Riemann Surfaces by Donaldson

introduction

• 113p
• 160p meaning of complex structure

Now suppose that \(\Sigma\) is a Riemann surface. Each tangent space is a one-dimensional complex vector space, and a Riemannian metric on \(\Sigma\) is compatible with the Riemann surface structure if multiplication by \(i\) preserves lengths of tangent vectors. Another way of expressing this is that at a point of a Riemann surface the angle between tangent vectors is intrinsically defined, and the compatibility condition is that this is the same as the angle furnished by the Euclidean geometry of the tangent space. In terms of a local holomorphic co-coordinate \(z=x+iy\), the compatibility condition is that the metric takes the form \[ ds^2=V(dx^2+dy^2), \] for a positive function \(V\), i.e. the matrix \(\left( \begin{array}{cc} g_{1,1} & g_{1,2} \\ g_{2,1} & g_{2,2} \end{array} \right)\) is \(V\) times the identity matrix.

• 182p