Structure on a slant submanifold of a contact manifold
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 manifold into another contact metric manifold preserving the characteristic vector fields. The authors then study the case where the ambient space is only an almost contact metric manifold. Suppose that one has an isometric immersion \(\varphi\) of an almost contact metric manifold \((M,\overline\varphi,\overline\xi,\overline\eta,\overline g)\) into an almost contact metric manifold \((\widetilde M,\varphi,\xi,\eta,g)\) such that \(\varphi_*\overline\xi=\xi\). Now if there exists a constant \(k\in[-1,1]\) such that \(\varphi^*\Phi=k\overline\Phi\), then \(\varphi\) is slant with slant angle \(\cos^{-1}|k|\). When \(M\) is 3-dimensional, \(\varphi\) is slant if and only if there exists a constant \(k\in[-1,1]\) such that \(\varphi^*\Phi=k\overline\Phi\).
scientific article; zbMATH DE number 1526413