The GHP Formalism and a Type N Twisting Vacuum Metric with Killing Vector

In this paper author present an investigation of some differential geometric properties of Para-Sasakian manifolds. Condition for a vector field to be Killing vector field in Para-Sasakian manifold is obtained. Mathematics Subject Classification (2010). 53B20, 53C15.

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Spacetime symmetries

10.1093/oso/9780198802877.003.0006 ◽

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.

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Cohomological localization of $$ \mathcal{N} $$ = 2 gauge theories with matter

Journal of High Energy Physics ◽

10.1007/jhep09(2020)133 ◽

2020 ◽

Vol 2020 (9) ◽

Author(s):

Guido Festuccia ◽

Anastasios Gorantis ◽

Antonio Pittelli ◽

Konstantina Polydorou ◽

Lorenzo Ruggeri

Keyword(s):

Vector Field ◽

Extended Supersymmetry ◽

General Framework ◽

Gauge Theories ◽

Killing Vector ◽

Killing Vector Field ◽

Explicit Result ◽

Yang Mills ◽

Self Duality ◽

Witten Theory

Abstract We construct a large class of gauge theories with extended supersymmetry on four-dimensional manifolds with a Killing vector field and isolated fixed points. We extend previous results limited to super Yang-Mills theory to general $$ \mathcal{N} $$ N = 2 gauge theories including hypermultiplets. We present a general framework encompassing equivariant Donaldson-Witten theory and Pestun’s theory on S4 as two particular cases. This is achieved by expressing fields in cohomological variables, whose features are dictated by supersymmetry and require a generalized notion of self-duality for two-forms and of chirality for spinors. Finally, we implement localization techniques to compute the exact partition function of the cohomological theories we built up and write the explicit result for manifolds with diverse topologies.

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

Symmetry ◽

10.3390/sym13020348 ◽

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.

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The geometry of null-like disformal transformations

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819501809 ◽

2019 ◽

Vol 16 (11) ◽

pp. 1950180 ◽

Cited By ~ 2

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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Extension of Killing vector fields beyond compact Cauchy horizons

Advances in Mathematics ◽

10.1016/j.aim.2021.107953 ◽

2021 ◽

Vol 391 ◽

pp. 107953

Author(s):

Oliver Lindblad Petersen

Keyword(s):

Vector Fields ◽

Killing Vector ◽

Killing Vector Fields

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Pseudo-Z symmetric space-times with divergence-free Weyl tensor and pp-waves

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887816500158 ◽

2016 ◽

Vol 13 (02) ◽

pp. 1650015 ◽

Cited By ~ 9

Author(s):

Carlo Alberto Mantica ◽

Young Jin Suh

Keyword(s):

Symmetric Space ◽

Weyl Tensor ◽

Killing Vector ◽

Complete Classification ◽

Space Time ◽

Conformal Killing ◽

Conformally Flat ◽

Conformal Curvature ◽

Wave Space ◽

Divergence Free

In this paper we present some new results about [Formula: see text]-dimensional pseudo-Z symmetric space-times. First we show that if the tensor Z satisfies the Codazzi condition then its rank is one, the space-time is a quasi-Einstein manifold, and the associated 1-form results to be null and recurrent. In the case in which such covector can be rescaled to a covariantly constant we obtain a Brinkmann-wave. Anyway the metric results to be a subclass of the Kundt metric. Next we investigate pseudo-Z symmetric space-times with harmonic conformal curvature tensor: a complete classification of such spaces is obtained. They are necessarily quasi-Einstein and represent a perfect fluid space-time in the case of time-like associated covector; in the case of null associated covector they represent a pure radiation field. Further if the associated covector is locally a gradient we get a Brinkmann-wave space-time for [Formula: see text] and a pp-wave space-time in [Formula: see text]. In all cases an algebraic classification for the Weyl tensor is provided for [Formula: see text] and higher dimensions. Then conformally flat pseudo-Z symmetric space-times are investigated. In the case of null associated covector the space-time reduces to a plane wave and results to be generalized quasi-Einstein. In the case of time-like associated covector we show that under the condition of divergence-free Weyl tensor the space-time admits a proper concircular vector that can be rescaled to a time like vector of concurrent form and is a conformal Killing vector. A recent result then shows that the metric is necessarily a generalized Robertson–Walker space-time. In particular we show that a conformally flat [Formula: see text], [Formula: see text], space-time is conformal to the Robertson–Walker space-time.

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