## Tohoku Mathematical Journal

### Holomorphic isometric embeddings of the projective line into quadrics

#### Abstract

We discuss holomorphic isometric embeddings of the projective line into quadrics using the generalisation of the theorem of do Carmo-Wallach in [14] to provide a description of their moduli spaces up to image and gauge equivalence. Moreover, we show rigidity of the real standard map from the projective line into quadrics.

#### Article information

Source
Tohoku Math. J. (2), Volume 69, Number 4 (2017), 525-545.

Dates
First available in Project Euclid: 2 December 2017

https://projecteuclid.org/euclid.tmj/1512183628

Digital Object Identifier
doi:10.2748/tmj/1512183628

Mathematical Reviews number (MathSciNet)
MR3732886

Zentralblatt MATH identifier
06850812

#### Citation

Macia, Oscar; Nagatomo, Yasuyuki; Takahashi, Masaro. Holomorphic isometric embeddings of the projective line into quadrics. Tohoku Math. J. (2) 69 (2017), no. 4, 525--545. doi:10.2748/tmj/1512183628. https://projecteuclid.org/euclid.tmj/1512183628

#### References

• S. Bando and Y. Ohnita, Minimal 2-spheres with constant curvature in $\mathbf P_n(\mathbf C)$, J. Math. Soc. Japan 39 (1987), 477–487.
• T. Bröcker and T. tom Dieck, Representations of compact Lie groups, Springer-Verlag, New York, 1985.
• E. Calabi, Isometric Imbedding of Complex Manifolds}, Ann. of Math. 58} (1953), 1–23.
• Q. S. Chi and Y. Zheng, Rigidity of pseudo-holomorphic curves of constant curvature in Grassmann manifolds, Trans. Amer. Math. Soc. 313 (1989), 393–406.
• M. P. do Carmo and N. R. Wallach, Minimal immersions of spheres into spheres, Ann. of Math. 93 (1971), 43–62.
• J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160.
• \textsc{J. Fei, X. Jiao, L. Xiao and X. Xu, On the Classification of Homogeneous 2-Spheres in Complex Grassmannians, Osaka J. Math 50 (2013), 135–152.
• P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, 1978.
• S. Kobayashi, Differential Geometry of Complex Vector Bundles, Iwanami Shoten and Princeton University, Tokyo, 1987.
• S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol.II, Interscience publishers, 1969.
• K. Kodaira, On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties)}, Ann. of Math. (2) 60 (1954), 28–48.
• Z. Q. Li and Z. H. Yu, Constant curved minimal 2-spheres in G(2,4), Manuscripta Math. 100 (1999), 305–316.
• O. Macia and Y. Nagatomo, Einstein–Hermitian harmonic mappings of the projective line into quadrics, in preparation (2015).
• Y. Nagatomo, Harmonic maps into Grassmannian manifolds, arXiv: mathDG/1408.1504.
• T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380–385.
• G. Toth, Moduli Spaces of Polynomial Minimal Immersions between Complex Projective Spaces, Michigan Math. J. 37 (1990), 385–396.
• J. G. Wolfson, Harmonic maps of the two-sphere into the complex hyperquadric, J. Differential. Geom. 24 (1986), 141–152.