Symmetric cut loci in Riemannian manifolds. (English) Zbl 0563.53037

Let M be a compact Riemannian manifold with \(H_ 1(M,Z)=0\). We show that, for a point \(p\in M\), the cut locus and conjugate locus of p must intersect if M admits a group of isometries which fixes p and has principal orbits of codimension at most 2. This is a classical theorem of S. B. Myers [Proc. Natl. Acad. Sci. USA 21, 225-227 (1935; Zbl 0011.22601)] in the case when M has dimension 2.


53C22 Geodesics in global differential geometry


Zbl 0011.22601
Full Text: DOI


[1] M. Berger, Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive, Ann. Scuola Norm. Sup. Pisa (3) 15 (1961), 179 – 246 (French). · Zbl 0101.14201
[2] Glen E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. · Zbl 0246.57017
[3] Isaac Chavel, A class of Riemannian homogeneous spaces, J. Differential Geometry 4 (1970), 13 – 20. · Zbl 0197.18302
[4] Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. North-Holland Mathematical Library, Vol. 9. · Zbl 0309.53035
[5] Sumner Byron Myers, Connections between differential geometry and topology. I. Simply connected surfaces, Duke Math. J. 1 (1935), no. 3, 376 – 391. · Zbl 0012.27502 · doi:10.1215/S0012-7094-35-00126-0
[6] Peter Orlik, Seifert manifolds, Lecture Notes in Mathematics, Vol. 291, Springer-Verlag, Berlin-New York, 1972. · Zbl 0263.57001
[7] H. Rauch, Geodesics and curvature in differential geometry in the large, Yeshiva Univ. Press, New York, 1959.
[8] Takashi Sakai, Cut loci of Berger’s spheres, Hokkaido Math. J. 10 (1981), no. 1, 143 – 155. · Zbl 0469.53041 · doi:10.14492/hokmj/1381758107
[9] Kunio Sugahara, On the cut locus and the topology of Riemannian manifolds, J. Math. Kyoto Univ. 14 (1974), 391 – 411. · Zbl 0289.53033
[10] Alan D. Weinstein, The cut locus and conjugate locus of a riemannian manifold, Ann. of Math. (2) 87 (1968), 29 – 41. · Zbl 0159.23902 · doi:10.2307/1970592
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