Let \(\tilde M(\phi,\xi,\eta,g)\) be an odd dimensional \(C^{\infty}\)- manifold whose structure tensors \(\phi,\xi,\eta\) are a (1,1)-tensor field, a vector field and 1-form respectively. If these tensors satisfy \(\phi^ 2=-Id+\eta \otimes \phi,\) \(\phi\xi =0,\) \(\eta \circ \phi =0,\) \(\eta (\xi)=1\) \(g(\phi X,\phi Y)=g(X,Y)-\eta(X)\eta (Y),\) \(\eta (X)=g(X,\xi)\) \(({\tilde \nabla}_ X\phi)Y+(\nabla_ y\phi)X=0\) then \(\tilde M(\phi,\xi,\eta,g)\) is defined as a nearly cosymplectic manifold [D. E. Blair, Pac. J. Math. 39, 285-292 (1971; Zbl 0239.53031)].
If on a submanifold M of \(\tilde M\) there exist two differentiable orthogonal distributions D and \(D^{\perp}\) such that: (i) \(TM=D\oplus D^{\perp}\oplus \{\xi\}\), (ii) D is invariant by \(\phi\), i.e. \(\phi (D)=D\), (iii) \(D^{\perp}\) is anti-invariant by \(\phi\), i.e. \(\phi(D^{\perp})\subset T^{\perp}M\). \((T^{\perp}M:\) the normal bundle to M), then the author calls M a CR-submanifold of \(\tilde M.\) Conditions for the integrability of \(D^{\perp}\) and \(D\oplus \{\xi \}\) are given in the present paper. We quote the simplest: The distribution \(D^{\perp}\) is integrable if and only if \[ A_{\phi y}z=A_{\phi z}y;\quad y,z\in \Gamma (D^{\perp}). \] In the above \(A_ V\) denotes the fundamental tensor of Weingarten with respect to the section \(V\in T'(T^{\perp}M)\) i.e. \({\tilde \nabla}_*V=-A_ VX+\nabla_ xV\).