Jacobi operators on a semi-invariant submanifold of codimension 3 in a complex projective space
Let \(M\) be a connected real hypersurface of \(\mathbb C\mathbb P^n\) with the sharp operator \(A\) and structure tensor field \(\phi.\) \textit{M. Okumura} [Trans. Am. Math. Soc. 212, 355--364 (1975; Zbl 0288.53043)] proved that, if \(A\phi=\phi A\), then \(M\) is locally congruent either to (\(A_1\))\ a geodesic hypersphere (that is, a tuble of radius \(r\) over a hyperplane \(\mathbb C\mathbb P^{n-1},\) where \(0<r<\frac\pi 2)\); or to (\(A_2\))\ a tube of radius \(r\) over a totally geodesic \(\mathbb C\mathbb P^k (1\leq k\leq n-2),\) where \(0<r<\frac\pi 2\). \textit{J. T. Cho} and \textit{U.-H. Ki} [Acta Math. Hung. 80, 155--167 (1998; Zbl 0910.53038)] gave another characterization of a connected real hypersurface of \(\mathbb C\mathbb P^n\) of type \(A_1\) and \(A_2\) by the \[ R_\xi\phi A=A\phi R_\xi, \] where \(R_\xi\) is the Jacobi operator. In this paper, the authors extend the result of Cho and Ki for a semi-invariant submanifold of codimension 3 in a complex projective space. Here is the main result: Theorem. Let \(M\) be a real \((2n-1)\)-dimensional semi-invariant \((n>2)\) submanifold of codimension 3 in a complex projective space \(\mathbb C\mathbb P^{n+1}\) such that the third fundamental form \(n\) satisfies \(dn=2\theta w\) for a certain scalar \(\theta(<\frac c2)\), where \(w(X,Y)=g(X,\phi Y)\) for any vectors \(X\) and \(Y\) on \(M.\) If \(M\) satisfies \(R_\xi\phi A=A\phi R_\xi,\) then \(M\) is locally congruent to one of the following spaces in \(\mathbb C\mathbb P^n:\) (\(A_1\))\ a geodesic hypersphere (that is, a tube of radius \(r\) over a hyperplane \(\mathbb C\mathbb P^{n-1},\) where \(0<r<\frac\pi 2);\) (\(A_2\))\ a tube of radius \(r\) over a totally geodesic \(\mathbb C\mathbb P^k (1\leq k\leq n-2),\) where \(0<r<\frac\pi 2\).
scientific article; zbMATH DE number 2075999