scholarly journals A study on killing vector fields in four-dimensional spaces

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1249-1257
Author(s):  
Bahar Kırık
Keyword(s):  
Ricci Tensor ◽  
Vector Fields ◽  
Killing Vector ◽  
Positive Definite ◽  
Killing Vector Field ◽  
Killing Vector Fields ◽  
Metric Tensor ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Weyl Conformal Curvature Tensor

In the present study, some properties of Killing vector fields are investigated on 4-dimensional manifolds in case of the signature of the metric tensor 1 is either Lorentz or positive definite or neutral. First of all, the notation and the main object of the study are introduced on these manifolds. Later on, some special subalgebras are examined for the members of the Killing algebra when the Killing vector field vanishes at a point of the manifold admitting any of these metric signatures. The constraints of this examination to the Weyl conformal curvature tensor and the Ricci tensor are then studied and some results are obtained. Finally, some examples related to these results are given for all metric signatures.

scholarly journals On a type of Sasakian space

1972 ◽  
Vol 13 (4) ◽  
pp. 508-510 ◽  
Author(s):  
M. C. Chaki ◽  
D. Ghosh
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Positive Definite ◽  
Killing Vector Field ◽  
Covariant Differentiation ◽  
Metric Tensor ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Image Position ◽  
Conformally Recurrent

A Sasakian space [1]Mn (n = 2m + 1) is a Riemannian n-space with a positive definite metric tensor gij and a unit Killing vector field η which satisfies where the comma denotes covariant differentiation with respect to the metic tensor. In a recent paper [2] M. C. Chaki and A. N. Roy Chowdhury studied conformally recurrent spaces of second order, or briefly conformally 2-recurrent spaces, that is, non-flat Riemannian spaces Vn (n > 3) defined by where is the conformal curvature tensor: and alm is a tensor not identically zero.

On Almost Contingent Manifolds of Second Class with Applications in Relativity

1978 ◽  
Vol 21 (3) ◽  
pp. 289-295 ◽  
Author(s):  
K. L. Duggal
Keyword(s):  
General Relativity ◽  
Electromagnetic Fields ◽  
Vector Fields ◽  
Killing Vector ◽  
Positive Definite ◽  
Geometric Proof ◽  
Sasakian Manifolds ◽  
Killing Vector Fields ◽  
Degenerate Metric ◽  
Frame Work

D. E. Blair [1] has introduced the notion of K-manifolds as an analogue of the even dimensional Kähler manifolds and of the odd dimensional quasi-Sasakian manifolds. These manifolds have been studied with respect to a positive definite metric. In this paper, we study a more general case of if-manifolds carrying an arbitrary non-degenerate metric, in particular, a metric of Lorentz signature. This theory is then applied within the frame-work of general relativity. Using the Ruse-Synge classification [8, 9] of non-null electromagnetic fields with source, we develop a geometric proof for the existence of either two space like or one space like and one time like Killing vector fields on the space-time manifold.

Some solutions of the Lanczos vacuum wave equation

1994 ◽  
Vol 447 (1931) ◽  
pp. 577-585 ◽  
Keyword(s):  
Wave Equation ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Spin Coefficient ◽  
Independent Components ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Non Local ◽  
Weyl Conformal Curvature Tensor ◽  
Gauge Conditions

In general relativity the non-local part of the gravitational field is described by the 10 degrees of freedom of the Weyl conformal curvature tensor C abcd . In every space-time the Weyl field C abcd is derivable from a potential L abc which has at most 16 algebraically independent components reducing to 10 degrees of freedom when the six gauge conditions L ab s ; s = 0 are imposed. The potential L abc discov­ered by Lanczos was shown by Illge to have an extremely simple vacuum wave equation, namely, □ L abc ≡ g sm L abc ; s ; m = 0. Using tensor, spinor and spin-coefficient methods we give some solutions of this new vacuum wave equation in some spacetimes containing one or more preferred vector fields.

SURFACES IN 𝔼3MAKING CONSTANT ANGLE WITH KILLING VECTOR FIELDS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250023 ◽  
Author(s):  
MARIAN IOAN MUNTEANU ◽  
ANA IRINA NISTOR
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Field ◽  
Killing Vector Fields ◽  
Curves And Surfaces ◽  
Constant Angle ◽  
Constant Angle Surfaces

In the present paper we classify curves and surfaces in Euclidean 3-space which make constant angle with a certain Killing vector field. Moreover, we characterize the catenoid and Dini's surface in terms of constant angle surfaces.

scholarly journals Pseudosymmetry properties of generalised Wintgen ideal Legendrian submanifolds

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1209-1215
Author(s):  
Aleksandar Sebekovic ◽  
Miroslava Petrovic-Torgasev ◽  
Anica Pantic
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Space Forms ◽  
Equality Case ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Sasakian Space Forms ◽  
Legendrian Submanifold ◽  
Legendrian Submanifolds ◽  
Weyl Conformal Curvature Tensor

For Legendrian submanifolds Mn in Sasakian space forms ?M2n+1(c), I. Mihai obtained an inequality relating the normalised scalar curvature (intrinsic invariant) and the squared mean curvature and the normalised scalar normal curvature of M in the ambient space ?M (extrinsic invariants) which is called the generalised Wintgen inequality, characterising also the corresponding equality case. And a Legendrian submanifold Mn in Sasakian space forms ?M2n+1(c) is said to be generalised Wintgen ideal Legendrian submanifold of ?M2n+1(c) when it realises at everyone of its points the equality in such inequality. Characterisations based on some basic intrinsic symmetries involving the Riemann-Cristoffel curvature tensor, the Ricci tensor and the Weyl conformal curvature tensor belonging to the class of pseudosymmetries in the sense of Deszcz of such generalised Wintgen ideal Legendrian submanifolds are given.

scholarly journals 2-Killing vector fields on warped product manifolds

2015 ◽  
Vol 26 (09) ◽  
pp. 1550065 ◽  
Author(s):  
Sameh Shenawy ◽  
Bülent Ünal
Keyword(s):  
Vector Field ◽  
Product Space ◽  
Warped Product ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Field ◽  
Product Manifold ◽  
Killing Vector Fields ◽  
Warped Product Manifold

This paper provides a study of 2-Killing vector fields on warped product manifolds as well as characterization of this structure on standard static and generalized Robertson–Walker space-times. Some conditions for a 2-Killing vector field on a warped product manifold to be parallel are obtained. Moreover, some results on the curvature of a warped product manifolds in terms of 2-Killing vector fields are derived. Finally, we apply our results to describe 2-Killing vector fields of some well-known warped product space-time models.

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.

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