Here is a silly mistake I am making: where exactly is the mistake?
I know that torus cannot hold a metric of constant curvature -1 ( hyperbolic metric ).
But what if I do this:
The upper half-plane and $\mathbb{C}$ are diffeomorphic by a diffeomorphism, say $\phi$; pull the hyperbolic metric from the upper half plane to the complex plane $\mathbb{C}$, and then quotient it by $ Z\oplus Z$ to get a hyperbolic metric on the torus. Impossible! But what am I missing?