Hostname: page-component-784d4fb959-2gr9v Total loading time: 0 Render date: 2025-07-14T18:34:02.231Z Has data issue: false hasContentIssue false
  • English
  • Français

Spherical submanifolds which are of 2-type via the second standard immersion of the sphere

Published online by Cambridge University Press:  22 January 2016

Manuel Barros  and
Bang-Yen Chen
Manuel Barros
Affiliation:
Departamento de Geometria y Topologia University de Granada, 18071—Granada, Spain
Bang-Yen Chen
Affiliation:
Department of Mathematics Michigan State University, East Lansing, Michigan 48824, USA
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.

Let Sm(r) be an m-sphere of constant sectional curvature 1/r2 and M an n-dimensional compact minimal submanifold of Sm(r). If Sm(r) is imbedded in Em+1 by its first standard imbedding, then, by a well-known result of Takahashi [11], the Euclidean coordinate functions restricted to M are eigenfunctions of Δ on M with the same eigenvalue n/r2. Moreover, the center of mass of M in Em+1 coincides with the center of the hypersphere Sm(r) in Em+1. Thus, M is mass-symmetric in Sm(r) ⊂ Em+\ Consequently, we see that if one wants to study the spectral geometry of a submanifold of Sm(r), it is natural to immerse Sm(r) by its k-th standard immersion, in particular, by its second standard immersion.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 108 , December 1987 , pp. 77 - 91
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1987

References

[ 1 ] Barros, M. and Chen, B. Y., Classification of stationary 2-type surfaces of hyper-spheres, C.R. Math. Rep. Acad. Sci. Canada, 7 (1985), 309–314.Google Scholar
[ 2 ] Barros, M. and Chen, B. Y., Finite type spherical submanifolds, Proc. II Intern. Symp. Diff. Geom., Lecture Notes in Math., Springer-Verlag, 1209 (1986), 73–93.Google Scholar
[ 3 ] Barros, M. and Ros, A., Spectral geometry of submanifolds, Note Mat., 4 (1984), 1–56.Google Scholar
[ 4 ] Chen, B. Y., Geometry of submanifolds, M. Dekker, 1973.Google Scholar
[ 5 ] Chen, B. Y., On total curvature of immersed manifolds, IV, Bull. Math. Acad. Sinica, 7 (1979), 301–311;, VI, ibid, 11 (1983), 309–328.Google Scholar
[ 6 ] Chen, B. Y., Total mean curvature and submanifolds of finite type, World Scientific, 1984.CrossRefGoogle Scholar
[ 7 ] Chen, B. Y., 2-type submanifolds and their applications, Chinese J. Math., 14 (1986),1–14.Google Scholar
[ 8 ] Itoh, T. and Ogiue, K., Isotropic immersions, J. Differential Geom., 8 (1973), 305–316.CrossRefGoogle Scholar
[ 9 ] Ros, A., Eigenvalue inequalities for minimal submanifolds and P-manifolds, Math. Z., 187 (1984), 393–404.CrossRefGoogle Scholar
[10] Sakamoto, K., Planar geodesic immersions, Tôhoku Math. J., 29 (1977), 25–56.CrossRefGoogle Scholar
[11] Takahashi, T., Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380–385.CrossRefGoogle Scholar