Structure on a slant submanifold of a contact manifold
type: Item
instance of: scholarly article
title: Structure on a slant submanifold of a contact manifold
zbMATH Open document ID: 0984.53034
author: R. Smith
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2002-02-27T00:00:00Z
review text: An almost contact metric structure on a \(C^\infty\) manifold \(\widetilde M\) consists of a tensor field \(\varphi\) of type (1,1), a vector field \(\xi\), a 1-form \(\eta\) and a Riemannian metric \(g\) satisfying \(\varphi^2=-I+\eta\otimes\xi\), \(\eta(\xi)=1\) and \(g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\eta(Y)\). The fundamental 2-form of such a structure is defined by \(\Phi(X,Y)=g(X,\varphi Y)\). The structure is a contact metric structure if \(\Phi=d\eta\) and \(\xi\) is then called the characteristic or Reeb vector field. A contact metric structure is a Sasakian structure if \((\widetilde\nabla X\varphi)Y=g(X,Y)\xi-\eta(Y)X\) where \(\widetilde\nabla\) is the Levi-Civita connection of \(g\). Let \(M\) be an immersed submanifold of \(\widetilde M\) and suppose that \(\xi\) is tangent to \(M\). For each non-zero vector \(X\) tangent to \(M\) at \(x\), such that \(X\) is not proportional to \(\xi_x\), denote by \(\theta(X)\) the Wirtinger angle of \(X\), i.e., the angle between \(\varphi X\) and \(T_xM\). The submanifold \(M\) is then said to be a slant submanifold if \(\theta(X)\) is a constant independent of the choice of \(x\in M\) and \(X\in T_xM-\langle \xi_x\rangle\). Invariant and anti-invariant submanifolds are characterized by \(\theta=0,{\pi\over 2}\) respectively. If \(TX\) denotes the tangential component of \(\varphi X\), then a submanifold of an almost contact metric manifold with \(\xi\) tangential is slant if and only if there exists a constant \(\lambda\in[0,1]\) such that \(T^2=-\lambda I+\lambda\eta\otimes\xi\) in which case the slant angle \(\theta\) satisfies \(\lambda=\cos^2\theta\) [see the authors' paper, Glasg. Math. J. 42, No. 1, 125-138 (2000; Zbl 0957.53022)]. It is well known that if \(M\) is an invariant submanifold of a Sasakian manifold with \(\xi\) tangent to \(M\), then \(M\) inherits a Sasakian structure. The main result of the present paper is that there is no non-invariant slant immersion of a 3-dimensional contact metric ma…
Mathematics Subject Classification ID: 53D15
Mathematics Subject Classification ID: 53B25
zbMATH DE Number: 1526413
zbMATH Keywords: almost contact metric manifold
zbMATH Keywords: contact metric manifold
zbMATH Keywords: slant submanifold
zbMATH Keywords: Sasakian structure
zbMATH Keywords: Wirtinger angle
zbMATH Keywords: characteristic vector fields
reviewed by: David E. Blair
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1526413