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A compatible almost complex structure is a smooth bundle map $J:TM\to TM$ such that, for all tangent vectors $u,v$,It determines the Riemannian metricIndeed, the invariance of the symplectic form under $J$ and its antisymmetry imply $\omega (v,Ju)=\omega (u,Jv)$, so $g$ is symmetric; the positivity condition makes it positive definite. A compatible triple consists of this symplectic form, this compatible almost complex structure, and this Riemannian metric. Useful equivalent identities are $\omega (u,v)=g(Ju,v)$ and $g(Ju,Jv)=g(u,v)$.

At a point $p$, write $W=T_{p}L$ and take the orthogonal complement with respect to the Riemannian metric $g$. The compatible triple identity givesIf $L$ is a Lagrangian submanifold, then $\omega (u,v)=0$ for $u,v\in W$, so $JW\subset W^{\perp }$. Both have dimension $\frac{1}{2}\dim M$, hence equality.

Conversely, if $JW=W^{\perp }$, equality of dimensions gives $\dim W=\frac{1}{2}\dim M$, and $g(Ju,v)=0$ for $u,v\in W$ gives $\omega {|}_W=0$. ThereforeA positive-definite inner product has $W\cap W^{\perp }=\{0\}$. Hence $W\cap JW=\{0\}$, proving that every Lagrangian submanifold is a totally real submanifold for a compatible almost complex structure.