Recently, Arslan et al. [K. Arslan, R. Ezentas, I. Mihai, C. Murathan, J. Korean Math. Soc., 42 (2005), 1101-1110] studied contact CR-warped product submanifolds of the form M T × f M ⊥ of a Kenmotsu manifold M, where M T and M ⊥ are invariant and anti-invariant submanifolds of M, respectively. In this paper, we study the warped product submanifolds by reversing these two factors, i.e., the warped products of the form M ⊥ × f M T which have not been considered in earlier studies. On the existence of such warped products, a characterization is given. A sharp estimation for the squared norm of the second fundamental form is obtained, and in the statement of inequality, the equality case is considered. Finally, we provide two examples of non-trivial warped product submanifolds.
This work examines some classical results of Bertrand curves for timelike ruled and developable surfaces using the E. Study map. This provides the ability to define two timelike ruled surfaces which are offset in the sense of Bertrand. It is shown that every timelike ruled surface has a Bertrand offset if and only if an equation should be satisfied between the dual geodesic curvatures. Some new results and theorems related to the developability of the Bertrand offsets of timelike ruled surfaces are also obtained.
<abstract><p>In this work, we introduce a line congruence as surface in the space of lines in terms of the E. Study map. This provides the ability to derive some formulae of surfaces theory into line spaces. In addition, the well known equation of the Plucker's conoid has been obtained and its kinematic-geometry are examined in details. At last, an example of application is investigated and explained in detail.</p></abstract>
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