An extension of Pizzetti's formula to riemmanian manifolds
Analyse sur les variétés (Metz, 1979), Astérisque, no. 80 (1980), pp. 53-56. 
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     author = {Willmore, T. J.},
     title = {An extension of {Pizzetti's} formula to riemmanian manifolds},
     booktitle = {Analyse sur les vari\'et\'es (Metz, 1979)},
     series = {Ast\'erisque},
     pages = {53--56},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {80},
     year = {1980},
     mrnumber = {620169},
     zbl = {0495.53041},
     language = {en},
     url = {http://archive.numdam.org/item/AST_1980__80__53_0/}
}
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Willmore, T. J. An extension of Pizzetti's formula to riemmanian manifolds, dans Analyse sur les variétés (Metz, 1979), Astérisque, no. 80 (1980), pp. 53-56. http://archive.numdam.org/item/AST_1980__80__53_0/

[BE] Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik, vol. 93, Springer Verlag, Berlin and New York, 1978 | MR | Zbl

[CH] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 2, Interscience, 1962. | Zbl

[FR] A. Friedman, "Function-theoretic characterization of Einstein spaces and harmonic spaces", Trans. Amer. Math. Soc. 101 (1961) 240-258 | DOI | MR | Zbl

[PI.1] P. Pizzetti, Sulla media dei valori che una funzione dei punti dello spazio assume alla superficie di una sfera", Rend. Acc. Naz. Lincei ser 5, 18 (1909), 182-185. | JFM

[PI-2] P. Pizzetti, Sul significato geometrico del secundo parametro differenziale 55 di una funzione sopra una superficie qualunque", Rend. Acc. Naz. Lincei ser. 5, 18 (1909), 309-316. | JFM

[WI] T. J. Willmore, "Mean-value theorems in harmonic Riemannian spaces", J. Lond. Math. Soc. 25 (1950), 54-57 | DOI | MR | Zbl

[ZA] L. Zalcman, "Mean values and differential equations", Israel J. Math. 14 (1973), 339-352 | DOI | MR | Zbl