2010 # Lie and Noether symmetries of geodesic equations and collineations

**Abstract:** The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric.The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces…

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(115 citation statements)

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“…Let us now apply the Noether Symmetry Approach [536] (see also [537]) to a general class of f (T ) gravity models where the corresponding Lagrangian of the field equations is given by (641). We start by performing the analysis for arbitrary space-times, and then we focus on static spherically-symmetric geometries.…”

confidence: 99%

“…Let us now apply the Noether Symmetry Approach [536] (see also [537]) to a general class of f (T ) gravity models where the corresponding Lagrangian of the field equations is given by (641). We start by performing the analysis for arbitrary space-times, and then we focus on static spherically-symmetric geometries.…”

confidence: 99%

“…In particular, the scope of the current article is (a) to investigate which of the available f (R) models admit extra Lie and Noether point symmetries, and (b) for these models to solve the system of the resulting field equations and derive analytically (for the first time to our knowledge) the main cosmological functions (the scale factor, the Hubble expansion rate etc.). We would like to remind the reader that a fundamental approach to derive the Lie and Noether point symmetries for a given dynamical problem living in a Riemannian space has been published recently by Tsamparlis & Paliathanasis [36] (a similar analysis can be found in [49][50][51][52][53][54][55]). …”

confidence: 99%

“…It turns out that general plane symmetric space-times can admit 4, 5, 6, 7, 8, 9, 11, or 17 Noether symmetries. The maximum number of Noether symmetries (i.e., 17) appears only for the Minkowski space-time and 10 of these symmetries correspond to the generators of isometries associated with this metric.…”

confidence: 99%

“…In the past few decades, a significant amount of relativistic literature has been devoted to the study of symmetries in general relativity [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. In [7], the authors considered Killing vectors (KVs) of spherically symmetric static space-times and concluded that they admit either ten KVs (corresponding to de Sitter, Minkowski, and anti-de Sitter metrics), seven KVs (corresponding to Einstein and anti-Einstein metrics), six KVs (incorporating the Bertotti-Robinson and two other metrics), or only four KVs (the minimal set of isometries).…”

confidence: 99%