Kähler manifolds and their relatives
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 9 (2010) no. 3, p. 495-501
Let M 1 and M 2 be two Kähler manifolds. We call M 1 and M 2 relatives if they share a non-trivial Kähler submanifold S, namely, if there exist two holomorphic and isometric immersions (Kähler immersions) h 1 :SM 1 and h 2 :SM 2 . Moreover, two Kähler manifolds M 1 and M 2 are said to be weakly relatives if there exist two locally isometric (not necessarily holomorphic) Kähler manifolds S 1 and S 2 which admit two Kähler immersions into M 1 and M 2 respectively. The notions introduced are not equivalent (cf. Example 2.3). Our main results in this paper are Theorem 1.2 and Theorem 1.4. In the first theorem we show that a complex bounded domain D n with its Bergman metric and a projective Kähler manifold (i.e. a projective manifold endowed with the restriction of the Fubini–Study metric) are not relatives. In the second theorem we prove that a Hermitian symmetric space of noncompact type and a projective Kähler manifold are not weakly relatives. Notice that the proof of the second result does not follows trivially from the first one. We also remark that the above results are of local nature, i.e. no assumptions are used about the compactness or completeness of the manifolds involved.
Classification:  53C55,  58C25
@article{ASNSP_2010_5_9_3_495_0,
     author = {Di Scala, Antonio and Loi, Andrea},
     title = {K\"ahler manifolds and their relatives},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {3},
     year = {2010},
     pages = {495-501},
     zbl = {1253.53066},
     mrnumber = {2722652},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2010_5_9_3_495_0}
}
Di Scala, Antonio; Loi, Andrea. Kähler manifolds and their relatives. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 9 (2010) no. 3, pp. 495-501. http://www.numdam.org/item/ASNSP_2010_5_9_3_495_0/

[1] D. V. Alekseevsky and M. M. Graev, Calabi-Yau metric on the Fermat surface. Isometries and totally geodesic submanifolds, J. Geom. Phys. 7 (1990), 21–43. | MR 1094729 | Zbl 0749.53028

[2] D. V. Alekseevsky and B. N. Kimel’fel’ d, Structure of homogeneous Riemannian spaces with zero Ricci curvature, Funktsional. Anal. i Prilozhen. 9 (1975), 5–11. | MR 402650 | Zbl 0316.53041

[3] M. Berger, Encounter with a geometer: Eugenio Calabi, Manifolds and geometry (Pisa, 1993), 20–60, Sympos. Math., XXXVI, Cambridge Univ. Press, Cambridge, 1996. | MR 1410067 | Zbl 0926.53001

[4] M. Berger, “A Panoramic View of Riemannian Geometry”, Springer Verlag, 2003. | MR 2002701 | Zbl 1038.53002

[5] A. Besse, “Einstein Manifolds”, Springer Verlag, 1987. | MR 867684 | Zbl 0613.53001

[6] E. Calabi, Isometric imbeddings of complex manifolds, Ann. of Math. 58 (1953), 1–23. | MR 57000 | Zbl 0051.13103

[7] J. E. D’Atri, Holomorphic sectional curvatures of bounded homogeneous domains and related questions, Trans. Amer. Math. Soc. 256 (1979), 405–413. | MR 546926 | Zbl 0387.32013

[8] A. J. Di Scala and A. Loi, Kähler maps of Hermitian symmetric spaces into complex space forms, Geom. Dedicata 25 (2007), 103–113. | MR 2322543 | Zbl 1132.53040

[9] D. Hulin, Kähler–Einstein metrics and projective embeddings, J. Geom. Anal. 10 (2000), 525–528. | MR 1794575 | Zbl 1057.53032

[10] A. Loi, Calabi’s diastasis function for Hermitian symmetric spaces, Differential Geom. Appl. 24 (2006), 311–319. | MR 2216943 | Zbl 1096.53043

[11] S. Kobayashi, Geometry of bounded domains, Trans. Amer. Math. Soc. 92 (1959), 267–290. | MR 112162 | Zbl 0136.07102

[12] I. I. Pyatetski-Shapiro, On a problem proposed by E. Cartan (Russian), Dokl. Akad. Nauk SSSR 124 (1959) 272–273. | MR 101922 | Zbl 0089.06201

[13] A. Spiro A remark on locally homogeneous Riemannian spaces, Results Math. 24 (1993), 318–325. | MR 1244285 | Zbl 0795.53016

[14] M. Umehara, Kähler submanifolds of complex space forms, Tokyo J. Math. 10 (1987), 203–214. | MR 899484 | Zbl 0679.53016