Totally real submanifolds of \(\mathbb{C} P^ n\) satisfying a basic equality
type: Item
instance of: scholarly article
title: Totally real submanifolds of \(\mathbb{C} P^ n\) satisfying a basic equality
zbMATH Open document ID: 0816.53034
DOI: 10.1007/BF01202073
author: Bang-Yen Chen
author: Franki Dillen
author: Leopold Verstraelen
author: Luc Vrancken
published in: Archiv der Mathematik
publication date: 1995-04-03T00:00:00Z
review text: Let \(M^ n\) be an \(n\)-dimensional totally real submanifold of an \(m\)- dimensional complex space form \(\widetilde M^ m(4c)\) with constant holomorphic sectional curvature \(4c\). We define a Riemannian invariant \(\delta_ m\) of \(M^ n\) by \(\delta_ M(p) = \tau(p) - \inf K(p)\), where \(\inf K\) is the function assigning to each \(p \in M^ n\) the infimum of \(K(\pi)\), where \(\pi\) runs over all planes in \(T_ pM\) and \(\tau\) is the scalar curvature, defined by \(\tau = \sum_{i < j} K(e_ i \wedge e_ j)\). Then the following inequality holds: \[ \delta_ M \leq {n^ 2(n-2) \over 2(n-1)} \| H\|^ 2 + {1\over 2}(n + 1)(n + 2)c \] where \(H\) is the mean curvature vector. In this paper we study totally real submanifolds which realize the equality in this inequality. Then we say that \(M\) satisfies Chen's equality. First we prove that, if \(M\) satisfies Chen's equality, then \(H(p)\) is orthogonal to \(J(T_ p M)\) at any point \(p\), where \(J\) denotes the complex structure of the complex space form. Hence any Lagrangian submanifold (\(n = m\)) which satisfies Chen's equality automatically is minimal. Next we study \(n\)-dimensional totally real submanifolds in \(\mathbb{C} P^ n(4)\) satisfying Chen's equality. We define for every \(p \in M\) the kernel of the second fundamental form by \[ {\mathcal D}(p) = \{X \in T_ p M \mid \forall Y \in T_ p M: h(X,Y) = 0\}. \] If the dimension of \({\mathcal D}(p)\) is constant, then it follows that either \(M\) is totally geodesic or that the distribution \(\mathcal D\) is an \((n-2)\)-dimensional completely integrable distribution. We give a complete classification of all 3- dimensional totally real submanifolds of \(\mathbb{C} P^ 3\) satisfying the dimension of \(\mathcal D\) is constant (and hence it is a completely integrable distribution), the distribution \({\mathcal D}^ \perp\) is also integrable. For the classification, we make use of the notion of warped product immersions, as introduced by S. Noelker.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 734498
zbMATH Keywords: totally real submanifolds
zbMATH Keywords: Chen's equality
zbMATH Keywords: Lagrangian submanifold
reviewed by: Franki Dillen
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01202073
OpenAlex ID: W2911743342
description: scientific article; zbMATH DE number 734498