On geometry of QS-hypersurfaces of Kählerian manifolds
If \((M,g)\) is a Riemannian manifold and \(J\) is an almost complex structure satisfying \(g(JX,JY)=g(X,Y)\), then \((M,g,J)\) is an almost Hermitian manifold. An almost Hermitian manifold \((M,g,J)\) is called an almost Kähler manifold if its Kähler form \(\Omega\) given by \(\Omega(X,Y)=g(JX,Y)\) is closed. An almost Hermitian manifold \(M\) is a Kähler manifold if \(J\) is integrable and \(\Omega\) is closed, or equivalently if \(\nabla J=0\) with respect to the Levi-Civita connection \(\nabla\) of \(g\). If \((\varphi,\xi,\eta)\) is a structure, where \(\varphi\) is an endomorphism, \(\xi\) is a vector field, and \(\eta\) is a \(1\)-form, satisfying \(\varphi^2=-\text{id}+\eta\otimes\xi\), \(\varphi(\xi)=0\), \(g(X,\xi)=\eta(X)\), and \(g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\eta(Y)\) on a Riemannian manifold \((M,g)\), then \((M,g,\varphi,\xi,\eta)\) is called an almost contact metric manifold. If the fundamental \(\Omega\) defined as \(\Omega(X,Y)=g(\varphi X,Y)\) of an almost contact metric manifold \(M\) is closed and \(M\) is normal, that is, \(N_\varphi+\frac12d\eta\otimes\xi=0\), where \(N_\varphi(X,Y)=\varphi^2[X,Y]+[\varphi X,\varphi Y]-\varphi[\varphi X,Y]-\varphi[X,\varphi Y]\) is the Nijenhuis tensor, then \(M\) is called a quasi-Sasakian manifold. If \(N\) is a submanifold of a Riemannian manifold \(M\), then the rank of its second fundamental form \(\sigma\) is called the type number of \(N\). An almost Kähler manifold \((M,g)\) induces an almost contact metric structure \(\Sigma=(\varphi,\xi,\eta,g)\) on a \((n-1)\)-dimensional hypersurface \(N\). In this paper the authors study properties of quasi-Sasakian hypersurfaces of Kählerian manifolds. If \(N\) is a hyperurface with the induced quasi-Sasakian structure \((\varphi,\xi,\eta,g)\) of the Kähler manifold \(M\) and \(\sigma\) is its second fundamental form, then the authors find the explicit form of the matrix of \(\sigma\) and the Cartan structure equations of \(N\). Also, they show that if \(N\) is minimal, then \(\sigma(\xi,\xi)=0\), \(N\) is a ruled hypersurface, and its type number is even.
scientific article; zbMATH DE number 6931838