Riemann curvature tensor

About: Riemann curvature tensor is a research topic. Over the lifetime, 6248 publications have been published within this topic receiving 138871 citations. The topic is also known as: Riemann–Christoffel tensor.

Twistor spaces with hermitian ricci tensor

Abstract: The twistor space Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics ht compatible with the almost-complex structures J, and J2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In the present note we describe the (real-analytic) manifolds M for which the Ricci tensor of (Z , ht) is ./"-Hermitian, n = 1 or 2. This is used to supply examples giving a negative answer to the Blair and Ianus question of whether a compact almost-Kahler manifold with Hermitian Ricci tensor is Kahlerian.

Type III and N universal spacetimes

Abstract: Universal spacetimes are spacetimes for which all conserved symmetric rank-2 tensors, constructed as contractions of polynomials from the metric, the Riemann tensor and its covariant derivatives of arbitrary order, are multiples of the metric. Consequently, metrics of universal spacetimes solve vacuum equations of all gravitational theories, with the Lagrangian being a polynomial curvature invariant constructed from the metric, the Riemann tensor and its derivatives of arbitrary order. In the literature, universal metrics are also discussed as metrics with vanishing quantum corrections and as classical solutions to string theory. Widely known examples of universal metrics are certain Ricci-flat pp waves. In this paper, we start a general study of the geometric properties of universal metrics in arbitrary dimension and arrive at a broader class of such metrics. In contrast with pp waves, these universal metrics also admit a non-vanishing cosmological constant and in general do not have to possess a covariant constant or recurrent null vector field. First, we show that a universal spacetime is necessarily a constant curvature invariant spacetime, i.e. all curvature invariants constructed from the Riemann tensor and its derivatives are constant. Then we focus on type N spacetimes, where we arrive at a simple necessary and sufficient condition: a type N spacetime is universal if and only if it is an Einstein Kundt spacetime. A class of type III Kundt universal metrics is also found. Several explicit examples of universal metrics are presented.

Algebraic computing and the Newman-Penrose formalism in general relativity

Abstract: The curvature tensor of space-time can be described most concisely by giving the components of the Weyl and Ricci tensors relative to a complex null tetrad. The Newman-Penrose equations provide a simple and direct algorithm for calculating these components. This paper describes a computer program, written in the symbolic manipulation language CAMAL, which performs this calculation. Comparisons are made with the classical tensorial method of calculation, and some applications are discussed.

Dirac fields in f(R)-gravity with torsion

Abstract: We study f(R)-gravity with torsion in the presence of Dirac massive fields. Using the Bianchi identities, we formulate the conservation laws of the theory and we check the consistency with the matter field equations. Further, we decompose the field equations in torsionless and torsional terms: we show that the nonlinearity of the gravitational Lagrangian reduces to the presence of a scalar field that depends on the spinor field; this additional scalar field gives rise to an effective stress?energy tensor and plays the role of a scale factor modifying the normalization of Dirac fields. Problems for fermions regarding the positivity of energy and the particle?antiparticle duality are discussed.

A visual introduction to Riemannian curvatures and some discrete generalizations

Abstract: We try to provide a visual introduction to some objects used in Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, Bianchi identities... We then explain some of the strategies used to define ana- logues of curvature in non-smooth or discrete spaces, beginning with Alexan- drov curvature and δ-hyperbolic spaces, and insisting on various notions of generalized Ricci curvature, which we briefly compare.