Some characterizations of quaternionic space forms. (English) Zbl 0997.53037

The authors prove that any of the following two properties characterize quaternionic space forms in the class of quaternionic Kähler manifolds \(M\):
1) \(R(X,JX)X\) is proportional to \(JX\) for all tangent vectors \(X\in T_pM\) to \(M\) and all complex structures \(J\) on \(T_pM\) subordinate to the quaternionic structure.
2) Geodesics in geodesic spheres \(S \subset M\) of sufficiently small radius are circles of positive curvature in \(M\) provided that their velocity vector field is perpendicular to the maximal quaternionic distribution in \(S\). The complex analogues of these two results were proven, respectively, by E. Kosmanek [C. R. Acad. Sci. Paris 259, 705-708 (1964; Zbl 0122.16704)] and in previous work of the authors [Bull. Aust. Math. Soc. 62, No. 2, 205-210 (2000; Zbl 0978.53071)].


53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
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[1] Adachi, T., and Maeda, S.: Space forms from the viewpoint of their geodesic spheres. Bull. Aus-tral. Math. Soc., 62 , 205-210 (2000). · Zbl 0978.53071
[2] Berndt, J. B.: Real hypersurfaces in quaternionic space forms. J. Reine Angew. Math., 419 , 9-26 (1991). · Zbl 0718.53017
[3] Chen, B.-y., and Vanhecke, L.: Differential geometry of geodesic spheres. J. Reine Angew. Math., 325 , 28-67 (1981). · Zbl 0503.53013
[4] Hamada, T.: On real hypersurfaces of a quaternionic projective space. Saitama Math. J., 11 , 29-39 (1993). · Zbl 0803.53036
[5] Ishihara, S.: Quaternion Kählerian manifolds. J. Differential Geom., 9 , 483-500 (1974). · Zbl 0297.53014
[6] Kosmanek, E.: Une propriété caractéristique des variétés Kählériennes à courbure holomorphe constante. C. R. Acad. Sci. Paris Sér. I. Math., 259 , 705-708 (1964). · Zbl 0122.16704
[7] Pak, J. S.: Real hypersurfaces in quaternionic Kaehlerian manifolds with constant Q-sectional curvature. Kodai Math. Sem. Rep., 29 , 22-61 (1977). · Zbl 0424.53012
[8] Tanno, S.: Constancy of holomorphic sectional curvature in almost Hermitian manifolds. Kodai Math. Sem. Rep., 25 , 190-201 (1973). · Zbl 0263.53019
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