  The Weil-Petersson metric for the moduli space of Riemann surfaces has
negative sectional curvature. Surfaces represented in the complement of a
compact set in the moduli space have short geodesics. At such surfaces the
Weil-Petersson metric is approximately a product metric. An almost product
metric has sections with almost vanishing curvature. We bound the sectional
curvature away from zero in terms of the product of lengths of short geodesics
on Riemann surfaces. We give examples and an expectation for the actual
vanishing rate.
