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Normal curvature of CR submanifolds of maximal CR dimension of the complex projective space

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Abstract

We prove that there do not exist CR submanifolds Mn of maximal CR dimension of a complex projective space \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) with flat normal connection D of M, when the distinguished normal vector field is parallel with respect to D. If D is lift-flat, then there exists a totally geodesic complex projective subspace \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\) of \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) such that M is a real hypersurface of \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\).

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References

  1. Bejancu A.: CR-submanifolds of a Kähler manifold I. Proc. Amer. Math. Soc. 69, 135–142 (1978)

    MathSciNet  MATH  Google Scholar 

  2. M. Djorić and M. Okumura, CR submanifolds of Complex Projective Space, Developments in Mathematics, vol. 19, Springer (2009).

  3. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. II, Interscience (New York, 1969).

  4. Lawson H. B. Jr.: Rigidity theorems in rank-1 symmetric spaces. J. Differential Geom. 4, 349–357 (1970)

    Article  MathSciNet  Google Scholar 

  5. Nirenberg R., Wells R. O. Jr.: Approximation theorems on differentiable submanifolds of a complex manifold. Trans. Amer. Math. Soc. 142, 15–35 (1965)

    Article  MathSciNet  Google Scholar 

  6. Okumura M.: Normal curvature and real submanifold of the complex projective space. Geom. Dedicata, 7, 509–517 (1978)

    Article  MathSciNet  Google Scholar 

  7. Okumura M.: Codimension reduction problem for real submanifolds of complex projective space. Colloq. Math. Soc. János Bolyai, 56, 574–585 (1989)

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, University of Belgrade, Studentski trg. 16, pb. 550, 11000, Belgrade, Serbia

    M. Djorić

  2. 5-25-25, Minami Ikuta, Tama-ku, Kawasaki, Japan

    M. Okumura

Authors
  1. M. Djorić
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  2. M. Okumura
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Correspondence to M. Djorić.

Additional information

The first author is partially supported by Ministry of Education, Science and Technological Development, Republic of Serbia, project 174012.

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Djorić, M., Okumura, M. Normal curvature of CR submanifolds of maximal CR dimension of the complex projective space. Acta Math. Hungar. 156, 82–90 (2018). https://doi.org/10.1007/s10474-018-0821-z

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  • DOI: https://doi.org/10.1007/s10474-018-0821-z

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