Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T16:19:55.749Z Has data issue: false hasContentIssue false
  • English
  • Français

Curvature Bounds for the Spectrum of Closed Einstein Spaces

Published online by Cambridge University Press:  20 November 2018

Udo Simon
Udo Simon*
Affiliation:
Technische Universitdt Berlin, Berlin, West Germany
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The following is our main result.

(A) THEOREM. Let (M, g) be a closed connected Einstein space, n = dim M ≧ 2 (with constant scalar curvature R). Let K0 be the lower bound of the sectional curvature. Then either (M, g) is isometrically diffeomorphic to a sphere and the first nonzero eigenvalue ƛ1of the Laplacian fulfils

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 5 , 01 October 1978 , pp. 1087 - 1091
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Do Carmo, M. P. and Wallach, N. R., Minimal immersions of spheres into spheres, Ann. Math. 2) 93 (1971), 43–62.Google Scholar
2. Hersch, J., Quatre propriétés isopérimétriques de membranes sphèriques homogènes, C. R. Acad. Sci. Paris, Ser. A 270 (1970), 1645–1648.Google Scholar
3. Kobayashi, S. and Nomizu, K., Foundations of differential geometry I, II (Interscience Publishers, New York, London, 1963, (1969).Google Scholar
4. Obata, M., Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333–340.Google Scholar
5. Simon, U., Isometries with spheres, Math. Z. 153 (1977), 23–27.Google Scholar
6. Simon, U., Submanifolds with parallel mean curvature vector and the curvature of minimal submanifolds of spheres, Archiv. Math. 29 (1977), 106–112.Google Scholar
7. Takahashi, T., Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380–385.Google Scholar
8. Tanno, S., On a lower bound of the second eigenvalue of the Laplacian on an Einstein space, to appear, Colloq. Math.Google Scholar
9. Yano, K. and Nagano, T., Einstein spaces admitting a one-parameter group of conformai transformations, Annals of Math. 69 (1959), 451–461.Google Scholar