arXiv subjects

Alfonso García-Parrado Gómez-Lobo

Publications and source records attributed to Alfonso García-Parrado Gómez-Lobo.

21 records · Page 2Linked to original sources

arXiv2005

Bi-conformal vector fields and the local geometric characterization of conformally separable pseudo-Riemannian manifolds II

In this paper we continue the study of bi-conformal vector fields started in {\em Class. Quantum Grav.} {\bf 21} 2153-2177. These are vector fields defined on a pseudo-Riemannian manifold by the differential conditions $\lie P_{ab}=ϕP_{ab}$, $\lieΠ_{ab}=χΠ_{ab}$ where $P_{ab}$, $Π_{ab}$ are orthogonal and complementary projectors with respect to the metric tensor $\rmg_{ab}$ and $\lie$ is the Lie derivative. In a previous paper we explained how the analysis of these differential conditions enabled us to derive local geometric characterizations of the most relevant cases of {\em conformally separable} (also called double twisted) pseudo Riemannian manifolds. In this paper we carry on this analysis further and provide local invariant characterizations of conformally separable pseudo-Riemannian manifolds with {\em conformally flat} leaf metrics. These characterizations are rather similar to that existing for conformally flat pseudo-Riemannian manifolds but instead of the Weyl tensor, we must demand the vanishing of certain four rank tensors constructed from the curvature of an affine, non-metric, connection (bi-conformal connection). We also speculate with possible applications to finding results for the existence of foliations by conformally flat hypersurfaces in any pseudo-Riemannian manifold.

math.DG↗

arXiv2005

Bi-conformal vector fields and the local geometric characterization of conformally separable (double-twisted) pseudo-Riemannian manifolds

This is the first of two companion papers in which a thorough study of the normal form and the first integrability conditions arising from {\em bi-conformal vector fields} is presented. These new symmetry transformations were introduced in {\em Class. Quantum Grav.} \textbf{21}, 2153-2177 and some of their basic properties were addressed there. In our calculations a new affine connection ({\em bi-conformal connection}) arises quite naturally and this connection enables us to find a local characterization of {\em conformally separable} pseudo-Riemannian manifolds (also called double twisted products) in terms of the vanishing of a rank three tensor $T_{abc}$. Similar local characterizations are found for the most important particular cases such as (double) warped products, twisted products and conformally reducible spaces (this classification will be enlarged in the forthcoming second paper).We speculate with potential applications of the bi-conformal connection to the invariant characterization of other types of pseudo-Riemannian manifolds in which the metric tensor has a part conformal to a metric of lesser rank.

math.DG↗

arXiv2004

Bi-conformal vector fields and their applications to the characterization of conformally separable pseudo-Riemannian manifolds: New criteria for the existence of conformally flat foliations in pseudo-Riemannian manifolds

In this paper a thorough study of the normal form and the first integrability conditions arising from {\em bi-conformal vector fields} is presented. These new symmetry transformations were introduced in {\em Class. Quantum Grav.}\textbf{21}, 2153-2177 and some of their basic properties were addressed there. Bi-conformal vector fields are defined on a pseudo-Riemannian manifold through the differential conditions $\lie P_{ab}=ϕP_{ab}$ and $\lieΠ_{ab}=χΠ_{ab}$ where $P_{ab}$ and $Π_{ab}$ are orthogonal and complementary projectors with respect to the metric tensor $\rmg_{ab}$. One of the main results of our study is the discovery of a new geometric characterization of {\em conformally separable spaces with conformally flat leaf metrics} similar to the vanishing of the Weyl tensor for conformally flat metrics. This geometric characterization seems to carry over to any pseudo-Riemannian manifold admitting conformally flat foliations which would open the door to the systematic searching of these type of foliations in a given pseudo-Riemannian metric. Other relevant aspects such as the existence of invariant tensors under the finite groups generated by these transformations are also addressed.

math-ph↗