Skip to main content
SpringerLink
Log in

Real Hypersurfaces in Quaternionic Space Forms Satyisfying Axioms of Planes

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

A Riemannian manifold satisfies the axiom of 2-planes if at each point, there are suficiently many totally geodesic surfaces passing through that point. Real hypersurfaces in quaternionic space forms admit nice families of tangent planes, namely, totally real, half-quaternionic and quaternionic. Several definitions of axiom of planes arise naturally when we consider such families of tangent planes. We are able to classify real hypersurfaces in quaternionic space forms satisfying these definitions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Berndt, Real hypersurfaces in quaternionic space forms, J. Reine. Angew. Math., 419 (1991), 9–26.

    Google Scholar 

  2. J. Berndt, Homogeneous hypersurfaces in hyperbolic spaces, Math. Z., 229 (1998), 589–600.

    Google Scholar 

  3. E. Cartan, Leçon sur la Géométrie des Espaces de Riemann, Gauthier–Villards (Paris, 1946).

    Google Scholar 

  4. S. Dragomir, The axiom of spheres in quaternionic geometry, Ann. Univ. Ferrara, Sez. VII, Sc Mat. Vol. XXXIV (1988), 15–20.

    Google Scholar 

  5. A. Gray, Pseudo–Riemannian almost product manifolds and submersions, J. Math. and Mech., 16 (1967), 715–737.

    Google Scholar 

  6. A. Gray, A note on manifolds whose holonomy group is a subgroup of Sp(m)·Sp(1), Michigan Math. J., 16 (1969), 125–128.

    Google Scholar 

  7. S. Ishihara, Quaternion Kälerian manifolds, J. Diff. Geom., 9 (1974), 483–500.

    Google Scholar 

  8. A. Martínez and J. D. Pérez, Real hypersurfaces in quaternionic projective space, Ann. Mat. Pura Appl., (IV) 145 (1986), 355–384.

    Google Scholar 

  9. K. Nomizu, Conditions for constancy of the holomorphic sectional curvature, J. Diff. Geom., 8 (1973), 335–339.

    Google Scholar 

  10. J. D. Pérez, F. G. Santos and F. Urbano, On the axioms of planes in quaternionic geometry, Ann. Mat. Pura Appl., (IV) 130 (1982), 215–221.

    Google Scholar 

  11. K. Tsukada, Parallel submanifolds in a quaternion projective space, Osaka J. Math., 22 (1985), 187–241.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Geometría Y Topología Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain

    M. Ortega

Authors
  1. M. Ortega
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ortega, M. Real Hypersurfaces in Quaternionic Space Forms Satyisfying Axioms of Planes. Acta Mathematica Hungarica 93, 225–242 (2001). https://doi.org/10.1023/A:1013934829024

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1013934829024

Search

Navigation