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

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• 作者：Alfonso Garcí ; a-Parrado Gó ; mez-Lobo
• 关键词：Differential geometry ; Symmetry transformations ; Invariant characterizations
• 刊名：Journal of Geometry and Physics
• 出版年：2006
• 出版时间：July 2006
• 年：2006
• 卷：56
• 期：7
• 页码：1069-1095
• 全文大小：282 K

文摘

This is the first of two companion papers in which a thorough study of the normal form and the first integrability conditions arising from bi-conformal vector fields is presented. These new symmetry transformations were introduced in Class. Quantum Grav. 21, 2153–2177 and some of their basic properties were addressed there. Bi-conformal vector fields are defined on a pseudo-Riemannian manifold V through the differential conditions and where P_ab and Π_ab are orthogonal and complementary projectors with respect to the metric tensor g_ab. In our calculations a new affine connection (bi-conformal connection) arises quite naturally and this connection enables us to find a local characterization of 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.