Consider real submanifolds, $M^n \subset \textbf{C}^n$. ($\textbf{C}^n \equiv (\textbf{R}^{2n}, J)$, where $J(x, y) = (-y, x)$). $M^n$ is totally real if $J(T_pM) \cap T_pM = 0$, for all $p \in M$.
Is $T(M)$ necessarily equivalent to the normal bundle of $M$ in $\textbf{R}^{2n}$, if $M$ is totally real?
Just to record here John Ma's comment (as an answer):
The normal bundle $NM$ of a submanifold $M$ in a manifold $X$ is the quotient $NM=i^*(TX)/TM$, where $i: M\to X$ is the inclusion map and $i^*$ is the pull-back map of bundles induced by $i$. Since $M\subset {\mathbb C}^n$ is totally real $J(T_mM)\cap T_mM=0$ for all $m\in M$, where $J$ is the almost complex structure. Therefore, the projection morphism $$ F: T_mM\to i^*(T_m{\mathbb C}^n)/T_mM, F(v):= [Jv] $$ is 1-1 for each $m\in M$. By the dimension count, $F$ is also surjective, hence, an isomorphism for each $m\in M$. Therefore, $F: TM\to NM$ is an isomorphism. qed