Minimal isometric immersions of inhomogeneous spherical space forms into spheres---

a necessary condition for existence

Author:
Christine M. Escher

Journal:
Trans. Amer. Math. Soc. **348** (1996), 3713-3732

MSC (1991):
Primary 53C42

DOI:
https://doi.org/10.1090/S0002-9947-96-01694-7

MathSciNet review:
1370639

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Although much is known about minimal isometric immersions into spheres of *homogeneous* spherical space forms, there are no results in the literature about such immersions in the dominant case of inhomogeneous space forms. For a large class of these, we give a necessary condition for the existence of such an immersion of a given degree. This condition depends only upon the degree and the fundamental group of the space form and is given in terms of an explicitly computable function. Evaluating this function shows that neither nor admit a minimal isometric immersion into any sphere if the degree of the immersion is less than , or less than , respectively.

**[C]**E. Calabi,*Minimal immersions of surfaces in euclidean spheres*, J. Diff. Geom.**1967**(1), 111-125. MR**38:1616****[DW1]**M. doCarmo and N.Wallach,*Representations of compact groups and minimal immersions into spheres*, J. Diff. Geom.**1970**, no. 4, 91 - 104. MR**42:1013****[DW2]**------,*Minimal immersions of spheres into spheres*, Annals of Math.**1971**, no. 93, 43 - 62. MR**43:4048****[DZ]**D. DeTurck and W.Ziller,*Minimal isometric immersions of spherical space forms into spheres*, Comm. Math. Helv.**1992**, no. 67, 428-458. MR**93f:53050****[E]**N.Ejiri,*Totally real submanifolds in a 6-sphere*, Proc. Amer. Math. Soc.**1981**, no. 83, 759-763. MR**83a:53033****[L]**P. Li,*Minimal immersions of compact irreducible homogeneous Riemannian manifolds*, J. Diff. Geom.**1984**, no. 16, 337- 358. MR**83a:53057****[Ma1]**K. Mashimo,*Minimal immersions of 3-dimensional spheres into spheres*, Osaka J. Math.**1984**, no. 2, 721 - 732. MR**86j:53040****[Ma2]**------,*Homogeneous totally real submanifolds in*, Tsukuba J. Math.**1985**, no. 9, 185 - 202. MR**86j:53083****[Mr]**J.D. Moore,*Isometric immersions of space forms into space forms*, Pacific J. Math.**1976**, no. 40, 157 - 166. MR**46:4442****[M]**Y. Muto,*The space of isometric minimal immersions of the three-dimensional sphere into spheres*, Tokyo J. Math.**1981**, no. 7, 105- 115. MR**86e:53042****[S]**P. Scott,*The geometries of 3-manifolds*, Bull. London. Math. Soc.**1983**, no. 5, 401- 487. MR**84m:57009****[T]**T. Takahashi,*Minimal immersions of Riemannian manifolds*, J. Math. Soc. Japan**1966**, no. 18, 380- 385. MR**33:6551****[To]**G. Toth,*Eigenmaps and the space of minimal immersions between spheres*, Indiana Univ. Math. J.**1994**, no. 4.**[TS]**W. Threlfall und H. Seifert,*Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes*, Math. Annalen**1930**, no. 104, 1-70.**[V]**N. J. Vilenkin,*Special Functions and the Theory of Group Representations*, American Mathematical Society, 1968. MR**37:5429****[W]**J. Wolf,*Spaces of Constant Curvature*, McGraw-Hill, 1967. MR**36:829****[Wa]**N. Wallach,*Extension of locally defined minimal immersions into spheres*, Arch. Math.**1970**, no. 21, 210-213. MR**42:6759****[WZ]**M. Wang and W. Ziller,*On isotropy irreducible Riemannian manifolds*, Acta Math.**1991**, no. 166, 223-261. MR**92b:53078**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
53C42

Retrieve articles in all journals with MSC (1991): 53C42

Additional Information

**Christine M. Escher**

Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 97331

Email:
tine@math.orst.edu

DOI:
https://doi.org/10.1090/S0002-9947-96-01694-7

Keywords:
Minimal isometric immersions,
inhomogeneous spherical space forms

Received by editor(s):
August 22, 1995

Article copyright:
© Copyright 1996
American Mathematical Society