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In this paper, we consider the class of square metrics F = α+ 2β + β/α where α = √ aijyy is a Riemannian metric and β = bi(x)y is a one-form on a manifold M . Let (M,F ) be a Douglas-square manifold. We show that F is a Berwald metric if and only if it a weakly stretch metric. It results that, a Douglas-square metric is R-quadratic if and only if it is a Berwald metric.

Douglas-Square Metrics with Vanishing Mean Stretch Curvature