Tangent Bundle of a Riemann Surface

Question

Let $g$ be the genus of a closed Riemann surface, what can be said about $g$ if the tangent bundle $T$ of that surface is trivial?

From the formula for the degree of a tangent bundle, $\deg(T)=2-2g$, I guess I can see that it's true for $g=1$. Is that it or am I understanding something wrong here?

Answer 1

Just to make the above observations into an answer:

The genus $g(M)$ of a closed (oriented) Riemann surface $M$ is related to the Euler characteristic $\chi(M)$ by the formula $2-2g(M) = \chi(M)$. Moreover for oriented manifolds, the Euler characteristic is a characteristic class, which means that in particular $\chi(M) = \chi(TM) = 0$ if $TM$ is a trivial bundle. Hence in your situation you see, that the only closed oriented Riemann surface with trivial tangent bundle is the torus $T^2$.