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Kahler Kähler potentials that depend only on geodesic distance

Hermitian symmetric spaces of constant curvature have the property that the potential for their Kahler Kähler metric can be expresed as some function of the geodesic distance. Does anyone know if there are any general results concerning Kahler Kähler manifolds with this property?

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Kahler potentials that depend only on geodesic distance

Hermitian symmetric spaces of constant curvature have the property that the potential for their Kahler metric can be expresed as some function of the geodesic distance. Does anyone know if there are any general results concerning Kahler manifolds with this property?