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Semi-invariant submanifold in a nearly Sasakian manifold. (Chinese. English summary) Zbl 1052.53037

Suppose \(\overline{M}\) is an almost contact manifold of dimensions \(2n+1\) with tensor structure \((\overline{\Phi}, \overline{\zeta}, \overline{\eta}, \overline{g})\), and \(\overline{\nabla}\) is its Levi-CivitĂ  connection on \(\overline{M}\). If the condition \[ (\overline{\nabla}_X\overline{\Phi})(Y)+ (\overline{\nabla}_Y\overline{\Phi})(X) =-\overline{\eta}(Y)X-\overline{\eta}(X)Y+2\overline{g}(X,Y) \overline{\zeta} \] holds for all vector fields \(X\), \(Y\) on \(\overline{M}\), then we call \(\overline{M}\) nearly Sasakian. Let \(M\) be a submanifold of dimensions \(2m+1\) in \(\overline{M}\), and suppose that \(\zeta\) is a distribution of dimension 1 such that
(1) \(TM=D\oplus D^{\bot}\oplus\{\zeta\}\);
(2) \(\overline {\Phi}(D_x)=D_x\), \(\forall x\in M\);
(3) \(\overline{\Phi}(D_x^{\bot})\subset TM^{\bot}\), then we call \(M\) a semi-invariant submanifold in \(\overline{M}\).
The main result of this paper is theorem 3: Suppose \(M\) is a semi-invariant submanifold in a nearly Sasakian manifold \(\overline{M}\), then \(M\) is trivial, i.e. \(D=0\), \(D^{\bot}= 0\).
The proof is direct in terms of the structure of \(M\) and \(\overline{M}\). There exists some ambiguity in this paper. For example, is there some relation between \(\zeta\) and \(\overline{\zeta}\)? The assumption in the theorems should be “invariant” or “semi-invariant”? In my opinion, this paper is not so well written.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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