  The Teichm\"{u}ller curve is the fiber space over Teichm\"{u}ller space of
closed Riemann surfaces, where the fiber over a point in Teichm\"{u}ller space
is the underlying surface. We derive formulas for sectional curvatures on the
Teichm\"{u}ller curve. In particular, our method can be applied to investigate
the geometry of the Weil-Petersson geodesic as a three-manifold, and the
degeneration of the curvatures near the infinity of the augmented
Teichm\"{u}ller space along a Weil-Petersson geodesic, as well as the
minimality of hyperbolic surfaces in this three-manifold.
