Continuous actions of compact Lie groups on Riemannian manifolds

Authors:
David Hoffman and L. N. Mann

Journal:
Proc. Amer. Math. Soc. **60** (1976), 343-348

MSC:
Primary 57E10

MathSciNet review:
0423386

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: M. H. A. Newman proved that if *M* is a connected topological manifold with metric *d*, there exists a number , depending only upon *M* and *d*, such that every compact Lie group acting effectively on *M* has at least one orbit of diameter at least . In this paper the authors consider the case where *M* is a Riemannian manifold and *d* is the distance function on *M* arising from the Riemannian metric. They obtain estimates for in terms of convexity and curvature invariants of *M*.

**[1]**Glen E. Bredon,*Introduction to compact transformation groups*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. MR**0413144****[2]**Andreas Dress,*Newman’s theorems on transformation groups*, Topology**8**(1969), 203–207. MR**0238353****[3]**R. C. Gunning,*Lectures on Riemann surfaces*, Princeton Mathematical Notes, Princeton University Press, Princeton, N.J., 1966. MR**0207977****[4]**David Hoffman,*The diameter of orbits of compact groups of isometries; Newman’s theorem for noncompact manifolds*, Trans. Amer. Math. Soc.**233**(1977), 223–233. MR**0494171**, 10.1090/S0002-9947-1977-0494171-0**[5]**Shoshichi Kobayashi and Katsumi Nomizu,*Foundations of differential geometry. Vol. II*, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR**0238225****[6]**M. C. Ku,*Newman's theorem for compact Riemannian manifolds*, University of Massachusetts (preprint).**[7]**L. N. Mann and J. L. Sicks,*Newman’s theorem in the Riemannian category*, Trans. Amer. Math. Soc.**210**(1975), 259–266. MR**0423388**, 10.1090/S0002-9947-1975-0423388-4**[8]**M. H. A. Newman,*A theorem on periodic transformations of spaces*, Quart. J. Math.**2**(1931), 1-9.**[9]**P. A. Smith,*Transformations of finite period. III. Newman’s theorem*, Ann. of Math. (2)**42**(1941), 446–458. MR**0004128**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
57E10

Retrieve articles in all journals with MSC: 57E10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0423386-7

Keywords:
Newman's theorem on periodic transformations,
diameter of orbits,
radius of convexity

Article copyright:
© Copyright 1976
American Mathematical Society