In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some conceptual issues on spacetimes provided with generic off-diagonal metrics and associated nonlinear connection structures are analyzed. The limit from gravity/Ricci flow models with nontrivial torsion to configurations with the Levi-Civita connection is allowed in some specific physical circumstances by constraining the class of integral varieties for the Einstein and Ricci flow equations.

The Schrödinger equation is investigated in the Euclidean Taub-NUT geometry. The bound states are degenerate and an extra degeneracy is due to the conserved Runge-Lenz vector. The existence of the extra conserved quantities, quadratic in four-velocities implies the possibility of separating variables in two different coordinate systems. The eigenvalues and the eigenvectors are given in both cases in explicit, closed form.Pacs 04.62.+v *

We investigate the SO(4, 1) gauge-invariant theory of the Dirac fermions in the external field of the Kaluza-Klein monopole, pointing out that the quantum modes can be recovered from a Klein-Gordon equation analogous to the Schrödinger equation in the Taub-NUT background. Moreover, we show that there is a large collection of observables that can be directly derived from those of the scalar theory. These offer many possibilities of choosing complete sets of commuting operators which determine the quantum modes. In addition there are some spin-like and Dirac-type operators involving the covariantly constant Killing-Yano tensors of the hyper-Kähler Taub-NUT space. The energy eigenspinors of the central modes in spherical coordinates are completely evaluated in explicit, closed form.Pacs 04..62.+v *

The SO(4, 1) gauge-invariant theory of the Dirac fermions in the external field of the Kaluza-Klein monopole is investigated. It is shown that the discrete quantum modes are governed by reducible representations of the o(4) dynamical algebra generated by the components of the angular momentum operator and those of the Runge-Lenz operator of the Dirac theory in Taub-NUT background. The consequence is that there exist central and axial discrete modes whose spinors have no separated variables.Pacs 04.62.+v

The geodesic motion of pseudo-classical spinning particles in Euclidean Taub-NUT space is analysed. The constants of motion are expressed in terms of Killing-Yano tensors. Some previous results from the literature are corrected. PACS number(s):04.20.Jb, 02.40.-K The configuration space of spinning particles (spinning space) is an supersymmetric extension of an ordinary Riemannian manifold, parametrized by local coordinates {x µ }, to a graded manifold parametrized by local coordinates {x µ , ψ µ }, with the first set of variables being Grassmann even (commuting) and the second set, Grassmann odd (anticommuting). The equation of motion of a spinning particle on a geodesic is derived from the action:The corresponding world-line Hamiltonian is given by:where Π µ = g µνẋ ν is the covariant momentum.

The authors consider the spontaneous compactification induced by a scalar sector in the form of a non-linear sigma model. A very general class of solutions is given by Riemannian submersions from the extra dimensional space onto the space in which the scalar fields take values. An explicit example is constructed taking for the extra dimensional space a generalised Hopf manifold. A massless gauge field is associated with a vertical Killing vector of the Lee type.

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The gravitational anomalies are investigated for generalized Euclidean Taub-NUT metrics which admit hidden symmetries analogous to the Runge-Lenz vector of the Kepler-type problem. In order to evaluate the axial anomalies, the index of the Dirac operator for these metrics with the APS boundary condition is computed. The role of the Killing-Yano tensors is discussed for these two types of quantum anomalies.