On the type number of nearly-cosymplectic hypersurfaces in nearly-K\"ahlerian manifolds

M. B. Banaru

Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 2, pp. 357-364.

Résumé

Nearly-cosymplectic hypersurfaces in nearly-Kählerian manifolds are considered. The following results are obtained. Theorem 1. The type number of a nearly-cosymplectic hypersurface in a nearly-Kählerian manifold is at most one. Theorem 2. Let $\sigma$ be the second fundamental form of the immersion of a nearly-cosymplectic hypersurface $(N,\{\Phi,\xi,\eta,g\})$ in a nearly-Kählerian manifold $M^{2n}$. Then $N$ is a minimal submanifold of $M^{2n}$ if and only if $\sigma(\xi,\xi)=0$. Theorem 3. Let $N$ be a nearly-cosymplectic hypersurface in a nearly-Kählerian manifold $M^{2n}$, and let $T$ be its type number. Then the following statements are equivalent: 1) $N$ is a minimal submanifold of $M^{2n}$; 2) $N$ is a totally geodesic submanifold of $M^{2n}$; 3) $T\equiv0$.

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