I have heard mentioned the following theorem:


If a Riemannian metric on a product of two Riemann surfaces has non-positive sectional curvature then the metric is a product metric.


And am trying to find a reference for it but have not found anything, any suggestions?



Non-positive metric on a product is a product metric
This is a very special case of Theorem 10.3.9 in Eberlein's book "Geometry of nonpositively curved manifolds".


