Let $f:M\rightarrow N$ be differential map. We denote tangent map $$f_*:TM\rightarrow TN$$ and cotangent map $$f^*:T^*N\rightarrow T^*M$$
Now let $M$, $N$ be Riemannian manifolds, and $i:M\rightarrow N$ be an isometric immersion.
In my textbook, it consider the $$i^*TN=TM\oplus T^\perp M$$
I think it has some mistakes, and the right form is $$TN=i_*TM\oplus T^\perp M$$
Can someone help me clarify the confusion?