## Tokyo Journal of Mathematics

### Compact Minimal $CR$ Submanifolds of a Complex Projective Space with Positive Ricci Curvature

Mayuko KON

#### Abstract

We give a reduction theorem for the codimension of a compact $n$-dimensional minimal proper $CR$ submanifold $M$ immersed in a complex projective space $CP^m$ with complex structure $J$, under the assumption that the Ricci curvature of $M$ is equal to or greater than $n-1$. Moreover, we classify compact $n$-dimensional minimal $CR$ submanifolds whose Ricci tensor $S$ satisfies $S(X,X)\geq (n-1)g(X,X)+kg(PX,PX)$, $k=0,1,2$, for any vector field $X$ tangent to $M$, where $PX$ is the tangential part of $JX$.

#### Article information

Source
Tokyo J. Math., Volume 33, Number 2 (2010), 415-434.

Dates
First available in Project Euclid: 31 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1296483480

Digital Object Identifier
doi:10.3836/tjm/1296483480

Mathematical Reviews number (MathSciNet)
MR2779427

Zentralblatt MATH identifier
1162.53316

Subjects
Primary: 53C40: Global submanifolds [See also 53B25]
Secondary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]

#### Citation

KON, Mayuko. Compact Minimal $CR$ Submanifolds of a Complex Projective Space with Positive Ricci Curvature. Tokyo J. Math. 33 (2010), no. 2, 415--434. doi:10.3836/tjm/1296483480. https://projecteuclid.org/euclid.tjm/1296483480

#### References

• K. Abe, Applications of Riccati type differential equation to Riemannian manifolds with totally geodesic distribution, Tôhoku Math. J. 25 (1973), 425–444.
• B-Y. Chen and K. Ogiue, A characterization of the complex sphere, Michigan Math. J. 21 (1974), 231–232.
• T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 (1982), 481–499.
• J. Erbacher, Reduction of the codimension of an isometric immersion, J.Differential Geom. 5 (1971), 333–340.
• S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. II, Interscience, New York, 1969.
• Masahiro Kon, On some complex submanifolds in Kaehler manifolds, Canad. J. Math. 26 (1974), 1442–1449.
• Masahiro Kon, Generic minimal submanifolds of a complex projective space, Bull. London Math. Soc. 12 (1980), 355–360.
• Masahiro Kon, Minimal $CR$ submanifolds immersed in a complex projective space, Geom. Dedicata 31 (1989), 357–368.
• S. Maeda, Real hypersurfaces of a complex projective space II, Bull. Austral. Math. Soc. 29 (1984), 123–127.
• S. Maeda, Ricci tensors of real hypersurfaces in a complex projective space, Proc. Amer. Math. Soc. 122 (1994), 1229–1235.
• K. Ogiue, Differential geometry of Kaehler submanifolds, Advances in Math. 13 (1974), 73–114.
• J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. 88 (1968), 62–105.
• B. Smith, Differential geometry of complex hypersurfaces, Ann. of Math. 85 (1967), 246–266.
• M. Yamagata and Masahiro Kon, Reduction of the codimension of a generic minimal submanifold immersed in a complex projective space, Coll. Math. 74 (1997), 185–190.
• K. Yano and Masahiro Kon, Structures on manifolds, World Scientific Publishing, Singapore, 1984.
• P. B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambridge. Philos. Soc. 53(1957), 568-575.