Submanifolds of codimension 3 in a complex space form with commuting structure Jacobi operator. (English) Zbl 1492.53023
Summary: Let \(M\) be a semi-invariant submanifold with almost contact metric structure \((\phi, \xi, \eta, g)\) of codimension 3 in a complex space form \(M_{n + 1} (c)\) for \(c \neq 0\). We denote by \(S\) and \(R_\xi\) the Ricci tensor of \(M\) and the structure Jacobi operator in the direction of the structure vector \(\xi \), respectively. Suppose that the third fundamental form \(t\) satisfies \(d t (X, Y) = 2 \theta g (\phi X, Y)\) for a certain scalar \(\theta \neq 2 c\) and any vector fields \(X\) and \(Y\) on \(M\). In this paper, we prove that if it satisfies \(R_\xi \phi = \phi R_\xi\) and at the same time \(S \xi = g (S \xi, \xi) \xi \), then \(M\) is a real hypersurface in \(M_n (c) (\subset M_{n + 1} (c))\) provided that \(\overline{r} - 2 (n - 1) c \leq 0\), where \(\overline{r}\) denotes the scalar curvature of \(M\).
MSC:
| 53B25 | Local submanifolds |
| 53C40 | Global submanifolds |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
Keywords:
semi-invariant submanifold; distinguished normal; complex space form; structure Jacobi operator; Ricci tensor; Hopf hypersurfacesReferences:
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