Abstract
We classify holomorphic isometric embeddings of the projective space into quadrics using the generalization of do Carmo–Wallach theory in Nagatomo [Harmonic maps into Grassmann manifolds, arXiv:mathDG/1408.1504[mathDG], Holomorphic maps into Grassmann manifolds (Harmonic maps into Grassmann manifolds III), Annals of Global analysis and Geometry 60, 33–63 (2021). An explicit description of their moduli spaces up to image and gauge equivalence is provided.
Similar content being viewed by others
Data Availability Statement
This manuscript has no associated data.
References
Calabi, E.: Isometric imbedding of complex manifolds. Ann. Math. 58, 1–23 (1953)
do Carmo, M.. P., Wallach, N.. R.: Minimal immersions of spheres into spheres. Ann. Math 93, 43–62 (1971)
Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. London Mathematical Society Student Texts (35), Cambridge University Press, Cambridge (1997)
Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Iwanami Shoten and Princeton University, Tokyo (1987)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. II. Interscience Publishers (1969)
Macia, O., Nagatomo, Y., Takahashi, M.: Holomorphic isometric embeddings of projective lines into quadrics. Tohoku Math. J. 69, 525–545 (2017)
Macia, O., Nagatomo, Y.: Moduli of Einstein-Hermitian harmonic mappings of the projective lines into quadrics. Ann. Glob. Anal. Geom. 53, 503–520 (2018)
Nagatomo, Y.: Harmonic maps into Grassmann manifolds, arXiv:mathDG/1408.1504 [mathDG]
Nagatomo, Y.: Holomorphic maps into Grassmann manifolds (Harmonic maps into Grassmann manifolds III), Annals of Global analysis and Geometry 60, 33–63 (2021)
Takahashi, T.: Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan 18, 380–385 (1966)
Acknowledgements
The author is grateful to Prof. O. Macia for useful conversations. This research is supported by JSPS KAKENHI Grant Number 17K05230.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Nagatomo, Y. Holomorphic isometric embeddings of the projective space into quadrics. Geom Dedicata 216, 31 (2022). https://doi.org/10.1007/s10711-022-00689-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10711-022-00689-4