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A Riemannian invariant and its applications to Einstein manifolds

Published online by Cambridge University Press:  17 April 2009

Bang-Yen Chen
Bang-Yen Chen
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824–1027, United States of America, e-mail: [email protected]
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We introduce a Riemannian invariant and establish general optimal inequalities involving the invariants and the squared mean curvature for Einstein manifolds isometrically immersed in real space forms. We show that these inequalities do not hold for arbitrary submanifolds in real space forms in general. We also provide some immediate applications of the inequalities.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 1 , August 2004 , pp. 55 - 65
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Chen, B.Y., Geometry of submanifolds (M. Dekker, New York, 1973).Google Scholar
[2]Chen, B.Y., ‘Some new obstructions to minimal and Lagrangian isometric immersions’, Japan. J. Math. 26 (2000), 105–127.Google Scholar
[3]Chen, B.Y., ‘On isometric minimal immersions from warped products into real space forms’, Proc. Edinburgh Math. Soc. 45 (2002), 579–587.CrossRefGoogle Scholar
[4]Chen, B.Y., ‘Non-immersion theorems for warped products in complex hyperbolic spaces’, Proc. Japan Acad. Ser. A Math. Sci. 79 (2002), 96–100.Google Scholar
[5]Chen, B.Y., ‘A general optimal inequality for warped products in complex projective spaces and its applications’, Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), 89–94.CrossRefGoogle Scholar
[6]Chen, B.Y., ‘A general inequality for conformally flat submanifolds and its applications’ (to appear).Google Scholar
[7]Gromov, M., ‘Isometric immersions of Riemannian manifolds, Elie Cartan et les Mathématiques d'Aujourd'hui’, Astérisque (1985), 129–133.Google Scholar
[8]Nagano, T., ‘On the minimum eigenvalues of the Laplacians in Riemannian manifolds’, Sci. Papers Coll. Gen. Edu. Univ. Tokyo 11 (1961), 177–182.Google Scholar
[9]Nash, J.F., ‘The imbedding problem for Riemannian manifolds’, Ann. of Math. 63 (1956), 20–63.Google Scholar
[10]Yau, S.T., ‘Mathematical research today and tomorrow’, in Viewpoints of Seven Fields Medalists (Berlin, Heidelberg, New York, 1991), pp. 30–39.Google Scholar