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A symplectic surface is a two-dimensional symplectic manifold. Every embedded curve in it is a Lagrangian submanifold.

For a smooth function $M(q)$, its differential defines a Lagrangian submanifold of the cotangent bundle: pullback of the canonical one-form on a cotangent bundle is $dM$, so pullback of the symplectic form $-d\theta$ is zero. The graph has half the ambient dimension. With $q=(S,V,N)$ and momentum coordinates $(T,-P,\mu )$, the first law of thermodynamics identifies the equilibrium graph with $dM$. This geometric statement does not require extensivity or thermodynamic stability.

In the maximal-dimension convention, a totally real submanifold of $(M,J)$ has half the real dimension of $M$ and $T_{p}L\cap J(T_{p}L)=0$. Every Lagrangian submanifold is totally real for a compatible almost complex structure, but the converse fails in complex dimension at least two.

With $\omega =-d\alpha$ and ${\omega }_{\sigma }=\omega +{\pi }^{\ast }\sigma$, the one-form graph of $\theta$ is a Lagrangian submanifold exactly when $d\theta =\sigma$. Indeed its pulled-back symplectic form is $\sigma -d\theta$ and its dimension is half that of the total space. This extends graph of a closed one-form is Lagrangian.

A neighborhood of a compact Lagrangian submanifold $L\subset (M,\omega )$ is symplectomorphic, by a map restricting to the identity on $L$, to a neighborhood of the zero section in $(T^{\ast }L,d\lambda )$, with the sign chosen to match the convention for the canonical form.