An Iterative Decomposition of Global Conformal Invariants: The First Step

2012 ◽  
Author(s):  
Spyros Alexakis
Keyword(s):  
Linear Combination ◽  
Curvature Tensor ◽  
Weyl Tensor ◽  
Conformal Invariants ◽  
Conformal Invariant ◽  
Free Part ◽  
Schouten Tensor ◽  
Iterative Decomposition

This chapter fleshes out the strategy of iteratively decomposing any P(g) = unconverted formula 1 for which ∫P(g)dVsubscript g is a global conformal invariant. It makes precise the notions of better and worse complete contractions in P(g) and then spells out (1.17), via Propositions 2.7, 2.8. In particular, using the well-known decomposition of the curvature tensor into its trace-free part (the Weyl tensor) and its trace part (the Schouten tensor), it reexpresses P(g) as a linear combination of complete contractions involving differentiated Weyl tensors and differentiated Schouten tensors, as in (2.47). The chapter also proves (1.17) when the worst terms involve at least one differentiated Schouten tensor.

scholarly journals WEYL COLLINEATIONS THAT ARE NOT CURVATURE COLLINEATIONS

2005 ◽  
Vol 14 (08) ◽  
pp. 1431-1437 ◽  
Author(s):  
IBRAR HUSSAIN ◽  
ASGHAR QADIR ◽  
K. SAIFULLAH
Keyword(s):  
Linear Combination ◽  
Lie Algebra ◽  
Curvature Tensor ◽  
Weyl Tensor ◽  
Ricci Tensor ◽  
Lie Symmetries ◽  
Ricci Scalar ◽  
Finite Dimensional ◽  
Fundamental Reason ◽  
Curvature Collineations

Though the Weyl tensor is a linear combination of the curvature tensor, Ricci tensor and Ricci scalar, it does not have all and only the Lie symmetries of these tensors since it is possible, in principle, that "asymmetries cancel." Here we investigate if, when and how the symmetries can be different. It is found that we can obtain a metric with a finite dimensional Lie algebra of Weyl symmetries that properly contains the Lie algebra of curvature symmetries. There is no example found for the converse requirement. It is speculated that there may be a fundamental reason for this lack of "duality."

scholarly journals Some aspects of Ricci flow on the 4-sphere

10.53733/152 ◽  
2021 ◽  
Vol 52 ◽  
pp. 381-402
Author(s):  
Sun-Yung Alice Chang ◽  
Eric Chen
Keyword(s):  
Scalar Curvature ◽  
Curvature Tensor ◽  
Ricci Flow ◽  
Weyl Tensor ◽  
Riemannian Metrics ◽  
Positive Scalar Curvature ◽  
Conformal Invariants ◽  
Positive Scalar ◽  
Gap Theorem ◽  
Normalized Ricci Flow

In this paper, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with L^2 norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the L^p norm for certain p>2 of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere.

scholarly journals On a Set of Conform-Invariant Equations of the Gravitational Field

1953 ◽  
Vol 10 (1) ◽  
pp. 16-20 ◽  
Author(s):  
H. A. Buchdahl
Keyword(s):  
Gravitational Field ◽  
Linear Combination ◽  
Curvature Tensor ◽  
Wide Class ◽  
Ricci Tensor ◽  
Empty Space ◽  
Image Position ◽  
Derivatives Of

Eddington has considered equations of the gravitational field in empty space which are of the fourth differential order, viz. the sets of equations which express the vanishing of the Hamiltonian derivatives of certain fundamental invariants. The author has shown that a wide class of such equations are satisfied by any solution of the equationswhere Gμν and gμν are the components of the Ricci tensor and the metrical tensor respectively, whilst λ is an arbitrary constant. For a V4 this applies in particular when the invariant referred to above is chosen from the setwhere Bμνσρ is the covariant curvature tensor. K3 has been included since, according to a result due to Lanczos3, its Hamiltonian derivative is a linear combination of and , i.e. of the Hamiltonian derivatives of K1 and K2. In fact

The Second Step: The Fefferman-Graham Ambient Metric and the Nature of the Decomposition

2012 ◽  
Author(s):  
Spyros Alexakis
Keyword(s):  
Weyl Tensor ◽  
Second Step ◽  
Conformal Invariant ◽  
Algebraic Properties

This chapter proves (1.17) when the worst terms in P(g) involve only factors of the differentiated Weyl tensor. This case is much harder than the previous one; in particular, in this case we need both a local conformal invariant W(g) and a divergence divᵢTⁱ(g) to prove (1.17). One obvious difficulty is how, upon inspection of P(g)subscript worst-piece, to separate the piece that must be cancelled out by a local conformal invariant from the piece that is cancelled out by a divergence. In a first step, we prove that we can first explicitly construct a local conformal invariant and a divergence and subtract them from P(g)subscript worst-piece, to be left with a new worst piece, which has some additional algebraic properties. In a second step, we show that this new worst piece can be cancelled out by subtracting a divergence.

