  We present ${\cal N}{=}\,4$ supersymmetric mechanics on $n$-dimensional
Riemannian manifolds constructed within the Hamiltonian approach. The structure
functions entering the supercharges and the Hamiltonian obey modified covariant
constancy equations as well as modified Witten-Dijkgraaf-Verlinde-Verlinde
equations specified by the presence of the manifold's curvature tensor.
Solutions of original Witten-Dijkgraaf-Verlinde-Verlinde equations and related
prepotentials defining ${\cal N}{=}\,4$ superconformal mechanics in flat space
can be lifted to $so(n)$-invariant Riemannian manifolds. For the Hamiltonian
this lift generates an additional potential term which, on spheres and
(two-sheeted) hyperboloids, becomes a Higgs-oscillator potential. In
particular, the sum of $n$ copies of one-dimensional conformal mechanics
results in a specific superintegrable deformation of the Higgs oscillator.
