Let $N\in\mathbb N$ and let $M\subset\mathbb C^N$ be a smooth embedded real submanifold of real codimension $k$, with $0\le k\le N$, equipped with the smooth structure induced from the embedding. Let $J:T\mathbb C^N\to T\mathbb C^N$ be the standard complex structure, and assume that, for every $p\in M$, one has $T_pM+J(T_pM)=T_p\mathbb C^N$. Let $J$ also denote its complex-linear extension to $T_p\mathbb C^N\otimes_{\mathbb R}\mathbb C$, and let $T_{1,0,p}\mathbb C^N$ and $T_{0,1,p}\mathbb C^N$ denote the $i$-eigenspace and $-i$-eigenspace of this extension. For each $p\in M$, define $T_{1,0,p}M:=T_{1,0,p}\mathbb C^N\cap (T_pM\otimes_{\mathbb R}\mathbb C)$. Then $T_{1,0}M=\bigsqcup_{p\in M}T_{1,0,p}M$ is a smooth complex subbundle of $TM\otimes_{\mathbb R}\mathbb C$ of complex rank $N-k$, satisfies $T_{1,0}M\cap \overline{T_{1,0}M}=\{0\}$, where the bar denotes fiberwise complex conjugation in $TM\otimes_{\mathbb R}\mathbb C$, and is formally integrable, meaning that for every [open set](/page/Open%20Set) $U\subset M$ and all smooth local sections $X,Y\in\Gamma(U,T_{1,0}M)$, where $\Gamma(U,E)$ denotes the space of smooth sections of a smooth vector bundle $E\to U$, one has $[X,Y]\in\Gamma(U,T_{1,0}M)$. Hence $T_{1,0}M$ defines an abstract CR structure on $M$.