Abstract
We study commutativity relations between differential operators and spherical means on Riemannian manifolds. The results show that all D'Atri spaces are ball-homogeneous. A further consequence characterizes complete nonpositively curved, simply connected D'Atri spaces by commuting mean value operators.
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References
Ballmann, W.: Axial isometries of manifolds of nonpositive curvature, Math. Ann. 259 (1982), 131–144.
Besson, G., Courtois, G. and Gallot, S.: Entropies et rigidités des espaces localement symétriques de courbure strictement négative, C. R. Acad. Sci. Paris. Sér. I Math. 319 (1994), 81–84.
Booss, B.: Topologie und Analysis: Eine Einführung in die Atiyah-Singer-Indexformel, Springer-Verlag, Berlin, 1977.
Chen, S. S. and Eberlein, P.: Isometry groups of simply connected manifolds of nonpositive curvature, Illinois J. Math. 24 (1980), 73–103.
Calvaruso, G. and Vanhecke, L.: Special ball-homogeneous spaces, Z. Anal. Anwendungen 16 (1997), 789–800.
D'Atri, J. E. and Nickerson, H. K.: Divergence-preserving geodesic symmetries, J. Differential Geom. 3 (1969), 467–476.
Gray, A. and Willmore, T. J.: Mean-value theorems for Riemannian manifolds, Proc. Roy. Soc. Edinburgh 92 (1982), 343–367.
Günther, P.: Sphärische Mittelwerte in kompakten harmonischen Riemannschen Mannigfaltigkeiten, Math. Ann. 165 (1966), 281–296.
Heber, J.: Homogeneous spaces of nonpositive curvature and their geodesic flow, Internat. J. Math. 6 (1995), 279–296.
Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.
Kowalski, O.: The second mean-value operator on Riemannian manifolds, in Kowalski, O. (ed.), Proc. of the ČSSR-GDR-Polish Conference on Differential Geometry and Its Applications, Nove Město, 1980, Univerzita Karlova, 1982, pp. 33–45.
Kowalski, O.: Some curvature identities for commutative spaces, Czechoslovak Math. J. 32 (1982), 389–396.
Kazdan, J. L.: personal communication with L. Vanhecke, 1985.
Kowalski, O. and Prüfer, F.: On probabilistic commutative spaces, Mh. Math. 107 (1989), 57–68.
Kowalski, O., Prüfer, F. and Vanhecke, L.: D'Atri spaces, in Gindikin, S. (ed.), Topics in Geometry: In Memory of Joseph D'Atri, Progress in Nonlinear Differential Equations and Their Applications, Vol. 20, Birkhäuser, Boston, 1996, pp. 241–284.
Kowalski, O. and Vanhecke, L.: Two point functions on Riemannian manifolds, Ann. Global Anal. Geom. 3 (1985), 95–119.
Prüfer, F.: On compact Riemannian manifolds with volume-preserving symmetries, Ann. Global Anal. Geom. 7 (1989), 133–140.
Prüfer, F., Tricerri, F. and Vanhecke, L.: Curvature invariants, differential operators and local homogeneity, Trans. Amer. Math. Soc. 348 (1996), 4643–4652.
Roberts, P. H. and Ursell, H. D.: Random walk on a sphere and on a Riemannian manifold, Philos. Trans. Roy. Soc. London, Ser. A 252 (1959–1960), 317–356.
Simon, U.: On differential operators of second order on Riemannian manifolds with nonpositive curvature, Colloq. Math. XXXI (1974), 223–229.
Sumitomo, T.: On the commutator of differential operators, Hokkaido Math. J. 1 (1972), 30–42.
Sunada, T.: Spherical means and geodesic chains on a Riemannian manifold, Trans. Amer. Math. Soc. 267 (1981), 483–501.
Szabó, Z. I.: Spectral theory for operator families on Riemannian manifolds, Proc. Sympos. Pure Math. 54 (1993), 615–665.
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Günther†, P., Prüfer, F. Mean Value Operators, Differential Operators and D'Atri Spaces. Annals of Global Analysis and Geometry 17, 113–127 (1999). https://doi.org/10.1023/A:1006502617787
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DOI: https://doi.org/10.1023/A:1006502617787