Cosymplectic Manifold And It's Semi-invariant Submanifold

To find simple relationships between the main extrinsic invariants and the main intrinsic invariants of a submanifold is one of the natural interests of the submanifold theory.1990s, B.Y.Chen proves that there exists a basis inequality on Ricci tensor S and thesquared mean curvature for any submanifold M~n in complex manifold M~m(c)-Chen'inequality.Furthermore, many authors extend the Chen' inequality to other space forms. And a Cosymplectic manifold is a important class of almost contact metric manifolds, in the present paper, we would like to introduce Cosymplectic manifold and it's semi-invariant submanifold, and extend the Chen' inequality to semi-invariant submanifold of Cosymplectic manifold, we prove:Theorem 1 Let M be a Cosymplectic manifold with almost contact metric structure, then we havefor any tangent vector fields X, Y, Z, W, which orthogonal to , on M.Theorem 2 Let M be a Cosymplectic manifold and M a semi-invariant submanifold, then the following assertions are equivalent to each other:1) the distribution D is integrable;2) the distribution D is integrable;Theorem 3 Let M be a (n+l)-dimensional semi-invariant submanifold of a (2m+l)-dimensional Cosymplectic space form M(4c). Then:(i)For each unit vector X M orthogonal to and if we haveand if(ii)If H(p) = 0, then a unit tangent vector X orthogonal to satisfies the equality case of a) and b) if and only if X N_p.(iii)The equality case of a) or b) holds identically for all unit tangent vectors orthogonal to at p if and only if P is a totally geodesic point.