Noether-Wald charges in six-dimensional Critical Gravity

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.

Sasaki–Ricci flow equation on five-dimensional Sasaki–Einstein space Yp,q

We analyze the transverse Kähler–Ricci flow equation on Sasaki-Einstein space [Formula: see text]. Explicit solutions are produced representing new five-dimensional Sasaki structures. Solutions which do not modify the transverse metric preserve the Sasaki–Einstein feature of the contact structure. If the transverse metric is altered, the deformed metrics remain Sasaki, but not Einstein.

Generalized quasi-Einstein GRW space-times

Recently, it is proven that generalized Robertson–Walker space-times in all orthogonal subspaces of Gray’s decomposition except one (unrestricted) are perfect fluid space-times. GRW space-times in the unrestricted subspace are identified by having constant scalar curvature. Generalized quasi-Einstein GRW space-times have a constant scalar curvature. It is shown that generalized quasi-Einstein GRW space-times reduce to Einstein space-times or perfect fluid space-times.

Sasaki–Ricci flow on Sasaki–Einstein space T1,1 and deformations

We study the transverse Kähler structure of the Sasaki–Einstein space [Formula: see text]. A set of local holomorphic coordinates is introduced and a Sasakian analogue of the Kähler potential is given. We investigate deformations of the Sasaki–Einstein structure preserving the Reeb vector field, but modifying the contact form. For this kind of deformations, we consider the Sasaki–Ricci flow which converges in a suitable sense to a Sasaki–Ricci soliton. Finally, it is described the constructions of Hamiltonian holomorphic vector fields and Hamiltonian function on the [Formula: see text] manifold.

Some Results on Projective Curvature Tensor of Nearly Cosymplectic Manifold

In the nearly cosymplectic manifold, dened a tensor of type (4,0), it's called a projective curvature tensor. In this article we discuss an interesting question; what the geometric meaning of this tensor when it's act on nearly cosymplectic manifold? The answer of this question leads to get an application on Einstein space. In particular, the necessary and sucient conditions that a projective tensor is vanishes are found.

Integrability of the geodesic flow on the resolved conifolds over Sasaki–Einstein space T1,1

Methods of Hamiltonian dynamics are applied to study the geodesic flow on the resolved conifolds (rcs) over Sasaki–Einstein space [Formula: see text]. We construct explicitly the constants of motion and prove complete integrability of geodesics in the five-dimensional Sasaki–Einstein space [Formula: see text] and its Calabi–Yau metric cone. The singularity at the apex of the metric cone can be smoothed out in two different ways. Using the small resolution, the geodesic motion on the rc remains completely integrable. Instead, in the case of the deformation of the conifold, the complete integrability is lost.