On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms
type: Item
instance of: scholarly article
title: On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms
zbMATH Open document ID: 1037.53041
DOI: 10.1007/PL00000420
published in: Archiv der Mathematik
publication date: 2000-01-01T00:00:00Z
review text: Let \((M^{n},g)\) be a Riemannian manifold isometrically immersed in \((\tilde{M}^{m},g).\) Let us denote by \(S,H^{2}\) the Ricci tensor and the squared mean curvature, respectively. In a previous paper [\textit{B.-Y. Chen}, Glasg. Math. J. 41, 33--41 (1999; Zbl 0962.53015)], the author proved the following basic inequality for any submanifold \((M^{n},g)\) of a real space form \(R^{m}(c):\) \(S\leq \left( \left( n-1)c\right) +\frac{n^{2}}{4}H^{2}\right) ,\) with equality holding, if and only if, either \((M^{n},g)\) is totally geodesic or \(n=2\) and \((M^{2},g)\) is a totally umbilical surface. Continuing a series of important results concerning basic inequalities for Riemannian submanifolds, the author first proves in this paper, a similar result to the above one for totally real submanifolds \((M^{n},g)\) of complex space forms \((\tilde{M}^{m}(4c),g).\) Then, he obtains that a Lagrangian submanifold \((M^{n},g)\) satisfying the equality \(\overline{\text{Ric}}=\left(\left( n-1)c\right) +\frac{n^{2}}{4}H^{2}\right) \) ( where \(\overline{\text{Ric}}\) is the maximum Ricci curvature function) must be a minimal submanifold. As a consequence of the proof, he shows that these submanifolds must have a relative null space of dimension one at least. The final part is devoted to the geometric description of the Lagrangian submanifolds satisfying the above equality and such that the dimension of the relative null space is constant.
reviewed by: Oscar J. Garay
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53D12
zbMATH DE Number: 1437102
zbMATH Keywords: Ricci curvature
zbMATH Keywords: isotropic submanifolds
zbMATH Keywords: Lagrangian submanifolds
zbMATH Keywords: minimal submanifolds
author: Bang-Yen Chen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1437102