New Properties of Euclidean Killing Tensors of Rank Two

A symmetric tensor field on a Riemannian manifold is called a Killing tensor field if the symmetric part of its covariant derivative is equal to zero. There exists a well-known bijection between Killing tensor fields and conserved quantities of the geodesic flow which depend polynomially on the momentum variables. In particular, Killing tensors of rank (or valence) two yields quadratic first integrals and we discuss some aspects of this process in Crasmareanu [7] from a dynamical point of view. Some classes of physical examples associated with the Euclidean 2D metric are provided in Crasmareanu and Baleanu [8].