EMIS ELibM Electronic Journals Publications de l'Institut Mathématique, Nouvelle Série
Vol. 88(102), pp. 53–65 (2010)

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Ryszard Deszcz, Miroslava Petrovic-Torgasev, Zerrin Sentürk, and Leopold Verstraelen

Department of Mathematics, Wroclaw University of Environmental and Life Sciences, Wroclaw, Poland} and Department of Mathematics, Faculty of Scence, Kragujevac, Serbia} and Mathematics Engineering Department, Faculty of Science and Letters, Istanbul Technical University, Istanbul, Turkey and Departement Wiskunde, Katholieke Universiteit Leuven, Fakulteit Wetenschappen, Heverlee, Belgium

Abstract: Recently, Choi and Lu proved that the Wintgen inequality $\rho\leq H^2-\rho^\bot+k$, (where $\rho$ is the normalized scalar curvature and $H^2$, respectively $\rho^\bot$, are the squared mean curvature and the normalized scalar normal curvature) holds on any $3$-dimensional submanifold $M^3$ with arbitrary codimension $m$ in any real space form $\widetilde M^{3+m}(k)$ of curvature $k$. For a given Riemannian manifold $M^3$, this inequality can be interpreted as follows: for all possible isometric immersions of $M^3$ in space forms $\widetilde M^{3+m}(k)$, the value of the intrinsic curvature $\rho$ of $M$ puts a lower bound to all possible values of the extrinsic curvature $H^2-\rho^\bot+k$ that $M$ in any case can not avoid to "undergo" as a submanifold of $\tilde M$. From this point of view, $M$ is called a Wintgen ideal submanifold of $\widetilde M$ when this extrinsic curvature $H^2-\rho^\bot+k$ actually assumes its theoretically smallest possible value, as given by its intrinsic curvature $\rho$, at all points of $M$. We show that the pseudo-symmetry or, equivalently, the property to be quasi-Einstein of such $3$-dimensional Wintgen ideal submanifolds $M^3$ of $\widetilde M^{3+m}(k)$ can be characterized in terms of the intrinsic minimal values of the Ricci curvatures and of the Riemannian sectional curvatures of $M$ and of the extrinsic notions of the umbilicity, the minimality and the pseudo-umbilicity of $M$ in $\widetilde M$.

Keywords: submanifold, Wintgen inequality, pseudo-symmetric manifold, quasi-Einstein space

Classification (MSC2000): 53B25, 53B35, 53A10, 53C42

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Electronic fulltext finalized on: 19 Nov 2010. This page was last modified: 6 Dec 2010.

© 2010 Mathematical Institute of the Serbian Academy of Science and Arts
© 2010 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition