  Convexity properties of Weil-Petersson geodesics on the Teichm\"{u}ller space
of punctured Riemann surfaces are investigated. A normal form is presented for
the Weil-Petersson Levi-Civita connection for pinched hyperbolic metrics. The
normal form is used to establish approximation of geodesics in boundary spaces.
Considerations are combined to establish convexity along Weil-Petersson
geodesics of the functions the distance between horocycles for a hyperbolic
metric.
