The authors proved in "On the volume of the intersection of two geodesic balls," Differential Geom. Appl. 29 (2011), 567–576, that in a complete, connected, and simply connected Riemannian manifold, the volume of the intersection of two small geodesic balls depends only on the distance between the centers and the radii if and only if the space is harmonic. In this paper, we show that this proposition remains true, if the condition is restricted to balls of equal radii.
"A Characterization of harmonic spaces." J. Differential Geom. 90 (3) 383 - 389, March 2012. https://doi.org/10.4310/jdg/1335273388