Isometric immersions into complex projective space. (English) Zbl 1043.53047

Morimoto, Tohru (ed.) et al., Lie groups, geometric structures and differential equations—one hundred years after Sophus Lie. Based on the international conference on the occasion of the centennial after the death of Sophus Lie (1842–1899), Kyoto and Nara, Japan, December 1999. Tokyo: Mathematical Society of Japan (ISBN 4-931469-21-3/hbk). Adv. Stud. Pure Math. 37, 367-396 (2002).
The fundamental existence and uniqueness theorems for isometric immersions of an \(n\)-dimensional smooth Riemannian manifold \((M^{n},g)\) into the complete, simply connected, constant curvature Riemannian manifolds of dimension \(n+1\), can be easily stated in terms of the Gauss and Codazzi equations. However, there is no such simple generalization to isometric immersions of hypersurfaces into Riemannian manifolds other than real space forms. In this paper the author studies isometric immersions in \({\mathbb C}P^{n}\) of spheres \(S^{2n-1}\) endowed with \(U(n)\)-invariant metrics. He first proves that these spheres can be realized as hypersurfaces of complex space forms. Then, he obtains his main result: The only \(U(n)\)-invariant metrics on \(S^{2n-1}\) which allow an isometric immersion in \(\mathbb{C}P^{n}\) are the Berger metrics.
The proof goes through a careful study of the Gauss equation. In fact a detailed analysis of the normal part in the Gauss equation suffices to take care of the case \(n\geq 4\). However for \(n=3\) consideration of Codazzi equations is also needed and the case \(n=2\) requires harder work. The techniques developed here seem to be useful to obtain similar results in the complex hyperbolic plane, quaternionic projective and hyperbolic spaces and other homogeneous spaces too.
For the entire collection see [Zbl 1011.00031].


53C40 Global submanifolds
32Q40 Embedding theorems for complex manifolds
53C30 Differential geometry of homogeneous manifolds