scholarly journals The Lanczos potential for the Weyl curvature tensor: existence, wave equation and algorithms

1997 ◽  
Vol 453 (1959) ◽  
pp. 835-851 ◽  
Author(s):  
S.B. Edgar ◽  
A. Höglund
Keyword(s):  
Wave Equation ◽  
Curvature Tensor ◽  
Weyl Curvature ◽  
Lanczos Potential ◽  
Weyl Curvature Tensor

scholarly journals On a New type of Tensor on Real Hypersurfaces in Non-Flat Complex Space Forms

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 559
Author(s):  
George Kaimakamis ◽  
Konstantina Panagiotidou
Keyword(s):  
Curvature Tensor ◽  
Complex Space ◽  
Three Dimensional ◽  
Real Hypersurfaces ◽  
Weyl Curvature ◽  
Space Forms ◽  
Hopf Hypersurfaces ◽  
Weyl Curvature Tensor ◽  
New Type ◽  
Complex Space Forms

In this paper the notion of ∗ -Weyl curvature tensor on real hypersurfaces in non-flat complex space forms is introduced. It is related to the ∗ -Ricci tensor of a real hypersurface. The aim of this paper is to provide two classification theorems concerning real hypersurfaces in non-flat complex space forms in terms of ∗ -Weyl curvature tensor. More precisely, Hopf hypersurfaces of dimension greater or equal to three in non-flat complex space forms with vanishing ∗ -Weyl curvature tensor are classified. Next, all three dimensional real hypersurfaces in non-flat complex space forms, whose ∗ -Weyl curvature tensor vanishes identically are classified. The used methods are based on tools from differential geometry and solving systems of differential equations.

scholarly journals CURVATURE STRUCTURE OF SELF-DUAL 4-MANIFOLDS

2008 ◽  
Vol 05 (07) ◽  
pp. 1191-1204 ◽  
Author(s):  
NOVICA BLAŽIĆ ◽  
PETER GILKEY ◽  
STANA NIKČEVIĆ ◽  
IVA STAVROV
Keyword(s):  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Weyl Curvature ◽  
Weyl Curvature Tensor ◽  
Algebraic Result ◽  
Curvature Structure

We show the existence of a modified Cliff(1,1)-structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we prove some new results regarding (Jordan) Osserman manifolds.

scholarly journals *-Weyl Curvature Tensor within the Framework of Sasakian and $(\kappa,\mu)$-Contact Manifolds

2021 ◽  
Vol 52 ◽  
Author(s):  
Venkatesha Venkatesha ◽  
H. Aruna Kumara
Keyword(s):  
Curvature Tensor ◽  
Contact Manifolds ◽  
Weyl Curvature ◽  
Weyl Curvature Tensor

The object of the present paper is to study $*$-Weyl curvature tensor within the framework of Sasakian and $(\kappa,\mu)$-contact manifolds.

scholarly journals On isospectral compactness in conformal class for 4-manifolds

2019 ◽  
Vol 21 (05) ◽  
pp. 1850041 ◽  
Author(s):  
Xianfu Liu ◽  
Zuoqin Wang
Keyword(s):  
Scalar Curvature ◽  
Curvature Tensor ◽  
Riemannian Metrics ◽  
Conformal Class ◽  
Weyl Curvature ◽  
Yamabe Invariant ◽  
Weyl Curvature Tensor

Let [Formula: see text] be a closed 4-manifold with positive Yamabe invariant and with [Formula: see text]-small Weyl curvature tensor. Let [Formula: see text] be any metric in the conformal class of [Formula: see text] whose scalar curvature is [Formula: see text]-close to a constant. We prove that the set of Riemannian metrics in the conformal class [Formula: see text] that are isospectral to [Formula: see text] is compact in the [Formula: see text] topology.

scholarly journals The Weyl Curvature Tensor of Hypersurfaces Under the Mean Curvature Flow

2020 ◽  
Vol 76 (1) ◽  
pp. 143-156
Author(s):  
Ghodrat Moazzaf ◽  
Esmaiel Abedi
Keyword(s):  
Curvature Tensor ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Conformally Flat ◽  
Weyl Curvature ◽  
Initial Hypersurface ◽  
Weyl Curvature Tensor ◽  
The Mean ◽  
Upper Estimate

AbstractIn this paper, we study the evolution of the Weyl curvature tensor W of hypersurfaces in 𝕉n+1 under the mean curvature flow. We find a bound for the Weyl curvature tensor of hypersurfaces during the evolution in terms of time. As a consequence, we suppose that the initial hypersurface is conformally flat, i.e., W =0 at t = 0 and then we find an upper estimate for W during the evolution in terms of time.

The wave equation for the Lanczos potential. I

1994 ◽  
Vol 447 (1931) ◽  
pp. 557-575 ◽  
Keyword(s):  
Wave Equation ◽  
Degrees Of Freedom ◽  
Wave Equations ◽  
Conformal Curvature ◽  
Lanczos Potential ◽  
Conformal Curvature Tensor ◽  
Radiation Theory ◽  
Non Local ◽  
Spinor Wave ◽  
Gauge Conditions

The non-local part of the gravitational field in general relativity is described by the 10 component conformal curvature tensor C abcd of Weyl. For this field Lanczos found a tensor potential L abc with 16 independent components. We can make L abc have only 10 effective degrees of freedom by imposing the 6 gauge conditions L ab s :s = 0. Both fields C abcd , L abc satisfy wave equations. The wave equation satisfied by C abcd is nonlinear, even in vacuo . However, a linear spinor wave equation for the Lanczos potential has been found by Illge but no correct tensor wave equation for L abc has yet been published. Here, we derive a correct tensor wave equation for L abc and when it is simplified with the aid of some four­-dimensional identities it is equivalent to Illge’s wave equation. We also show that the nonlinear spinor wave equation of Penrose for the Weyl field can be derived from Illge’s spinor wave equation. A set of analogues of well-known results of classical electromagnetic radiation theory can now be given. We indicate how a Green’s function approach to gravitational radiation could be based on our tensor wave equation, when a global study of space-time is attempted.

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