scholarly journals A rigidity theorem for Killing vector fields on compact manifolds with almost nonpositive Ricci curvature

2021 ◽  
Vol 149 (3) ◽  
pp. 1215-1224
Author(s):  
Atsushi Katsuda ◽  
Takuya Nakamura
Keyword(s):  
Ricci Curvature ◽  
Vector Fields ◽  
Killing Vector ◽  
Rigidity Theorem ◽  
Compact Manifolds ◽  
Killing Vector Fields

scholarly journals On the symmetries of Lorentzian four-dimensional generalized symmetric spaces of type C

Mathematica ◽  
2020 ◽  
Vol 62 (85) (2) ◽  
pp. 133-147
Author(s):  
Lakehal Belarbi
Keyword(s):  
Ricci Curvature ◽  
Symmetric Spaces ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields ◽  
Lorentzian Metric ◽  
Generalized Symmetric Spaces ◽  
Type C ◽  
Full Classification

We consider the four-dimensional generalized symmetric spaces of type C, equipped with a left-invariant Lorentzian metric. We completely describe its affine, homothetic and Killing vector fields. We also obtain a full classification of its Ricci, curvature and matter collineations.

Spacetime symmetries

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Riemannian Manifold ◽  
Conservation Laws ◽  
Vector Fields ◽  
Killing Vector ◽  
Geodesic Equation ◽  
Spacetime Symmetries ◽  
Tensor Fields ◽  
Killing Vector Fields ◽  
Covariant Derivatives ◽  
Killing Equation

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.

scholarly journals Off-Shell Noether Currents and Potentials for First-Order General Relativity

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 348
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez ◽  
Rodrigo Romero ◽  
Mariano Celada
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Arbitrary Vector ◽  
Four Dimensions ◽  
First Order ◽  
Direct Form ◽  
Definition Of

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

scholarly journals The geometry of null-like disformal transformations

2019 ◽  
Vol 16 (11) ◽  
pp. 1950180 ◽  
Author(s):  
I. P. Lobo ◽  
G. G. Carvalho
Keyword(s):  
Vector Field ◽  
Conformal Transformation ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields ◽  
Spinor Structure ◽  
Schwarzschild Metric ◽  
Killing Vectors ◽  
Symmetry Properties ◽  
Geometric Objects

Motivated by the hindrance of defining metric tensors compatible with the underlying spinor structure, other than the ones obtained via a conformal transformation, we study how some geometric objects are affected by the action of a disformal transformation in the closest scenario possible: the disformal transformation in the direction of a null-like vector field. Subsequently, we analyze symmetry properties such as mutual geodesics and mutual Killing vectors, generalized Weyl transformations that leave the disformal relation invariant, and introduce the concept of disformal Killing vector fields. In most cases, we use the Schwarzschild metric, in the Kerr–Schild formulation, to verify our calculations and results. We also revisit the disformal operator using a Newman–Penrose basis to show that, in the null-like case, this operator is not diagonalizable.

Extension of Killing vector fields beyond compact Cauchy horizons

2021 ◽  
Vol 391 ◽  
pp. 107953
Author(s):  
Oliver Lindblad Petersen
Keyword(s):  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields

scholarly journals PROPER CURVATURE COLLINEATIONS IN NONSTATIC SPHERICALLY SYMMETRIC SPACE–TIMES

2008 ◽  
Vol 23 (05) ◽  
pp. 749-759 ◽  
Author(s):  
GHULAM SHABBIR ◽  
M. RAMZAN
Keyword(s):  
Symmetric Space ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Infinite Dimensional ◽  
Riemann Matrix ◽  
Integration Techniques ◽  
Curvature Collineations ◽  
Proper Curvature

A study of nonstatic spherically symmetric space–times according to their proper curvature collineations is given by using the rank of the 6×6 Riemann matrix and direct integration techniques. Studying proper curvature collineations in each case of the above space–times it is shown that when the above space–times admit proper curvature collineations, they turn out to be static spherically symmetric and form an infinite dimensional vector space. In the nonstatic cases curvature collineations are just Killing vector fields.

Killing and Affine Killing Vector Fields

1999 ◽  
pp. 79-102
Author(s):  
Krishan L. Duggal ◽  
Ramesh Sharma
Keyword(s):  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields

scholarly journals On the symmetries of three-dimensional generalized symmetric spaces

2021 ◽  
Vol 13(62) (2) ◽  
pp. 451-462
Author(s):  
Lakehal Belarbi
Keyword(s):  
Symmetric Spaces ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Riemannian Metric ◽  
Killing Vector Fields ◽  
Generalized Symmetric Space ◽  
Generalized Symmetric Spaces ◽  
Full Classification

In this work we consider the three-dimensional generalized symmetric space, equipped with the left-invariant pseudo-Riemannian metric. We determine Killing vector fields and affine vectors fields. Also we obtain a full classification of Ricci, curvature and matter collineations

Solutions of the Ricci soliton equation for a large class of Siklos spacetimes

2021 ◽  
pp. 2150052
Author(s):  
Giovanni Calvaruso
Keyword(s):  
Large Class ◽  
Vector Fields ◽  
Killing Vector ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Killing Vector Fields ◽  
Soliton Equation

We determine and describe all the Ricci solitons within a very large class of Siklos metrics. As an application, the Ricci soliton equation is completely solved for several classes of Siklos metrics admitting additional Killing vector fields (in particular, for several homogeneous ones).

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