×

Transversal symmetries on real hypersurfaces in a complex space form. (English) Zbl 1278.53021

The authors consider orientable hypersurfaces \(M\) in a complex non-flat space form \(\tilde M(c)\) of constant holomorphic curvature \(c\) (which is either the complex projective space \(P_n\mathbb C\) or the complex hyperbolic space \(H_n\mathbb C\)). These \(M\) have an almost contact metric structure. In the contact geometry there is a notion of transversal symmetry (or a \(\phi\)-symmetry). Also the structural reflections (i.e. the reflections with respect to the integral curves of the structure vector \(\xi\) of the contact structure mentioned above) are considered. Two main results in this paper are:
Theorem 1. Let \(M\) be a real hypersurface in a non-flat complex space form \(\tilde M_n(c)\) (\(c \neq 0\)). Then the structural reflections on \(M\) are isometries iff \(M\) is locally congruent to a homogeneous hypersurface of type (A) or (B) in \(P_n\mathbb C\) or \(H_n\mathbb C\).
There are some special classes \((A)\) (in fact \((A_1),(A_2)),(B),(C),(D),(E)\) of hypersurfaces in \(P_n\mathbb C\) and \(H_n\mathbb C\). They may be characterized geometrically. So Theorem 1 gives a classification of some real hypersurface \(M\) in a non-flat complex space form \(\tilde M_n(c)\).
Theorem 2. Let \(M\) be a real hypersurface in a non-flat complex space form \(\tilde M_n(c)\) (\(c \neq 0\)). Then the eigenvalues of the transversal Jacobi operators are constant and their eigenspaces are parallel along each transversal geodesic iff \(M\) is locally congruent to a ruled hypersurface in \(P_n\mathbb C\) or \(H_n\mathbb C\).

MSC:

53B25 Local submanifolds
53C30 Differential geometry of homogeneous manifolds
PDFBibTeX XMLCite
Full Text: Euclid

References:

[1] S.-S. Ahn, S.-B. Lee and Y.J. Suh, On ruled real hypersurfaces in a complex space form, Tsukuba J. Math., 17 (1993), 311-322. · Zbl 0804.53024
[2] J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math., 395 (1989), 132-141. · Zbl 0655.53046 · doi:10.1515/crll.1989.395.132
[3] J. Berndt and L.Vanhecke, Two natural generalizations of locally symmetric spaces, Diff. Geom. Appl. 2 (1992), 57-80. · Zbl 0747.53013 · doi:10.1016/0926-2245(92)90009-C
[4] D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Math. 203 , Birkhäuser, Boston, Basel, Berlin, 2002. · Zbl 1011.53001
[5] D. E. Blair and L. Vanhecke, Symmetries and \(\varphi\)-symmetric spaces, Tôhoku Math. J., 39 (1987), 373-383. · Zbl 0632.53039 · doi:10.2748/tmj/1178228284
[6] E. Boeckx and L. Vanhecke, Characteristic reflections on unit tangent sphere bundles, Houston J. Math., 23 (1997), 427-448. · Zbl 0897.53010
[7] E. Cartan, Leçons sur la géométrie des espaces de Riemann, 2nd edition, Gaythier-Villars, Paris 1946. · JFM 54.0755.01
[8] T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc., 269 (1982), 481-499. · Zbl 0492.53039 · doi:10.2307/1998460
[9] B.Y. Chen and L. Vanhecke, Isometric, holomorphic and symplectic reflections, Geom. Dedicata, 29 (1989), 259-277. · Zbl 0673.53035 · doi:10.1007/BF00572443
[10] J.T. Cho and M. Kimura, Ricci solitons of compact real hypersurfaces in Kähler manifolds, Math. Nach. 284 (2011), 1385-1393. · Zbl 1241.53036 · doi:10.1002/mana.200910186
[11] J.T. Cho and L. Vanhecke, Hopf hypersurfaces of D’Atri- and C-type in a complex space form, Rendconti di Matematica Serie VII, Roma, 18 (1998), 601-613. · Zbl 0939.53011
[12] A. Gray, Classification des variétés approximativement kählériennes de courbure sectionelle holomorphe constante, J. Reine Angew. Math., 279 (1974), 797-800. · Zbl 0301.53016
[13] U-H. Ki, Real hypersurfaces with parallel Ricci tensor of a complex space form, Tsukuba J. Math., 13 (1989), 73-81. · Zbl 0678.53046
[14] U-H. Ki and Y.J. Suh, On real hypersurfaces of a complex space form, Math. J. Okayama, 32 (1990), 207-221. · Zbl 0734.53040
[15] U.K. Kim, Nonexistence of Ricci-parallel real hypersurfaces in \(P_2 \mathbb C\) or \(H_2 \mathbb C\), Bull. Korean Math. Soc. 41 (2004), 699-708. · Zbl 1070.53029 · doi:10.4134/BKMS.2004.41.4.699
[16] M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc., 296 (1986), 137-149. · Zbl 0597.53021 · doi:10.2307/2000565
[17] M. Kimura, Sectional curvatures of holomorphic planes on a real hypersurfaces in \(P^n(\mathbb C)\), Math. Ann. 276 (1987), 487-497. · Zbl 0605.53023 · doi:10.1007/BF01450843
[18] M. Kimura and S. Maeda, On real hypersurfaces of a complex projective space, Math. Z., 202 (1989), 299-311. · Zbl 0661.53015 · doi:10.1007/BF01159962
[19] M. Kon, Pseudo-Einstein real hypersurfaces of complex space forms, J. Diff. Geometry, 14 (1979), 339-354. · Zbl 0461.53031
[20] J.G. Lee, J. D. Pérez and Y.J. Suh, On real hypersurfaces with \(\eta\)-parallel curvature tensor in complex space forms, Acta Math. Hungar., 101 (2003), 1-12. · Zbl 1055.53042 · doi:10.1023/B:AMHU.0000003886.15401.76
[21] Y. Maeda, On real hypersurfaces of a complex projective space, J. Math. Soc. Japan, 28 (1976), 529-540. · Zbl 0324.53039 · doi:10.2969/jmsj/02830529
[22] S. Montiel, Real hypersurfaces of a complex hyperbolic space, J. Math. Soc. Japan, 37 (1985), 515-535. · Zbl 0554.53021 · doi:10.2969/jmsj/03730515
[23] S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata, 20 (1986), 245-261. · Zbl 0587.53052 · doi:10.1007/BF00164402
[24] S. Nagai, Real hypersurfaces of a complex projective space in which the reflections with respect to \(\xi\)-curves are isometric, Hokkaido Math. J. 30 (2001), 631-643. · Zbl 1002.53008
[25] M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc., 212 (1975), 355-364. · Zbl 0288.53043 · doi:10.2307/1998631
[26] Y.J. Suh, On real hypersurfaces of a complex space form with \(\eta\)-parallel Ricci tensor, Tsukuba J. Math., 14 (1990), 27-37. · Zbl 0721.53029
[27] R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math., 10 (1973), 495-506. · Zbl 0274.53062
[28] R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures I, II, J. Math. Soc. Japan, 27 (1975), 43-53, 507-516. · Zbl 0311.53064 · doi:10.2969/jmsj/02740507
[29] T. Takahashi, Sasakian \(\phi\)-symmetric spaces, Tôhoku Math. J., 29 (1977), 91-113. · Zbl 0343.53030 · doi:10.2748/tmj/1178240699
[30] L. Vanhecke and T. J. Willmore, Interaction of tubes and spheres, Math. Ann., 263 (1983), 31-42. · Zbl 0491.53034 · doi:10.1007/BF01457081
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.