Their existence guarantees the integrability of the geodesic motions and is intimately related to separability of Hamilton-Jacobi (see, e.g., ) and the Klein-Gordon equation at the quantum level. The corresponding conserved quantities are quadratic, or, more general, polynomial in momenta. These symmetries are known as hidden symmetries and the typical example is the Runge-Lenz vector in the Kepler/Coulomb problem. Sometimes a spacetime could admit higher-order symmetries described by symmetric Killing tensors, called Stackel-Killing (S-K). For the geodesic motions on a space-time the usual conserved quantities are related to the isometries which correspond to Killing vectors. One of the key concepts in physics is that of symmetries, Noether's theorem giving a correspondence between symmetries and conserved quantities. HIDDEN SYMMETRIES AND KILLING TENSORS ON CURVED SPACES The existence of the conformal Killing-Yano tensors is investigated for some spaces with mixed 3-Sasakian structures. One presents the infinite dimensional superalgebra of Dirac type operators on the 4-dimensional Euclidean Taub-NUT space that can be seen as a twisted loop algebra.

The general results are applied to space-times which appear in modern studies. In the Dirac theory on curved spaces, Killing-Yano tensors generate Dirac-type operators involved in interesting algebraic structures as dynamical algebras or even infinite dimensional algebras or superalgebras. The intimate relation between Killing-Yano tensors and nonstandard supersymmetries is pointed out. Higher-order symmetries corresponding to Killing tensors are investigated.

Tensor field curved space