Abstract
In this paper we investigate $(n+1)(n \geq 5)$-dimensional contact $CR$-submanifolds $M$ of $(n-1)$ contact $CR$-dimension in a $(2m+1)$-dimensional unit sphere $S^{2m+1}$ which satisfy the condition $h(FX,Y)-h(X,FY)=g(FX, Y)\zeta $ for a normal vector field $\zeta$ to $M$, where $h$ and $F$ denote the second fundamental form and a skew-symmetric endomorphism (defined by (2.3)) acting on tangent space of $M$, respectively..
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