Conformal class of a Riemannian metric ($[g]=\{e^{2\phi }g:\phi \in C^{\infty }(M,\mathbb{R})\}$)

Two Riemannian metrics lie in the same conformal class when one is a positive smooth scalar multiple of the other. They measure the same angles, while lengths may change from point to point. On an oriented Riemannian surface, rotation of each tangent plane through $\pi /2$ determines a complex structure which depends only on this conformal class.