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

MathSciNet review:
1370639

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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]**Eugenio Calabi,*Minimal immersions of surfaces in Euclidean spheres*, J. Differential Geometry**1**(1967), 111–125. MR**0233294****[DW1]**Manfredo P. do Carmo and Nolan R. Wallach,*Representations of compact groups and minimal immersions into spheres.*, J. Differential Geometry**4**(1970), 91–104. MR**0266104****[DW2]**Manfredo P. do Carmo and Nolan R. Wallach,*Minimal immersions of spheres into spheres*, Ann. of Math. (2)**93**(1971), 43–62. MR**0278318****[DZ]**Dennis DeTurck and Wolfgang Ziller,*Minimal isometric immersions of spherical space forms in spheres*, Comment. Math. Helv.**67**(1992), no. 3, 428–458. MR**1171304**, 10.1007/BF02566512**[E]**Norio Ejiri,*Totally real submanifolds in a 6-sphere*, Proc. Amer. Math. Soc.**83**(1981), no. 4, 759–763. MR**630028**, 10.1090/S0002-9939-1981-0630028-6**[L]**Peter Li,*Minimal immersions of compact irreducible homogeneous Riemannian manifolds*, J. Differential Geom.**16**(1981), no. 1, 105–115. MR**633629****[Ma1]**K. Mashimo,*Minimal immersions of 3-dimensional spheres into spheres*, Osaka J. Math.**1984**, no. 2, 721 - 732. MR**86j:53040****[Ma2]**Katsuya Mashimo,*Homogeneous totally real submanifolds of 𝑆⁶*, Tsukuba J. Math.**9**(1985), no. 1, 185–202. MR**794670****[Mr]**John Douglas Moore,*Isometric immersions of space forms in space forms*, Pacific J. Math.**40**(1972), 157–166. MR**0305312****[M]**Yosio Mutō,*The space 𝑊₂ of isometric minimal immersions of the three-dimensional sphere into spheres*, Tokyo J. Math.**7**(1984), no. 2, 337–358. MR**776943**, 10.3836/tjm/1270151731**[S]**Peter Scott,*The geometries of 3-manifolds*, Bull. London Math. Soc.**15**(1983), no. 5, 401–487. MR**705527**, 10.1112/blms/15.5.401**[T]**Tsunero Takahashi,*Minimal immersions of Riemannian manifolds*, J. Math. Soc. Japan**18**(1966), 380–385. MR**0198393****[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. Ja. Vilenkin,*Special functions and the theory of group representations*, Translated from the Russian by V. N. Singh. Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R. I., 1968. MR**0229863****[W]**Joseph A. Wolf,*Spaces of constant curvature*, McGraw-Hill Book Co., New York-London-Sydney, 1967. MR**0217740****[Wa]**Nolan R. Wallach,*Extension of locally defined minimal immersions into spheres*, Arch. Math. (Basel)**21**(1970), 210–213. MR**0271878****[WZ]**McKenzie Wang and Wolfgang Ziller,*On isotropy irreducible Riemannian manifolds*, Acta Math.**166**(1991), no. 3-4, 223–261. MR**1097024**, 10.1007/BF02398887

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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