Projectively flat immersions of Hermitian symmetric spaces of compact type (Q2810580)

The author defines a projectively flat map of a compact Kähler manifold \(M\) into the complex Grassmannian manifold \(\mathrm{Gr}_p(\mathbb C^n)\). He compares this map to a holomorphic map into the complex projective space. In the first main theorem of the paper, a rigidity theorem of an isometric projectively flat immersion of Hermitian symmetric spaces of compact type is proven. This result leads to the second main result of the paper, namely a description of the holomorphic isometric projectively flat immersions of compact Kähler manifolds with parallel second fundamental form.NEWLINENEWLINEThe groundwork for proving Theorem 1 can be found in [\textit{Y. Nagatomo}, in: Real and complex submanifolds. Proceedings of the ICM 2014 satellite conference and of the 18th international workshop on differential geometry, Daejeon, Korea, August 10--12, 2014. Tokyo: Springer. 453--463 (2014; Zbl 1319.53068)]. The decomposition of the Hermitian symmetric manifold into irreducible components is another key element of the proof of Theorem 1.NEWLINENEWLINEA result from the paper by the author and \textit{Y. Nagatomo} [Tokyo J. Math. 39, No. 1, 173--185 (2016; Zbl 1357.53061)] together with Theorem 1 of this paper, leads to the proof of Theorem 2, taking into account the classification of isometric immersions with parallel second fundamental form for \(f_0\) from \(M\) to \(\mathrm{Gr}_p(\mathbb C^n)\) which is found in [\textit{H. Nakagawa} and \textit{R. Takagi}, J. Math. Soc. Japan 28, 638--667 (1976; Zbl 0328.53009)].