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Submanifolds of codimension two attaining equality in an extrinsic inequality

Published online by Cambridge University Press:  01 March 2009

MARCOS DAJCZER  and
RUY TOJEIRO
MARCOS DAJCZER
Affiliation:
IMPA—Estrada Dona Castorina, 110 22460-320 Rio de Janeiro, Brazil. e-mail: marcos@impa.br
RUY TOJEIRO
Affiliation:
Universidade Federal de São Carlos, 13565-905 São Carlos, Brazil. e-mail: tojeiro@dm.ufscar.br
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Abstract

We provide a parametric construction in terms of minimal surfaces of the Euclidean submanifolds of codimension two and arbitrary dimension that attain equality in an inequality due to De Smet, Dillen, Verstraelen and Vrancken. The latter involves the scalar curvature, the norm of the normal curvature tensor and the length of the mean curvature vector.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 2 , March 2009 , pp. 461 - 474
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Bryant, R.Some remarks on the geometry of austere manifolds. Bol. Soc. Bras. Mat. 21 (1991), 122–157.CrossRefGoogle Scholar
[2]Chen, B.Y.Mean curvature and shape operators of isometric immersions into real-space-forms. Glasgow Math. J. 38 (1996), 87–97.CrossRefGoogle Scholar
[3]Choi, T. and Lu, Z. On the DDVV conjecture and the comass in calibrated geometry. Preprint (available at arXiv:math.DG/0610709).Google Scholar
[4]Dajczer, M. and Florit, L.On Chen's basic equality. Illinois J. Math. 42 (1) (1998), 97–106.CrossRefGoogle Scholar
[5]Dajczer, M. and Florit, L.A class of austere submanifolds. Illinois J. Math. 45 (2001), 735–755.CrossRefGoogle Scholar
[6]Dajczer, M. and Tojeiro, R. All superconformal surfaces in in terms of minimal surfaces. Preprint (available at arXiv:math.DG/0710.5317).Google Scholar
[7]Dajczer, M. and Tojeiro, R.Submanifolds with nonparallel first normal bundle. Canad. Math. Bull. 37 (1994), 330–337.CrossRefGoogle Scholar
[8]Dillen, F., Fastenakels, J. and Van der Veken, J. Remarks on an inequality involving the normal scalar curvature. Preprint (available at arXiv:math.DG/0610721).Google Scholar
[9]Ge, J. and Tang, Z. A proof of the DDVV conjecture and its equality case. Preprint (available at arXiv:math.DG/0801.0650).Google Scholar
[10]Guadalupe, I. and Rodríguez, L.Normal curvature of surfaces in space forms. Pacific J. Math. 106 (1983), 95–103.CrossRefGoogle Scholar
[11]Hoffman, D. and Osserman, R.The geometry of the generalized Gauss map. Mem. Amer. Math. Soc. 28 (1980), no. 236.Google Scholar
[12]Lawson, B. Jr.Lectures on Minimal Submanifolds. Mathematics Lecture Series, vol. 9 (Publish or Perish Inc., 1980).Google Scholar
[13]Lu, Z. On the DDVV conjecture and the comass in calibrated geometry (II). Preprint (available at arXiv:math.DG/0708.2921).Google Scholar
[14]Lu, Z. Proof of the normal scalar curvature conjecture. Preprint (available at arXiv:math.DG/0711.3510).Google Scholar
[15]de Smet, P. J., Dillen, F., Verstraelen, L. and Vrancken, L.A pointwise inequality in submanifold theory. Arch. Math. (Brno) 35 (1999), 115–128.Google Scholar