almost-complex structure on $S$. If $g_1=\lambda g$ is another, conformally equivalent to $g$ metric, then this construction leads to the same almost-complex structure.*

Vice-versa, given $J$ on $S$ , denote by $J^{\cc }$ the $\cc $-linear extension of $J$ onto the complexified tangent bundle $T^{\cc }S:=TS\otimes
\cc $. Here $J^{\cc }$ is defined by $J^{\cc }(v\otimes \lambda )=Jv\otimes
\lambda $ for $v\in TS$ and $\lambda \in \cc $. Take an eigenvector $w$ of $J^{\cc }$ in $T^{\cc }S$ with an eigenvalue $-i$. Write $w=u+iv$, with $u,v\in TS$ (more accurately $w=u\otimes 1 + v\otimes i$), and put $g(u,v)=0,
g(u)=g(v)=1$. This defines a Riemannian metric, which corresponds to $J$. The statement $J^{\cc }w=-iw=v-iu$ means that $Ju=v, Jv=-u$.

The freedom in choosing $w$ here permits us to define $g$ up to a real factor, , $J$ corresponds to a class of conformally equivalent metrics.

Exercise. Prove the last assertion. More precisely: if $\lambda w$ is another eigenvector of $J^{\cc }$ with an eigenvalue $-i$ and a metric $g_{\lambda }$ is constructed from $\lambda w$ as $g$ from $w$ above, then $g=\vert \lambda \vert^2g_{\lambda }$.

We shall see later in [*Corollary 2.2.2*]{} that any almost-complex structure on the Riemann surface is complex.

Definition 1.1.3.

