  Fix a number $g>1$, let $S$ be a close surface of genus $g$, and $Teich(S)$
be the Teichm\"uller space of $S$ endowed with the Weil-Petersson metric. In
this paper we show that the Riemannian sectional curvature operator of
$Teich(S)$ is non-positive definite. As an application we show that any twist
harmonic map from rank-one hyperbolic spaces $H_{Q,m}=Sp(m,1)/Sp(m)\cdot Sp(1)$
or $H_{O,2}=F_{4}^{-20}/SO(9)$ into $Teich(S)$ is a constant.
