Submanifolds of an even-dimensional manifold structured by a $\mathcal T$-parallel connection

Even-dimensional manifolds $N$ structured by a $\mathcal T$-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 $x : M\to N$. 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 $x:M\to N$ are discussed.