Summary
This paper is devoted to make a systematic study of real hypersurfaces of quaternionic projective space using focal set theory. We obtain three types of such real hypersurfaces. Two of them are known. Third type is new and in its study the first example of proper quaternion CR-submanifold appears. We study real hypersurfaces with constant principal curvatures and classify such hypersurfaces with at most two distinct principal curvatures. Finally we study the Ricci tensor of a real hypersurface of quaternionic projective space and classify pseudo-Einstein, almost-Einstein and Einstein real hypersurfaces.
Article PDF
Similar content being viewed by others
References
M. Barros -B. Y. Chen -F. Urbano,Quaternion CR-submanifolds of quaternion manifolds, Kodai Math. J.,4 (1981), pp. 399–417.
M.Berger,Sur les variétés compactes, C. R. III Réunion Math. Expression Latine, Namur, (1965), pp. 35–55.
W. M. Boothby,An introduction to differentiable manifolds and Riemannian geometry. Academic Press, New York, 1975.
E. Cartan,Sur des familles remarquables d'hypersurfaces isoparametriques dans les espaces sphériques, Math. Zeit.,45 (1939), pp. 335–367.
T. E. Cecil -P. J. Ryan,Focal sets and real hypersurfaces in complexprojective space, Trans. Amer. Math. Soc.,269 (1982), pp. 481–499.
B. Y. Chen,Geometry of submanifolds, Marcel Dekker, New York, 1973.
B. Y. Chen,Totally umbilical submanifolds of quaternion-space-forms, J. Austral. Math. Soc. (Series A),26 (1978), pp. 154–162.
B. Y. Chen,On the first eigenvalue of Laplacian of compact minimal submanifolds of rank one Symmetric spaces, Chinese J. Math.,11 (1984), pp. 259–273.
A. Fialkow,Hypersurfaces of a space of constant curvature, Ann. of Math.,39 (1938), pp. 762–785.
S. Funabashi,Totally complex submanifolds of quaternion Kaehler manifolds, Kodai Math. J.,2 (1979), pp. 314–336.
A. Gray,A note on manifolds whose holonomy group is a subgroup of Sp (tm)· Sp (1), Michigan Math. J.,16 (1969), pp. 125–128.
A. Gray,Weak holonomy groups, Math. Z.,123 (1971), pp. 290–300.
S. Ishihara,Quaternion Kählerian manifolds and fibred Riemannian spaces with Sasakian 3-structure, Kodai Math. Sem. Rep.,25 (1973), pp. 321–329.
S. Ishihara,Quaternion Kählerian manifolds, J. Differential Geometry,9 (1974), pp. 483–500.
G. R. Jensen,Einstein metrics on principal fibre bundles, J. Differential Geometry,8 (1973), pp. 599–614.
E. Martinelli,Modello metrico reale dello spazio proiettivo quaternionale, Ann. Mat. Pura Appl.,49 (1960), pp. 73–90.
H. F. Munzer,Isoparametrische Eyperflächen in Sphären, I, Math. Ann.,251 (1980), pp. 57–71; II, Math. Ann.,256 (1981), pp. 215–232.
J. S. Pak,Real hypersurfaces in quaternionic Kaehlerian manifolds with constant Q-sectional curvature, Kodai Math. Sem. Rep.,29 (1977), pp. 22–61.
J. D. Perez -F. G. Santos,On pseudo-Einstein real hypersurfaces of the quaternionic projective space, Kyungpook Math. J.,25 (1985), pp. 15–28.
K. Sakamoto,Planar geodesic immersions, Tohoku Math. J.,29 (1977), pp. 25–56.
S. Tai,Minimal imbeddings of compact symmetric spaces of rank one, J. Differential Geometry,2 (1968), pp. 55–66.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Martínez, A., Pérez, J.D. Real hypersurfaces in quaternionic projective space. Annali di Matematica pura ed applicata 145, 355–384 (1986). https://doi.org/10.1007/BF01790548
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01790548