Abstract
For any irreducible compact homogeneous Kähler manifold, we classify the compact tight Lagrangian submanifolds which have the ℤ2-homology of a sphere.
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D. Alekseevsky, Flag manifolds, 11th Yugoslav Geometric Seminar (Divčibare, 1996), Beograd Matematički Institute. Zbornik Radova 6(14) (1997), 3–35.
A. L. Besse, Einstein Manifolds, Ergebnisse, deer Mathematik und ihrer Grenzgebiete, Vol. 10, Springer-Verlag, Berlin, 1987.
M. Bordermann, M. Forger, and H. Römer, Homogeneous Kähler manifolds: paving the way towards new supersymmetric sigma models, Communications in Mathematical Physics 102 (1986), 605–617.
L. Bedulli and A. Gori, Homogeneous Lagrangian submanifolds, Communications in Analysis and Geometry 16 (2008), 591–615.
A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, American Journal of Mathematics 80 (1958), 458–538.
T. E. Cecil and S.-S. Chern (eds.), Tight and Taut Submanifolds, in memory of Nicolaas H. Kuiper. Papers from the Workshop on Differential Systems, Submanifolds and Control Theory held in Berkeley, CA, March 1–4, 1994, Mathematical Sciences Research Institute Publications, Vol. 32, Cambridge University Press, Cambridge, 1997.
S. S. Chern and R. Lashof, On the total curvature of immersed manifolds, American Journal of Mathematics 79 (1957), 306–318.
N. Ejiri and K. Tsukada, Another natural lift of a Kähler submanifold of a quaternionic Kähler manifold to the twistor space, Tokyo Journal of Mathematics 28 (2005), 71–78.
T. Gotoh, The nullity of a compact minimal real hypersurface in a quaternion projective space, Geometriae Dedicata 76 (1999), 53–64.
C. Gorodski and G. Thorbergsson, The classification of taut irreducible representations, Journal für die Reine und Angewandte Mathematik 555 (2003), 187–235.
E. Heintze, R. Palais, C.-L. Terng and G. Thorbergsson, Hyperpolar actions and k-flat homogeneous spaces, Journal für die Reine und Angewandte Mathematik 454 (1994), 163–179.
H. Iriyeh, H. Ono and T. Sakai, Integral geometry and Hamiltonian volume minimizing property of a totally geodesic Lagrangian torus in S 2 ×S 2, Japanese Academy. Proceedings. Series A. Mathematical Sciences 79 (2003), 167–170.
H. Iriyeh, Uniqueness of Hamiltonian volume minimizing Lagrangian submanifolds which are Hamiltonian isotopic to ℝP n in ℂP n, arXiv:math/0509091 [math.DG], 2005.
H. Iriyeh and T. Sakai, Tight Lagrangian surfaces in S 2 × S 2, Geometriae Dedicata 145 (2010), 1–17.
A. Kollross, A classification of hyperpolar and cohomogeneity one actions, Transactions of the American Mathematical Society 354 (2002), 571–612.
M. Nishiyama, Classification of invariant complex structures on irreducible compact simply connected coset spaces, Osaka Journal of Mathematics 21 (1984), 35–58.
K. Nomizu and L. Rodríguez, Umbilical submanifolds and Morse functions, Nagoya Mathematical Journal 48 (1972), 197–201.
Y.-G. Oh, Tight Lagrangian submanifolds in ℂPn, Mathematische Zeitschrift 207 (1991), 409–416.
Y. Ohnita, On stability of minimal submanifolds in compact symmetric spaces, Compositio Mathematica 64 (1987), 157–189.
A. L. Onishchik, Topology of Transitive Transformation Groups, Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994.
M. Takeuchi and S. Kobayashi, Minimal imbeddings of R-spaces, Journal of Differential Geometry 2 (1968), 203–215.
M. S. Tanaka and H. Tasaki, The intersection of two real forms in Hermitian symmetric spaces of compact type, Journal of the Mathematical Society of Japan 64 (2012), 1297–1332.
G. Warner, Harmonic Analysis on Semi-simple Lie Groups. I, Die Grundlehren der mathematischen Wissenschaften, Vol. 188, Springer-Verlag, New York-Heidelberg, 1972.
J. A. Wolf and A. Gray, Homogeneous spaces defined by Lie group automorphisms. II, Journal of Differential Geometry 2 (1968), 115–159.
J. A. Wolf, The action of a real semisimple group on a complex flag manifold. I, Orbit structure and holomorphic arc components, Bulletin of the American Mathematical Society 75 (1969), 1121–1237.
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The first author was partially supported by the CNPq Federal Grant 303038/2013-6 and Fapesp project 2011/21362-2.
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Gorodski, C., Podestà, F. Tight Lagrangian homology spheres in compact homogeneous Kähler manifolds. Isr. J. Math. 206, 413–429 (2015). https://doi.org/10.1007/s11856-014-1145-5
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DOI: https://doi.org/10.1007/s11856-014-1145-5