Superstring theory on pp waves with ADE geometries

In this paper, partially invariant solutions (PISs) method is applied in order to obtain new four-dimensional Einstein Walker manifolds. This method is based on subgroup classification for the symmetry group of partial differential equations (PDEs) and can be regarded as the generalization of the similarity reduction method. For this purpose, those cases of PISs which have the defect structure δ = 1 and are resulted from two-dimensional subalgebras are considered in the present paper. Also it is shown that the obtained PISs are distinct from the invariant solutions that obtained by similarity reduction method.

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“…If u = f (x) is a function, then the manifold HΓ f can be expressed in terms of the invariants (8). So, we have:…”

Section: Partially Invariant Solutions Methodsmentioning

confidence: 99%

“…A Walker manifold is called Einstein Walker manifold if its Ricci tensor is a scaler multiple of the metric at each point. Four-dimensional Einstein Walker manifolds form the underling structure of many geometric and physical models such as; hh-space in general relativity, pp-wave model and other areas [6][7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Computation of partially invariant solutions for the Einstein Walker manifolds’ identifying equations

Nadjafikhah

Jafari

2013

Communications in Nonlinear Science and Numerical Simulation

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“…According to modern physics, we do not live in a Riemannian universe, but rather in a Lorentzian universe or even in the higher signature setting according to some string theories. Indefinite signatures also appears in supergravity theory (see, for example, Abounasr, Belhaj, Rasmussen, and Saidi [2]). Walker metrics have indefinite signatures and are important as well (see, for example, the discussion in Davidov, Diaz-Ramos, Garcia-Rio, Matsushita, Muskarov, and Vazquez-Lorenzo [7]).…”

Section: Higher Signature Geometrymentioning

confidence: 99%

Non-closed curves in ℝn with finite total first curvature arising from the solutions of an ODE

Gilkey¹,

Kim²,

Matsuda³

et al. 2017

Hokkaido Math. J.

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We use the solution set of a real ordinary differential equation which has order n ≥ 2 to construct a smooth curve σ in R n . We describe when σ is a proper embedding of infinite length with finite total first curvature. Statement of resultsLet σ be an immersion of R into R n for n ≥ 2; if σ is proper, then the length of σ is then necessarily infinite. The first curvature κ and the total first curvature κ[σ] are given, respectively, by: κ := ||σ ∧σ|| ||σ|| 3 and κ[σ] := ∞ −∞ κ||σ||dx . (1.a) 1.1. History. Fenchel [16] showed that a simple closed curve in R 3 had κ[σ] ≥ 2π. Fáry [15] and Milnor [18] showed the total curvature of any knot is greater than 4π. Castrillón López and Fernández Mateos [6], and Kondo and Tanaka [17] have examined the global properties of the total curvature of a curve in an arbitrary Riemannian manifold. The total curvature of open plane curves of fixed length in R 2 was studied by Enomoto [11]. The analogous question for S 2 was examined by Enomoto and Itoh [12, 13]. Enomoto, Itoh, and Sinclair [14] examined curves in R 3 . We also refer to related work of Sullivan [19]. Borisenko and Tenenblat [4] studied the problem in Minkowski space. Buck and Simon [5] and Diao and Ernst [9] studied this invariant in the context of knot theory, and Ekholm [10] used this invariant in the context of algebraic topology. Alexander, Bishop, and Ghrist [1]extended these notions to more general spaces than smooth manifolds. The total curvature also appears in the study of Plateau's problem -see the discussion in Desideri and Jakob [8].The literature on the subject is a vast one and we have only cited a few representative papers to give a flavor for the subject.The papers cited above focused on closed curves, polygonal curves, knotted curves, curves with fixed endpoints, curves with finite length and pursuit curves. This present paper deals, by contrast, with properly embedded curves which arise from ordinary differential equations. We take as our starting point a real constant coefficient ordinary differential operator P of degree n ≥ 2 of the form: P (y) := y (n) + c n−1 y (n−1) + · · · + c 0 y .2010 Mathematics Subject Classification. 53A04.

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“…Lorentzian Walker manifolds have been studied extensively in physics since they constitute the background metric of the pp-wave models. A pp-wave spacetime admits a covariantly constant null vector field U [1].…”

Section: Introductionmentioning

confidence: 99%

Some Geometric Properties of a Non-Strict Eight Dimensional Walker Manifold

Longwap

Bitrus

Chibuisi

2021

JAMCS

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An 8 dimensional Walker manifold (M; g) is a strict walker manifold if we can choose a coordinate system fx1; x2; x3; x4; x5; x6; x7; x8g on (M,g) such that any function f on the manfold (M,g), f(x1; x2; x3; x4; x5; x6; x7; x8) = f(x5; x6; x7; x8): In this work, we dene a Non-strict eight dimensional walker manifold as the one that we can choose the coordinate system such that for any f in (M; g); f(x1; x2; x3; x4; x5; x6; x7; x8) = f(x1; x2; x3; x4): We derive cononical form of the Levi-Civita connection, curvature operator, (0; 4)-curvature tansor, the Ricci tensor, Weyl tensorand study some of the properties associated with the class of Non-strict 8 dimensionalWalker manifold. We investigate the Einstein property and establish a theorem for the metric to be locally conformally at.

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Superstring theory on pp waves with ADE geometries

Cited by 6 publications

References 41 publications

Computation of partially invariant solutions for the Einstein Walker manifolds’ identifying equations

Computation of partially invariant solutions for the Einstein Walker manifolds’ identifying equations

Non-closed curves in ℝn with finite total first curvature arising from the solutions of an ODE

Some Geometric Properties of a Non-Strict Eight Dimensional Walker Manifold

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