  We review the geodesic motion of pseudo-classical spinning particles in
curved spaces. Investigating the generalized Killing equations for spinning
spaces, we express the constants of motion in terms of Killing-Yano tensors.
Passing from the spinning spaces to the Dirac equation in curved backgrounds we
point out the role of the Killing-Yano tensors in the construction of the
Dirac-type operators. The general results are applied to the case of the
four-dimensional Euclidean Taub-Newman-Unti-Tamburino space. The gravitational
and axial anomalies are studied for generalized Euclidean Taub-NUT metrics
which admit hidden symmetries analogous to the Runge-Lenz vector of the
Kepler-type problem. Using the Atiyah-Patodi-Singer index theorem for manifolds
with boundaries, it is shown that the these metrics make no contribution to the
axial anomaly.

High Energy Physics - Theory