Conformal geometry

Remarks

The angle between two tangent vectors at a point is the same for all metrics in $[g]$, so angles are defined in a conformal manifold, while lengths are not. Accordingly, conformal diffeomorphisms are exactly the diffeomorphisms that are conformal maps, that is, that preserve angles. Example: stereographic projection from the north pole identifies $S^n$ minus a point with $\R^n$, and the round metric becomes $\dfrac{4}{(1 + \abs{x}^2)^2} \abs{dx}^2$. This is $e^{2f}$ times the flat metric with $f(x) = \ln 2 - \ln(1 + \abs{x}^2)$, so the round sphere minus a point is conformally equivalent to Euclidean space. Non-example: the map $(x, y) \mapsto (2x, y)$ of $\R^2$ pulls back $dx^2 + dy^2$ to $4\, dx^2 + dy^2$, which is not a multiple of $dx^2 + dy^2$; it does not preserve the angle between the vectors $(1, 1)$ and $(1, -1)$. On an oriented surface, a conformal structure is the same thing as a structure of Riemann surface.