When is a submanifold of a Sasakian manifold anti-invariant (Q2754342)

Let \((\varphi,\xi,\eta,g)\) be an almost contact structure on a manifold \(\overline{M}\). If \(\xi\) is a Killing vector field and \((\nabla_X\varphi)(Y)=g(X,Y)\xi-\eta(Y)X\), then \(\overline{M}\) is a Sasakian manifold. Let \(M\) be a submanifold of \(\overline{M}\). For any vector field \(X\) tangent to \(M\) let \(\varphi X=PX+FX\), where \(PX\) and \(FX\) are the projections of \(\varphi X\) on \(TM\) and \(TM^\perp\), respectively. The submanifold \(M\) is said to be anti-invariant, that is \(\varphi(TM)\subset TM^\perp\), if \(P=0\), and \(M\) is invariant if \(F=0\). In this paper, the authors show that if \(M\) is a submanifold of the Sasakian manifold \(\overline M\) tangent to the vector field \(\xi\), then \(M\) is anti-invariant if and only if \(\nabla P=0\). Also, they show that if a non-invariant submanifold \(M\) of the Sasakian manifold \(\overline M\) is tangent to the vector field \(\xi\), then \(\nabla F=0\) implies that \(M\) is anti-invariant.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00008].