as on a related class of Riemannian homogeneous spaces having an additional principal bundle structure.
