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Chen’s inequality in the Lagrangian case. (English) Zbl 1118.53035
To establish the relationships between the main intrinsic invariants and the main extrinsic invariants of submanifolds, B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken proved that every totally real submanifolds $$M$$ of real dimension $$n$$ in a complex space form $$N(c)$$ of real dimension $$2m$$ satisfies Chen’s inequality $\delta_M\leq {{n-2}\over{2}}\left\{{{n^2}\over{n-1}}\| H\|^2+(n+1){{c}\over{4}}\right\},$ where $$H$$ is the mean curvature vector of $$M$$.
In this paper, the author proves this result in Lagrangian case
Theorem. Let $$M$$ be a Lagrangian submanifold in a complex space form $$N(c)$$ of real dimension $$2n$$, $$n\geq 3$$. Then $\delta_M\leq {(n-2)(n+1)\over{2}}{c\over 4}+{n^2\over 2}{{2n-3}\over{2n+3}}\|H\|^2.$

##### MSC:
 53C40 Global submanifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds
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