Cyclic parallel structure Jacobi operator for real hypersurfaces in the complex quadric. (English) Zbl 1542.53064
The paper under review deals with real hypersurfaces in the complex quadric \(Q^m=\mathrm{SO}_{m+2}/\mathrm{SO}_m\mathrm{SO}_2\). In the main result of the paper, the authors obtain a complete classification of real hypersurfaces in \(Q^m\) with \(\eta\)-parallel shape operator as follows.
Theorem. Let \(M\) be a Hopf real hypersurface in the complex quadric \(Q^m\), \(m\geq3\). The shape operator \(S\) of \(M\) is \(\eta\)-parallel if and only if \(M\) is locally congruent to an open part of a tube around the \(m\)-dimensional sphere \(S^m\) which is embedded in \(Q^m\) as a real form.
Moreover, as an application of this result, the authors derive a classification of Hopf real hypersurfaces with cyclic parallel structure Jacobi operator in \(Q^m\), proving the next result.
Theorem. Let \(M\) be a Hopf real hypersurface in the complex quadric \(Q^m\), \(m \geq 3\). Then, the structure Jacobi operator \(R_{\xi}\) on \(M\) is cyclic parallel if and only if \(M\) is locally congruent to an open part of the following hypersurfaces in the complex quadric \(Q^m\):
(i) a tube of radius \(r = \frac{\pi}{4}\) around a totally geodesic \(CP^k\) in \(Q^{2k}\), \(m = 2k\);
(ii) a tube of radius \(0 < r < \frac{\pi}{2\sqrt{2}}\) around the \(m\)-dimensional sphere \(S^m\) satisfying \(\tan^2(\sqrt{2}r) = 2\).
Theorem. Let \(M\) be a Hopf real hypersurface in the complex quadric \(Q^m\), \(m\geq3\). The shape operator \(S\) of \(M\) is \(\eta\)-parallel if and only if \(M\) is locally congruent to an open part of a tube around the \(m\)-dimensional sphere \(S^m\) which is embedded in \(Q^m\) as a real form.
Moreover, as an application of this result, the authors derive a classification of Hopf real hypersurfaces with cyclic parallel structure Jacobi operator in \(Q^m\), proving the next result.
Theorem. Let \(M\) be a Hopf real hypersurface in the complex quadric \(Q^m\), \(m \geq 3\). Then, the structure Jacobi operator \(R_{\xi}\) on \(M\) is cyclic parallel if and only if \(M\) is locally congruent to an open part of the following hypersurfaces in the complex quadric \(Q^m\):
(i) a tube of radius \(r = \frac{\pi}{4}\) around a totally geodesic \(CP^k\) in \(Q^{2k}\), \(m = 2k\);
(ii) a tube of radius \(0 < r < \frac{\pi}{2\sqrt{2}}\) around the \(m\)-dimensional sphere \(S^m\) satisfying \(\tan^2(\sqrt{2}r) = 2\).
Reviewer: Gabriel Eduard Vilcu (București)
Keywords:
complex quadrics; Hopf real hypersurfaces; \(\eta\)-parallel shape operator; Killing structure Jacobi operator; cyclic parallel structure Jacobi operatorReferences:
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