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B. Csik\'os and M. Horv\'ath proved that if a connected Riemannian manifoldof dimension at least $4$ is harmonic, then the total scalar curvatures oftubes of small radius about a regular curve depend only on the length of thecurve and the radius of the tube, and conversely, if the latter condition holdsfor cylinders, i.e., for tubes about geodesic segments, then the manifold isharmonic. In the present paper, we show that in contrast to the higherdimensional case, a connected 3-dimensional Riemannian manifold has the abovementioned property of tubes if and only if the manifold is a D'Atri space,furthermore, if the space has bounded sectional curvature, then it is enough torequire the total scalar curvature condition just for cylinders to imply thatthe space is D'Atri. This result gives a negative answer to a question posed byL. Gheysens and L. Vanhecke. To prove these statements, we give acharacterization of D'Atri spaces in terms of the total scalar curvature ofgeodesic hemispheres in any dimension.

D'Atri spaces and the total scalar curvature of hemispheres, tubes and cylinders