A Geometric Inequality for Warped Product Semi-slant Submanifolds of Nearly Cosymplectic Manifolds

Recently, we have shown that there do not exist warped product semi-slant submanifolds of cosymplectic manifolds [K.A. Khan, V.A. Khan and Siraj Uddin, Balkan J. Geom. Appl. 13 (2008), 55–65]. The nearly cosymplectic structure generalizes the cosymplectic one. Therefore the nearly Kaehler structure generalizes the Kaehler structure in almost Hermitian setting. It is interesting that the warped product semi-slant submanifolds exist in the nearly cosymplectic case while in the cosymplectic case they do not. In the beginning, we prove some preparatory results and finally we obtain an inequality such as ‖h‖2 ≥ 4q csc2 θ{1+ 1 9 cos2 θ}‖∇ ln f‖2 in terms of intrinsic and extrinsic invariants. The equality case is also considered.