We extend the Chern-Heinz inequalities to graphs of C 2-functions defined on open subsets of Riemannian manifolds. We generalize results about foliations of 3-dimensional Riemannian manifolds by constant mean curvature surfaces due to Barbosa-Kenmotsu-Oshikiri [3], Barbosa-Gomes-Silveira [2].

We study the forms of curvatures of lightlike hypersurfaces M of an indefinite Kenmotsu manifold M̄ subject to the conditions: (1) M is locally symmetric, i.e., the curvature tensor R of M be parallel on TM , or (2) M is a semi-symmetric manifold, i.e., R(X, Y )R = 0 on TM . M.S.C. 2010: 53C25, 53C40, 53C50.

nowadays “design” is considered as an independent discipline which is neither a science, nor an art, but an interdiscipline with strong relations with both art and science. therefore, this new understanding demands reidentifying all design sub-disciplines. “space” as one of this sub-disciplines, has a pivotal role in shaping form in both design and architecture. while concrete understanding of ...

let $(m^{2n},g)$ be a real hypersurface with recurrent shapeoperator and tangent to the structure vector field $xi$ of the sasakian space form$widetilde{m}(c)$. we show that if the shape operator $a$ of $m$ isrecurrent then it is parallel. moreover, we show that $m$is locally a product of two constant $phi-$sectional curvaturespaces.