Pinching theorems for C-totally real submanifolds of Sasakian space forms
type: Item
instance of: scholarly article
title: Pinching theorems for C-totally real submanifolds of Sasakian space forms
zbMATH Open document ID: 0654.53061
DOI: 10.1007/BF01230616
author: Leopold Verstraelen
author: Luc Vrancken
published in: Journal of Geometry
publication date: 1988-01-01T00:00:00Z
review text: A C-totally real submanifold of a Sasakian manifold \(\tilde M^{2n+1}\) is an integral submanifold of the contact subbundle of \(\tilde M^{2n+1}\); such submanifolds have dimension at most \(n\). A Sasakian space form, \(\tilde M^{2n+1}(c)\), is a Sasakian manifold of constant \(\phi\)-sectional curvature, \(c\); this is analogous to the notion of a Kähler manifold of constant holomorphic curvature. For a reference to Sasakian manifolds and these ideas, see the paper or the reviewer's monograph [Contact manifolds in Riemannian geometry. Lect. Notes Math. 509. Berlin etc.: Springer-Verlag (1976; Zbl 0319.53026)] The main results of the paper are the following: If M is an n-dimensional compact, minimal, C-totally real submanifold of \(\tilde M^{2n+1}(c)\) and if all sectional curvatures \(K\) of \(M\) satisfy \(K\geq 0\), then M has constant scalar curvature. Moreover if \(n=2\), then the Gaussian curvature \(K\) of \(M\) is either 0 or \((c+3)/4\) and an example is given. These results are stronger than previously known ones due to the method of proof. The method is to use the integral formulas of \textit{A. Ros} [Ann. Math. (2) 121, 377--382 (1983; Zbl 0589.53065)] on the unit tangent bundle of a compact Riemannian manifold.
reviewed by: David E. Blair
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 4067812
zbMATH Keywords: minimal submanifold
zbMATH Keywords: C-totally real submanifold
zbMATH Keywords: Sasakian space form
zbMATH Keywords: sectional curvatures
zbMATH Keywords: scalar curvature
zbMATH Keywords: integral formulas
MaRDI profile type: Publication
cites work: Contact manifolds in Riemannian geometry
cites work: On Totally Real 3-Dimensional Submanifolds of the Nearly Kaehler 6-Sphere
cites work: A characterization of seven compact Kähler submanifolds by holomorphic pinching
cites work: CR submanifolds of Kaehlerian and Sasakian manifolds
cites work: Q3222791
cites work: Minimal submanifolds in Sasakian space forms
full work available at URL: https://doi.org/10.1007/bf01230616
OpenAlex ID: W2006490397
description: scientific article; zbMATH DE number 4067812