SU(2$|$1) supersymmetric mechanics on curved spaces

We present SU$(2|1)$ supersymmetric mechanics on $n$-dimensional Riemannian manifolds within the Hamiltonian approach. The structure functions including prepotentials entering the supercharges and the Hamiltonian obey extended curved WDVV equations specified by the manifold's metric and curvature tensor. We consider the most general $u(2)$-valued prepotential, which contains both types (with and without spin variables), previously considered only separately. For the case of real K\"{a}hler manifolds we construct all possible interactions. For isotropic ($so(n)$-invariant) spaces we provide admissible prepotentials for any solution to the curved WDVV equations. All known one-dimensional SU$(2|1)$ supersymmetric models are reproduced.