Ricci decomposition

About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.

A metric with no symmetries or invariants

Abstract: A metric is given which has no scalar invariants formed from its Riemann tensor or derivatives of its Riemann tensor and which admits, in its general form, no local homotheties or isometries. It is conformally flat and describes pure radiation. Subcases of this metric are the plane wave metric and a metric of Wils.

Cosmological applications of the Lanczos Lagrangian

Abstract: There is a unique Lagrangian quadratic in the curvature tensor which yields second-order field equations in dimensions greater than four. This Lagrangian is applied to a Kaluza-Klein theory and its cosmological implications are investigated.

Einstein structures: Existence versus uniqueness

Abstract: Two well-known questions in differential geometry are “Does every compact manifold of dimension greater than four admit an Einstein metric?” and “Does an Einstein metric of a negative scalar curvature exist on a sphere?” We demonstrate that these questions are related: For everyn≥5 the existence of metrics for which the deviation from being Einstein is arbitrarily small on every compact manifold of dimensionn (or even on every smooth homology sphere of dimensionn) implies the existence of metrics of negative Ricci curvature on the sphereS n for which the deviation from being Einstein is arbitrarily small. Furthermore, assuming either a version of the Palais-Smale condition or the plausible looking existence of an algorithm deciding when a given metric on a compact manifold is close to an Einstein metric, we show for anyn≥5 that: 1) If everyn-dimensional smooth homology sphere admits an Einstein metric thenS n admits infinitely many Einstein structures of volume one and of negative scalar curvature; 2) If every compactn-dimensional manifold admits an Einstein metric then every compactn-dimensional manifold admits infinitely many distinct Einstein structures of volume one and of negative scalar curvature.

Singular value decomposition for the 2D fan-beam Radon transform of tensor fields

Abstract: In this article we study the fan-beam Radon transform Dm of symmet- rical solenoidal 2D tensor fields of arbitrary rank m in a unit disc D as the operator, acting from the object space L2(D ; Sm) to the data space L2((0, 2π) × (0, 2π)). The orthogonal polynomial basis s (±m) n,k of solenoidal tensor fields on the disc D was built with the help of Zernike polynomials and then a singular value decomposition (SVD) for the operator Dm was obtained. The inversion formula for the fan-beam tensor transform Dm follows from this decomposition. Thus obtained inversion formula can be used as a tomographic filter for splitting a known tensor field into potential and solenoidal parts. Numerical results are presented.