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On an inequality of Oprea for Lagrangian submanifolds. (English) Zbl 1176.53030
Summary: We show that a Lagrangian submanifold of a complex space form attaining equality in the inequality obtained by T. Oprea in [Rocky Mountain J. Math. 38, 567–581 (2008; Zbl 1195.53072)], must be totally geodesic.

MSC:
 53B25 Local submanifolds 53D12 Lagrangian submanifolds; Maslov index
Zbl 1195.53072
Full Text:
References:
 [1] Bolton J., Dillen F., Fastenakels J., Vrancken L., A best possible inequality for curvature-like tensor fields, preprint · Zbl 1175.53023 [2] Bolton J., Rodriguez Montealegre C., Vrancken L., Characterizing warped product Lagrangian immersions in complex projective space, Proc. Edinb. Math. Soc., 2008, 51, 1-14 · Zbl 1166.53010 [3] Bolton J., Vrancken L., Lagrangian submanifolds attaining equality in the improved Chen’s inequality, Bull. Belg. Math. Soc., 2007, 14, 311-315 · Zbl 1130.53016 [4] Chen B.Y., Some pinching and classification theorems for minimal submanifolds, Arch. Math., 1993, 60, 568-578 http://dx.doi.org/10.1007/BF01236084 · Zbl 0811.53060 [5] Chen B.Y., Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J., 1999, 41, 33-41 http://dx.doi.org/10.1017/S0017089599970271 · Zbl 0962.53015 [6] Chen B.Y., Riemannian geometry of Lagrangian submanifolds, Taiwan. J. Math., 2001, 5, 681-723 · Zbl 1002.53053 [7] Oprea T., Chen’s inequality in Lagrangian case, Colloq. Math., 2007, 108, 163-169 http://dx.doi.org/10.4064/cm108-1-15 · Zbl 1118.53035 [8] Oprea T., On a Riemannian invariant of Chen type, Rocky Mountain J. Math., 2008, 38, 567-581 http://dx.doi.org/10.1216/RMJ-2008-38-2-567 · Zbl 1195.53072
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