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

CHARACTERIZATION OF THE PSEUDOSYMMETRIES OF IDEAL WINTGEN SUBMANIFOLDS OF DIMENSION 3Ryszard Deszcz, Miroslava PetrovicTorgasev, Zerrin Sentürk, and Leopold VerstraelenDepartment 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, BelgiumAbstract: 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 pseudosymmetry or, equivalently, the property to be quasiEinstein 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 pseudoumbilicity of $M$ in $\widetilde M$. Keywords: submanifold, Wintgen inequality, pseudosymmetric manifold, quasiEinstein space Classification (MSC2000): 53B25, 53B35, 53A10, 53C42 Full text of the article: (for faster download, first choose a mirror)
Electronic fulltext finalized on: 19 Nov 2010. This page was last modified: 6 Dec 2010.
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