Recently, we have used the symmetric bracket of vector fields, and developed the notion of the symmetric derivation. Using this machinery, we have defined the concept of symmetric curvature. This concept is natural and is related to the notions divergence and Laplacian of vector fields. This concept is also related to the derivations on the algebra of symmetric forms which has been discu...

In this note we wish to complement some recent work in the cosmological literature concerning the Weyl conformal curvature tensor and its parts. In particular, we shall give a clear–cut definition of the Newtonian limits of electric and magnetic parts of the Weyl tensor. We also discuss that only a subset of the relativistic equations is needed to obtain a closed system of equations in the Newt...

We study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension n ≤ 24. The Weyl Vanishing Theorem is also established under these hypotheses, and we provide counter-examples to compactness when n ≥ 25. Lastly, our methods...

We define and compute the energy of higher curvature gravity theories in arbitrary dimensions. Generically, these theories admit constant curvature vacua (even in the absence of an explicit cosmological constant), and asymptotically constant curvature solutions with non-trivial energy properties. For concreteness, we study quadratic curvature models in detail. Among them, the one whose action i...

On a Riemannian almost product manifold, the notion of a componentwise conformal vector field is introduced and several examples are exhibited. We show that this class of vector fields is a conformal invariant. For a compact manifold, a Bochner type integral formula for the Ricci tensor on such vector fields is obtained. Then, integral inequalities which link a curvature condition with the exis...

In this paper it is shown that a Lanczos potential for the Weyl curvature tensor does not exist for all spaces of dimension n ≥ 7. Whether there exists a Lanczos potential [1] for Weyl curvature tensors in dimensions n > 4 has still not been determined. Although Lanczos's original proof [1] for existence was flawed [2], there have subsequently been complete proofs for existence in four dimensio...

New relations involving curvature components for the various connections appearing in the theory of almost product manifolds are given and the conformal behaviour of these connections are studied. New identities for the irreducible parts of the deformation tensor are derived. Some direct physical applications in Kaluza–Klein and gauge theory are discussed. [email protected] [email protected]

In this note, we prove an existence result for the Einstein conformal constraint equations metrics with vanishing Yamabe invariant assuming that mean curvature satisfies explicit near-CMC condition and TT-tensor is small in $$L^2$$ .