  The quantum oscillator and Kepler-Coulomb problems in $d$-dimensional spaces
with constant curvature are analyzed from several viewpoints. In a deformed
supersymmetric framework, the corresponding nonlinear potentials are shown to
exhibit a deformed shape invariance property. By using the point canonical
transformation method, the two deformed Schr\"odinger equations are mapped onto
conventional ones corresponding to some shape-invariant potentials, whose
rational extensions are well known. The inverse point canonical transformations
then provide some rational extensions of the oscillator and Kepler-Coulomb
potentials in curved space. The oscillator on the sphere and the Kepler-Coulomb
potential in a hyperbolic space are studied in detail and their extensions are
proved to be consistent with already known ones in Euclidean space. The
partnership between nonextended and extended potentials is interpreted in a
deformed supersymmetric framework. Those extended potentials that are
isospectral to some nonextended ones are shown to display deformed shape
invariance, which in the Kepler-Coulomb case is enlarged by also translating
the degree of the polynomial arising in the rational part denominator.
