Hostname: page-component-cb9f654ff-c75p9 Total loading time: 0 Render date: 2025-09-08T19:47:12.236Z Has data issue: false hasContentIssue false
  • English
  • Français

WARPED PRODUCTS IN RIEMANNIAN MANIFOLDS

Part of: Global differential geometry

Published online by Cambridge University Press:  09 September 2014

KWANG-SOON PARK
KWANG-SOON PARK*
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea email parkksn@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

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.

In this paper we prove two inequalities relating the warping function to various curvature terms, for warped products isometrically immersed in Riemannian manifolds. This extends work by B. Y. Chen [‘On isometric minimal immersions from warped products into real space forms’, Proc. Edinb. Math. Soc. (2) 45(3) (2002), 579–587 and ‘Warped products in real space forms’, Rocky Mountain J. Math.34(2) (2004), 551–563] for the case of immersions into space forms. Finally, we give an application where the target manifold is the Clifford torus.

Keywords

warped productsquared mean curvaturewarping functionsecond fundamental form

MSC classification

Secondary: 53C40: Global submanifolds 53C42: Immersions (minimal, prescribed curvature, tight, etc.) 53C50: Lorentz manifolds, manifolds with indefinite metrics

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 3 , December 2014 , pp. 510 - 520
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Chen, B. Y., ‘Some pinching and classification theorems for minimal submanifolds’, Arch. Math. 60(6) (1993), 568–578.Google Scholar
Chen, B. Y., ‘On isometric minimal immersions from warped products into real space forms’, Proc. Edinb. Math. Soc. (2) 45(3) (2002), 579–587.CrossRefGoogle Scholar
Chen, B. Y., ‘Warped products in real space forms’, Rocky Mountain J. Math. 34(2) (2004), 551–563.Google Scholar
Chen, B. Y., ‘A survey on geometry of warped product submanifolds’, J. Adv. Math. Stud. 6(2) (2013), 1–43.Google Scholar
Cheng, Q. M. and Ishikawa, S., ‘A characterization of the Clifford torus’, Proc. Amer. Math. Soc. 127 (1999), 819–828.Google Scholar
Chern, S. S., do Carmo, M. and Kobayashi, S., ‘Minimal submanifolds of a sphere with second fundamental form of constant length’, in: Functional Analysis and Related Fields (ed. Browder, F.) (Springer, Berlin, 1970), 59–75.Google Scholar
Ejiri, N., ‘Some compact hypersurfaces of constant scalar curvature in a sphere’, J. Geom. 19 (1982), 197–199.CrossRefGoogle Scholar
Lawson, H. B. Jr, ‘Local rigidity theorems for minimal hypersurfaces’, Ann. of Math. (2) 89 (1969), 167–179.CrossRefGoogle Scholar
Nash, J. F., ‘The imbedding problem for Riemannian manifolds’, Ann. of Math. (2) 63 (1956), 20–63.Google Scholar
Suh, Y. J. and Yang, H. Y., ‘The scalar curvature of minimal hypersurfaces in a unit sphere’, Commun. Contemp. Math. 9(2) (2007), 183–200.CrossRefGoogle Scholar
Zhang, Y., ‘Rigidity theorems of Clifford torus’, Acta Math. Sci. 30B(3) (2010), 890–896.Google Scholar