Erratum to: On the geometry of spaces of oriented geodesics (original) (raw)

On the geometry of spaces of oriented geodesics

Annals of Global Analysis and Geometry, 2011

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space other than OH 2 , and consider the space L(M) of oriented geodesics of M. The space L(M) is a smooth homogeneous manifold and in this paper we describe all invariant symplectic structures, (para)complex structures, pseudo-Riemannian metrics and (para)Kähler structure on L(M).

Geodesics in generalized Wallach spaces

Journal of Geometry, 2015

We study geodesics in generalized Wallach spaces which are expressed as orbits of products of three exponential terms. These are homogeneous spaces M = G/K whose isotropy representation decomposes into a direct sum of three submodules m = m 1 ⊕ m 2 ⊕ m 3 , satisfying the relations [m i , m i ] ⊂ k. Assuming that the submodules m i are pairwise non isomorphic, we study geodesics on such spaces of the form γ(t) = exp(tX) exp(tY) exp(tZ) • o, where X ∈ m 1 , Y ∈ m 2 , Z ∈ m 3 (o = eK), with respect to a Ginvariant metric. Our investigation imposes certain restrictions on the G-invariant metric, so the geodesics turn out to be orbits of two exponential terms. We give a point of view using Riemannian submersions. As an application, we describe geodesics in generalized flag manifolds with three isotropy summands and with second Betti number b 2 (M) = 2, and in the Stiefel manifolds SO(n + 2)/S(n). We relate our results to geodesic orbit spaces (g.o. spaces).

The Lorentzian oscillator group as a geodesic orbit space

Journal of Mathematical Physics, 2012

We prove that the four-dimensional oscillator group Os endowed with any of its usual left-invariant Lorentzian metrics, is a Lorentzian geodesic (so in particular null-geodesic) orbit space with some of its homogeneous descriptions, corresponding to certain homogeneous Lorentzian structures. Each time that Os is endowed with a suitable metric and an appropriate homogeneous Lorentzian structure, it is a candidate for constructing solutions in eleven-dimensional supergravity with at least twenty-four of the thirty-two possible supersymmetries.

Special Structures on General Rotational Surfaces in Pseudo-Euclidean 4-Space With Neutral Metric

Research Square (Research Square), 2023

On general rotational surfaces in the pseudo-Euclidean 4-dimensional space of neutral signature, we describe the behavior of geometric objects, such as Killing vector fields (and in particular homothetic vector fields), divergence-free vector fields, co-closed and harmonic one-forms, and also harmonic functions. We classify geodesic and parallel vector fields, geodesic curves, concircular vector fields and concircular functions, and also concurrent vector fields and functions whose gradient is concurrent. Our results are new, as they have not been obtained in the Euclidean and Minkowski framework. The tools here are taken from both differential geometry and partial and ordinary differential equations. This topic could be of interest to many fields of mathematics, physics, engineering, architecture.

The Geodesic Curves on the Oscillator Group of Dimension Four

Journal of Science and Arts

Our work is study the geometry of oscillator groups, they are the only non commutative simply connected solvable Lie groups which have a biinvariant Lorentzian metric. The oscillator group has been generalized to one dimension equals , and several aspects of its geometry have been intensively studied, both in differential geometry and in mathematical physics. In this paper, we find geodesic curves on the oscillator group of dimension four.

Novel Invariants for Almost Geodesic Mappings of the Third Type

2020

Two kinds of invariance for geometrical objects under transformations are involved in this paper. With respect to these kinds, we obtained novel invariants for almost geodesic mappings of the third type of a non-symmetric affine connection space in this paper. Our results are presented in two sections. In the Section 3, we obtained the invariants for the equitorsion almost geodesic mappings which do not have the property of reciprocity.