## On Chen invariant of CR-submanifolds in a complex hyperbolic space.(English)Zbl 1129.53302

Recently B. Y. Chen [Jap. J. Math., New Ser. 26, No. 1, 105–127 (2000; Zbl 1026.53009)] introduced an invariant for a Riemannian manifold and obtained a sharp inequality between his invariant and the squared mean curvature for a CR-submanifolds $$M$$ in real spaceforms.
In the present paper, by applying the Chen invariant the author has obtained some interesting new results for CR-submanifolds which satisfy $\delta(n_1, ...,n_k) = c(n_1, ..., n_k)H^2 - b(n_1, ..., n_k) -3n + \frac{3}{2} \Sigma^k_{i = 1} n_i$ where $$\delta (n_1, ..., n_k)$$ and $$H^2$$ are the Chen invariant and the square of the mean curvature of $$M$$.

### MSC:

 53C40 Global submanifolds

### Keywords:

submanifold; complex hyperbolic space; invariants

Zbl 1026.53009
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