Conformal structure ($[g]=\{e^{2u}g:u\in C^{\infty }(M)\}$)

An equivalence class of Riemannian metrics under multiplication by positive smooth functions. Such rescalings preserve angles. On an oriented smooth surface, a conformal structure is equivalent to a complex structure, with local holomorphic coordinates in which its metrics are positive multiples of $d{x}^2+d{y}^2$.