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Two-point functions on Riemannian manifolds. (English) Zbl 0571.53031
Let (M,g) be a smooth Riemannian manifold of dimension $$n\geq 2$$. A smooth function F on a neighborhood U of the diagonal in $$M\times M$$ is called a two-point function if for every (x,y)$$\in U$$ there exists a shortest geodesic joining x to y. The authors give the basic elements of the theory of two-point functions and then obtain results with respect to harmonic spaces and spaces with volume-preserving symmetries. The clarity of the paper is noteworthy.
Reviewer: A.Bejancu

MSC:
 53C22 Geodesics in global differential geometry 53C20 Global Riemannian geometry, including pinching
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References:
 [1] Berger, M., Gauduchon, P. and Mazet, E.:Le spectre d’une vari?t? riemannienne, Lecture Notes in Mathematics, 194, Springer-Verlag, Berlin, 1971. · Zbl 0223.53034 [2] Besse, A.L.:Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik, 93, Springer-Verlag, Berlin, 1978. · Zbl 0387.53010 [3] Chen, B.Y. and Vanhecke, L.: Differential geometry of geodesic spheres,J. Reine Angew. Math. 325 (1981), 28-67. · Zbl 0503.53013 [4] D’atri, J.E. and Nickerson, H.K.: Geodesic symmetries in spaces with special curvature tensor,J. Differential Geometry 9 (1974), 251-262. · Zbl 0285.53019 [5] D’atri, J.E.: Geodesic spheres and symmetries in naturally reductive homogeneous spaces,Michigan Math. J. 22 (1975), 71-76. · Zbl 0317.53045 [6] Gray, A. and Vanhecke, L.: Riemannian geometry as determined by the volumes of small geodesic balls,Acta M Math. 142 (1979), 157-198. · Zbl 0428.53017 [7] Kowalski, O. and Vanhecke, L.: Op?rateurs diff?rentiels invariants et symm?tries g?odesiques pr?servant le volume,C.R. Acad. Sc. Paris, S?rie I 296 (1983), 1001-1003. [8] Kowalski, O. and Vanhecke, L. : A generalization of a theorem on naturally reductive homogeneous spaces,Proc. Amer. Math. Soc., to appear. · Zbl 0542.53029 [9] Lichnerowicz, A.: Op?rateurs diff?rentiels invariant sur un espace homog?ne,Ann. Sc. Ecole Norm. Sup. 81 (1964), 341-385. [10] Roberts, P.H. and Ursell, H.D.: Random walk on a sphere and on a Riemannian manifold,Phil. Trans. Royal Soc. London A 252 (1960), 317-356. · Zbl 0094.31901 [11] Ruse, H.S.: On commutative Riemannian manifolds,Tensor 26 (1972), 180-184. · Zbl 0259.53037 [12] Ruse, H.S., Walker, A.G. and Willmore, T.J. :Harmonic spaces, Cremonese, Roma, 1961. · Zbl 0134.39202 [13] Sumitomo, T.: On a certain class of Riemannian homogeneous spaces,Colloq. Math. 26 (1971), 129-133. · Zbl 0249.53042 [14] Sumitomo, T.: On the commutator of differential operators,Hokkaido Math. J. 1 (1972), 30-42. · Zbl 0248.53050 [15] Vanhecke, L.: 1-harmonic spaces are harmonic,Bull. London Math. Soc. 13 (1981), 409-411. · Zbl 0474.53025 [16] Vanhecke, L.: A note on harmonic spaces,Bull. London Math. Soc. 13 (1981), 545-546. · Zbl 0472.53048 [17] Vanhecke, L.: A conjecture of Besse on harmonic manifolds,Math. Z. 178 (1981), 555-557. · Zbl 0465.53030 [18] Vanhecke, L.: Some solved and unsolved problems about harmonic and commutative spaces,Bull. Soc. Math. Belg. B34 (1982), 1-24. · Zbl 0518.53042 [19] Vanhecke, L. and Willmore, T.J.: Riemannian extensions of D’Atri spaces,Tensor 38 (1982), 154-158. · Zbl 0505.53016 [20] Vanhecke, L.: The canonical geodesic involution and harmonic spaces,Ann. Global Analysis and Geometry 1 (1983), 131-136. · Zbl 0532.53015 [21] Vanhecke, L. and Willmore, T.J.: Interaction of tubes and spheres,Math. Ann. 263 (1983), 31-42. · Zbl 0502.53032 [22] Willmore, T.J.: 2-point invariant functions and k-harmonic manifolds,Rev. Rouwaine Math. Pures Appl. 13 (1968), 1051-1057. · Zbl 0169.53302 [23] Willmore, T.J. and El Hadi, K.: k-harmonic symmetric manifolds,Rev. Roumaine Math. Pures Appl. 15 (1970), 1573-1577. · Zbl 0209.25302
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