Advertisement

Archiv der Mathematik

, Volume 110, Issue 5, pp 523–531 | Cite as

Einstein submanifolds with parallel mean curvature

  • Christos-Raent Onti
Article

Abstract

We provide a classification of Einstein submanifolds in space forms with flat normal bundle and parallel mean curvature. This extends a previous result due to Dajczer and Tojeiro (Tohoku Math J (2) 45:43–49, 1993) for isometric immersions of Riemannian manifolds with constant sectional curvature.

Keywords

Einstein submanifolds Parallel mean curvature Flat normal bundle Principal normals 

Mathematics Subject Classification

Primary 53B25 Secondary 53C40 53C42 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Cartan, Sur les variétés de courbure constante d’un espace euclidien ou non-euclidien, Bull. Soc. Math. France 47 (1919), 125–160.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    E. Cartan, Sur les variétés de courbure constante d’un espace euclidien ou non-euclidien, Bull. Soc. Math. France 48 (1920), 132–208.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. Dajczer and R. Tojeiro, Submanifolds of constant sectional curvature with parallel or constant mean curvature, Tohoku Math. J. (2) 45 (1993), 43–49.Google Scholar
  4. 4.
    M. Dajczer and R. Tojeiro, Isometric immersions and the generalized Laplace and elliptic sinh-Gordon equations, J. Reine Angew. Math. 467 (1995), 109–147.MathSciNetzbMATHGoogle Scholar
  5. 5.
    M. Dajczer and R. Tojeiro, On compositions of isometric immersions, J. Differential Geom. 36 (1992), 1–18.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. J. Di Scala, Minimal immersions of Kähler manifolds into Euclidean spaces, Bull. Lond. Math. Soc. 35 (2003), 825–827.CrossRefzbMATHGoogle Scholar
  7. 7.
    A. Fialkow, Hypersurfaces of a space of constant curvature, Ann. of Math. (2) 39 (1938), 762–785.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. D. Moore, Isometric immersions of riemannian products, J. Differential Geom. 5 (1971), 159–168.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    J. D. Moore, Submanifolds of constant positive curvature I, Duke Math. J. 44 (1977), 449–484.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. Molzan, Extrinsische Produkte und symmetrische Untermannigfaltigkeiten in Standardraumen konstanter und konstanter holomorpher Krümmung, Doctoral Thesis, Köln, 1983.Google Scholar
  11. 11.
    S. Nölker, Isometric immersions with homothetical Gauss map, Geom. Dedicata 34 (1990), 271–280.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    H. Reckziegel, Krümmungsflächen von isometrischen Immersionen in Räume konstanter Krümmung, Math. Ann. 223 (1976), 169–181.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    H. Reckziegel, Hypersurfaces with parallel Ricci tensor in spaces of constant curvature, Results Math. 27 (1995), 113–116. Festschrift dedicated to Katsumi Nomizu on his 70th birthday (Leuven, 1994; Brussels, 1994).Google Scholar
  14. 14.
    P. J. Ryan, Homogeneity and some curvature conditions for hypersurfaces, Tohoku Math. J. (2) 21 (1969), 363–388.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    T. Y. Thomas, On closed spaces of constant mean curvature, Amer. J. Math. 58 (1936), 701–704.MathSciNetCrossRefGoogle Scholar
  16. 16.
    R. Tojeiro, A decomposition theorem for immersions of product manifolds, Proc. Edinb. Math. Soc. 59 (2016), 247–269.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IoanninaIoanninaGreece

Personalised recommendations