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

Fermions on curved spaces, symmetries, and quantum anomalies