Real hypersurfaces with isometric Reeb flow in Kähler manifolds. (English) Zbl 1456.53042
The paper under review consists of two main parts. In the first part of the article, the authors develop a general structure theory for real hypersurfaces in Kähler manifolds for which the Reeb flow preserves the induced metric. In the second part of the article, the authors apply this theory to classify real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces of compact type, obtain the following interesting classification result:
Theorem. Let \(M\) be a connected orientable real hypersurface in an irreducible Hermitian symmetric space \(\bar{M}\) of compact type. If the Reeb flow on \(M\) is an isometric flow, then \(M\) is congruent to an open part of a tube of radius \(0 < t < \pi/\sqrt{2}\) around the totally geodesic submanifold \(\Sigma\) in \(\bar{M}\), where
(i) \(\bar{M} = \mathbb{C} P^r = \mathrm{SU}_{r+1}/\mathrm{S}(\mathrm{U}_1\mathrm{U}_r)\) and \(\Sigma = \mathbb{C} P^k\), \(0 \leq k \leq r-1\);
(ii) \(\bar{M} = G_k(\mathbb{C}^{r+1}) = \mathrm{SU}_{r+1}/\mathrm{S}(\mathrm{U}_k\mathrm{U}_{r+1-k})\) and \(\Sigma = G_k(\mathbb{C}^r)\), \(2 \leq k \leq \frac{r+1}{2}\);
(iii) \(\bar{M} = G_2^+(\mathbb{R}^{2r}) = \mathrm{SO}_{2r}/\mathrm{SO}_{2r-2}\mathrm{SO}_2\) and \(\Sigma = \mathbb{C} P^{r-1}\), \(3 \leq r\);
(iv) \(\bar{M} = \mathrm{SO}_{2r}/\mathrm{U}_r\) and \(\Sigma = \mathrm{SO}_{2r-2}/\mathrm{U}_{r-1}\), \(5 \leq r\).
Conversely, the Reeb flow on any of these hypersurfaces is an isometric flow.
Theorem. Let \(M\) be a connected orientable real hypersurface in an irreducible Hermitian symmetric space \(\bar{M}\) of compact type. If the Reeb flow on \(M\) is an isometric flow, then \(M\) is congruent to an open part of a tube of radius \(0 < t < \pi/\sqrt{2}\) around the totally geodesic submanifold \(\Sigma\) in \(\bar{M}\), where
(i) \(\bar{M} = \mathbb{C} P^r = \mathrm{SU}_{r+1}/\mathrm{S}(\mathrm{U}_1\mathrm{U}_r)\) and \(\Sigma = \mathbb{C} P^k\), \(0 \leq k \leq r-1\);
(ii) \(\bar{M} = G_k(\mathbb{C}^{r+1}) = \mathrm{SU}_{r+1}/\mathrm{S}(\mathrm{U}_k\mathrm{U}_{r+1-k})\) and \(\Sigma = G_k(\mathbb{C}^r)\), \(2 \leq k \leq \frac{r+1}{2}\);
(iii) \(\bar{M} = G_2^+(\mathbb{R}^{2r}) = \mathrm{SO}_{2r}/\mathrm{SO}_{2r-2}\mathrm{SO}_2\) and \(\Sigma = \mathbb{C} P^{r-1}\), \(3 \leq r\);
(iv) \(\bar{M} = \mathrm{SO}_{2r}/\mathrm{U}_r\) and \(\Sigma = \mathrm{SO}_{2r-2}/\mathrm{U}_{r-1}\), \(5 \leq r\).
Conversely, the Reeb flow on any of these hypersurfaces is an isometric flow.
Reviewer: Gabriel Eduard Vilcu (Ploieşti)
MSC:
| 53C40 | Global submanifolds |
| 32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |
| 37C10 | Dynamics induced by flows and semiflows |
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 53D15 | Almost contact and almost symplectic manifolds |
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