A note on generalized Robertson–Walker space-times

2016 ◽  
Vol 13 (06) ◽  
pp. 1650079 ◽  
Author(s):  
Carlo Alberto Mantica ◽  
Young Jin Suh ◽  
Uday Chand De
Keyword(s):  
Vector Field ◽  
Scalar Field ◽  
Perfect Fluid ◽  
Curvature Tensor ◽  
Weyl Tensor ◽  
Space Time ◽  
Stiff Matter ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Mass Less Scalar Field

A generalized Robertson–Walker (GRW) space-time is the generalization of the classical Robertson–Walker space-time. In the present paper, we show that a Ricci simple manifold with vanishing divergence of the conformal curvature tensor admits a proper concircular vector field and it is necessarily a GRW space-time. Further, we show that a stiff matter perfect fluid space-time or a mass-less scalar field with time-like gradient and with divergence-free Weyl tensor are GRW space-times.

scholarly journals Geometry of Nonholonomic Kenmotsu Manifolds

2021 ◽  
pp. 84-87
Author(s):  
A.V. Bukusheva
Keyword(s):  
Curvature Tensor ◽  
Einstein Manifold ◽  
Lie Derivative ◽  
Riemann Curvature Tensor ◽  
Intrinsic Geometry ◽  
Kenmotsu Manifold ◽  
Metric Tensor ◽  
Schouten Tensor ◽  
Kenmotsu Manifolds ◽  
Formal Properties

The concept of the intrinsic geometry of a nonholonomic Kenmotsu manifold M is introduced. It is understood as the set of those properties of the manifold that depend only on the framing  of the D^ distribution D of the manifold M, on the parallel transformation of vectors belonging to the distribution D along curves tangent to this distribution. The invariants of the intrinsic geometry of the nonholonomic Kenmotsu manifold are: the Schouten curvature tensor; 1-form η generating the distribution D; the Lie derivative  of the metric tensor g along the vector field ; Schouten — Wagner tensor field P, whose components in adapted coordinates are expressed using the equalities . It is proved that, as in the case of the Kenmotsu manifold, the Schouten — Wagner tensor of the manifold M vanishes. It follows that the Schouten tensor of a nonholonomic Kenmotsu manifold has the same formal properties as the Riemann curvature tensor. It is proved that the alternation of the Ricci — Schouten tensor coincides with the differential of the structural form. This property of the Ricci — Schouten tensor is used in the proof of the main result of the article: a nonholonomic Kenmotsu manifold cannot carry the structure of an η-Einstein manifold.

On the uniqueness of the quadratic action Lagrangian in general relativity

1970 ◽  
Vol 48 (16) ◽  
pp. 1924-1932
Author(s):  
Borut Gogala
Keyword(s):  
General Relativity ◽  
Linear Combination ◽  
Curvature Tensor ◽  
Lagrangian Density ◽  
Computer Language ◽  
Riemann Curvature ◽  
Field Equations ◽  
Riemann Curvature Tensor ◽  
Quadratic Action ◽  
Uniqueness Proof

A procedure, suitable for translation into computer language and analogous to the uniqueness proof of Einstein's field equations in vierbein formulation, is developed for finding the most general quadratic action Lagrangian density in general relativity that satisfies the requirement of covariance and of invariance under coordinate dependent frame reorientation. It is found that this Lagrangian is identical with the most general linear combination that can be made up out of the three scalars formed by bilinear combinations of the Riemann curvature tensor.

scholarly journals Noether-Wald charges in six-dimensional Critical Gravity

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Giorgos Anastasiou ◽  
Ignacio J. Araya ◽  
Cristóbal Corral ◽  
Rodrigo Olea
Keyword(s):  
Weyl Tensor ◽  
Einstein Space ◽  
Gravity Theory ◽  
Compact Form ◽  
Conformal Invariants ◽  
Einstein Manifolds ◽  
Conformal Gravity ◽  
Covariant Derivatives ◽  
The Difference ◽  
Derivatives Of

Abstract It has been recently shown that there is a particular combination of conformal invariants in six dimensions which accepts a generic Einstein space as a solution. The Lagrangian of this Conformal Gravity theory — originally found by Lu, Pang and Pope (LPP) — can be conveniently rewritten in terms of products and covariant derivatives of the Weyl tensor. This allows one to derive the corresponding Noether prepotential and Noether-Wald charges in a compact form. Based on this expression, we calculate the Noether-Wald charges of six-dimensional Critical Gravity at the bicritical point, which is defined by the difference of the actions for Einstein-AdS gravity and the LPP Conformal Gravity. When considering Einstein manifolds, we show the vanishing of the Noether prepotential of Critical Gravity explicitly, which implies the triviality of the Noether-Wald charges. This result shows the equivalence between Einstein-AdS gravity and Conformal Gravity within its Einstein sector not only at the level of the action but also at the level of the charges.

Recognition by Linear Combination of Models

1989 ◽  
Author(s):  
Shimon Ullman ◽  
Ronen Basri
Keyword(s):  
Linear Combination
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