Anholonomic frames, generalized Killing equations, and anisotropic Taub-NUT spinning spaces

Abstract

By using anholonomic frames in (pseudo)-Riemannian spaces we define anisotropic extensions of Euclidean Taub-NUT spaces. With respect to coordinate frames such spaces are described by offdiagonal metrics which could be diagonalized by corresponding anholonomic transforms. We define the conditions when the 5D vacuum Einstein equations have as solutions anisotropic Taub-NUT spaces. The generalized Killing equations for the configuration space of anisotropically spinning particles (anisotropic spinning space) are analyzed. Simple solutions of the homogeneous part of these equations are expressed in terms of some anisotropically modified Killing-Yano tensors. The general results are applied to the case of the four-dimensional locally anisotropic Taub-NUT manifold with Euclidean signature. We emphasize that all constructions are for (pseudo)-Riemannian spaces defined by vacuum solutions, with generic anisotropy, of 5D Einstein equations, the solutions being generated by applying the moving frame method. 

RICCI FLOWS AND SOLITONIC pp-WAVES

“…[17] locally anisotropic Taub-NUT, soliton-spinor waves, wormholes and solitonically moving black holes in higher dimensions [18,8]. Here, we emphasize that the approach elaborated in this work contains general off-diagonal terms (so called N-coefficients) not restricted to the Kaluza-Klein conditions (related to linearizations and compactifications on extra-dimension coordinates).…”

Nonholonomic Ricci Flows and Running Cosmological Constant I: 4d Taub-Nut Metrics

“…[13,14,15,16,17] for various models of gravity theory. The idea of work [18] was to use the former method and some integral varieties of those solutions in order to subject the metric and N-connection coefficients additionally to the conditions (19) and (20) and generate Ricci flows of offdiagonal metrics.…”

“…The idea is to use certain classes of nonholnomic (equivalently, anholonomic) deformations of the frame, metric and connection structures and superpositions of generalized conformal maps in order to generate a class of off-diagonal metric ansatz solving exactly the vacuum or nonvacuum Einstein equations. 1 The method was considered, for instance, for constructing locally anisotropic Taub-NUT solutions [16] and investigating self-consistent propagations of three dimensional Dirac and/or solitonic waves in such spacetimes [17].…”

Nonlinear connections and spinor geometry

“…We emphasize that the theory of anisotropic spinors may be related not only to generalized Finsler, Lagrange, Cartan, and Hamilton spaces or their higher-order generalizations, but also to anholonomic frames with associated nonlinear connections which appear naturally even in (pseudo-) Riemannian and Riemann-Cartan geometries if off-diagonal metrics are considered [94,96,97,98,102,103,104,105,110]. In order to construct exact solutions of the Einstein equations in general relativity and extradimensional gravity (for lower dimensions see [85,96,107,108]), it is more convenient to diagonalize space-time metrics by using some anholonomic transforms.…”

“…Various types of Finsler-like structures can be parametrized by generic off-diagonal metrics, which cannot be diagonalized by coordinate transforms but only by anholonomic maps with associated nonlinear connection (in brief, N-connection). Such structures may be defined as exact solutions of gravitational field equations in the Einstein gravity and its generalizations [75,79,80,94,95,96,97,98,99,100,102,103,104,105,109,110,111], for instance, in the metric-affine [19,23,56] Riemann-Cartan gravity [24,25]. Finsler-like configurations are considered in locally anisotropic thermodynamics, kinetics, related stochastic processes [85,96,107,108], and (super-) string theory [84,87,90,91,92].…”