  Let $X$ be a compact connected Riemann surface of genus at least two. The
Abel-Jacobi map $\varphi: {\rm Sym}^d(X) \rightarrow {\rm Pic}^d(X)$ is an
embedding if $d$ is less than the gonality of $X$. We investigate the curvature
of the pull-back, by $\varphi$, of the flat metric on ${\rm Pic}^d(X)$. In
particular, we show that when $d=1$, the curvature is strictly negative
everywhere if $X$ is not hyperelliptic, and when $X$ is hyperelliptic, the
curvature is nonpositive with vanishing exactly on the points of $X$ fixed by
the hyperelliptic involution.
