On mean value theorems for small geodesic spheres in Riemannian manifolds

Kôzaki, Masanori

Czechoslovak Mathematical Journal, Tome 42 (1992) no. 3, pp. 519-547

Détail

DOI : 10.21136/CMJ.1992.128352

Classification : 53C25, 60D05

Bibliographie

[1] K. B. Athreya and T. G. Kuryz: A generalization of Dynkin’s identity. Ann. Probability 1 (1973), 570–579. | DOI | MR

[2] A. L. Besse: Manifolds all of whose Geodesics are Closed. Springer-Verlag, Berlin Heidelberg New York, 1978. | MR | Zbl

[3] Q. S. Chi: A curvature characterization of certain locally rank-one symmetric spaces. J. Differential Geometry 28 (1988), 187–202. | DOI | MR | Zbl

[4] D. M. DeTurck and J. L. Kazdan: Some regularity theorems in Riemannian geometry. Ann. Scient. Éc. Norm. Sup., 4$^{\text{e}}$ série 14 (1981), 249–260. | MR

[5] E. B. Dynkin: Markov Processes, vol. 2. Springer-Verlag, Berlin Heidelberg New York, 1965. | MR

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

[7] A. Gray and M. Pinsky: The mean exit time from a small geodesic ball in a Riemannian manifold. Bulletin des Sciences Mathematiques, 2$^{\text{e}}$ série 107 (1983), 345–370. | MR

[8] A. Gray and L. Vanhecke: Riemannian geometry as determined by the volume of small geodesic balls. Acta Math. 142 (1979), 157–198. | DOI | MR

[9] A. Gray and T. J. Willmore: Mean-value theorems for Riemannian manifolds. Proc. Roy. Soc. Edinburgh 92A (1982), 343–364. | MR

[10] O. Kowalski: The second mean-value operator on Riemannian manifolds. Proceedings of the CSSR-GDR-Polish Conference on Differential Geometry and its Applications, Nove Mesto 1980, pp. 33–45, Universita Karlova, Praha, 1982. | MR

[11] O. Kowalski: A comparison theorem for spherical mean-value operators in Riemannian manifolds. Proc. London Math. Soc. (3) 47 (1983), 1–14. | MR | Zbl

[12] M. Kôzaki and Y. Ogura: On geometric and stochastic mean values for small geodesic spheres in Riemannian manifolds. Tsukuba J. Math. 11 (1987), 131–145. | DOI | MR

[13] M. Kôzaki and Y. Ogura: On the independence of exit time and exit position from small geodesic balls for Brownian motions on Riemannian manifolds. Math. Z. 197 (1988), 561–581. | DOI | MR

[14] M. Kôzaki and Y. Ogura: Riemannian manifolds with stochastic independence conditions are rich enough. Probability theory and Math. statistics (Kyoto 1986), pp. 206–213 vol. 1299, Springer, Berlin-Heidelberg-New York, 1988. | MR

[15] M. Liao: Hitting distributions of small geodesic spheres. Ann. Probability 16 (1988), 1039–1050. | DOI | MR | Zbl

[16] E. M. Patterson: A class of critical Riemannian metrics. J. London Math. Soc. 23 (1981), no. 2, 349–358. | MR | Zbl

[17] M. Pinsky: Moyenne stochastique sur une variété riemannienne. C. R. Acad. Sci. Paris, Série I 292 (1981), 991–994. | MR | Zbl

[18] M. Pinsky: Brownian motion in a small geodesic ball. Asterique 132, Actes du Colloque Laurant Schwartz, 1985, pp. 89–101. | MR

[19] M. Pinsky: Independence implies Einstein metric, preprint.

[20] T. Sakai: On eigen-values of Laplacian and curvature of Riemannian manifold. Tôhoku Math. J. 23 (1971), 589–603. | DOI | MR | Zbl

[21] K. Sekigawa and L. Vanhecke: Volume-preserving geodesic symmetries on four-dimensional 2-stein spaces. Kodai Math. J. 9 (1986), 215–224. | DOI | MR

[22] T. J. Willmore: Mean value theorems in harmonic Riemannian spaces. J. London Math. Soc. 25 (1950), 54–57. | DOI | MR | Zbl