Geodesic graphs on the 13–dimensional group of Heisenberg type

Dušek, Zdeněk; Kowalski, Oldřich

doi:10.1002/mana.200310054

Cited by 14 publications

(1 citation statement)

References 17 publications

(35 reference statements)

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“…A counterexample, however, was found by Kaplan [1], initiating the extensive study of g.o. spaces [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Pseudo-Riemannian g.o.…”

Section: Introductionmentioning

confidence: 99%

On Lagrangian and Hamiltonian systems with homogeneous trajectories

Tóth¹

2010

J. Phys. A: Math. Theor.

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Motivated by various results on homogeneous geodesics of Riemannian spaces, we study homogeneous trajectories, i.e. trajectories which are orbits of a one-parameter symmetry group, of Lagrangian and Hamiltonian systems. We present criteria under which an orbit of a one-parameter subgroup of a symmetry group G is a solution of the Euler-Lagrange or Hamiltonian equations. In particular, we generalize the 'geodesic lemma' known in Riemannian geometry to Lagrangian and Hamiltonian systems. We present results on the existence of homogeneous trajectories of Lagrangian systems. We study Hamiltonian and Lagrangian g.o. spaces, i.e. homogeneous spaces G/H with G-invariant Lagrangian or Hamiltonian functions on which every solution of the equations of motion is homogeneous. We show that the Hamiltonian g.o. spaces are related to the functions that are invariant under the coadjoint action of G. Riemannian g.o. spaces thus correspond to special Ad * (G)-invariant functions. An Ad * (G)-invariant function that is related to a g.o. space also serves as a potential for the mapping called 'geodesic graph'. As illustration we discuss the Riemannian g.o. metrics on SU (3)/SU (2).

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Section: Introductionmentioning

confidence: 99%

On Lagrangian and Hamiltonian systems with homogeneous trajectories

Tóth¹

2010

J. Phys. A: Math. Theor.

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How Far Does Hyperbolic Geometry Generalize?

Szenthe

Non-Euclidean Geometries

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Homogeneous Geodesics in Homogeneous Affine Manifolds

2009

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For studying homogeneous geodesics in Riemannian and pseudoRiemannian geometry (on reductive homogeneous spaces) there is a simple algebraic formula which involves the reductive decomposition g = h +m of the Lie algebra g of the isometry group G and the scalar product on m induced by the metric. In the affine differential geometry, there is not such a universal formula. In the present paper, we propose a simple method of investigation of affine homogeneous geodesics. As an application, we describe homogeneous geodesics for homogeneous affine connections in dimension 2 and we find families of affine g.o. spaces in dimension 2. We also solved the problem of the canonical re-parametrization of affine homogeneous geodesics. Mathematics Subject Classification (2000). 53B05, 53C30.

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Geodesic graphs on the 13–dimensional group of Heisenberg type

Cited by 14 publications

References 17 publications

On Lagrangian and Hamiltonian systems with homogeneous trajectories

On Lagrangian and Hamiltonian systems with homogeneous trajectories

How Far Does Hyperbolic Geometry Generalize?

Homogeneous Geodesics in Homogeneous Affine Manifolds

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