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The quantum oscillator and Kepler-Coulomb problems in $d$-dimensional spaceswith constant curvature are analyzed from several viewpoints. In a deformedsupersymmetric framework, the corresponding nonlinear potentials are shown toexhibit a deformed shape invariance property. By using the point canonicaltransformation method, the two deformed Schr\"odinger equations are mapped ontoconventional ones corresponding to some shape-invariant potentials, whoserational extensions are well known. The inverse point canonical transformationsthen provide some rational extensions of the oscillator and Kepler-Coulombpotentials in curved space. The oscillator on the sphere and the Kepler-Coulombpotential in a hyperbolic space are studied in detail and their extensions areproved to be consistent with already known ones in Euclidean space. Thepartnership between nonextended and extended potentials is interpreted in adeformed supersymmetric framework. Those extended potentials that areisospectral to some nonextended ones are shown to display deformed shapeinvariance, which in the Kepler-Coulomb case is enlarged by also translatingthe degree of the polynomial arising in the rational part denominator.

Quantum oscillator and Kepler-Coulomb problems in curved spaces: deformed shape invariance, point canonical transformations, and rational extensions