Even-dimensional manifolds $N$ structured by a ${\cal T}$-parallel connection have been defined and studied in [DR], [MRV]. In the present paper, we assume that $N$ carries a (1,1)-tensor field $J$ of square $-1$ and we consider an immersion $x:M\rightarrow N$. It is proved that any such $M$ is a CR-product [B] and one may decompose $M$ as $% M=M_D\times M_{D^{\perp }}$, where $M_D$ is an invariant submanifold of $M$ and $M_{D^{\perp }}$ is an antiinvariant submanifold of $M$. Some other properties regarding the immersion $x:M\rightarrow N$ are discussed.
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