## 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.

### MSC:

 53C22 Geodesics in global differential geometry

### Keywords:

cut locus; conjugate locus; group of isometries

Zbl 0011.22601
Full Text:

### References:

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