Assume given a smooth manifold $(\mathbb{R}^n, g)$, where the metric is a scaled identity $g = e^{2f}I$. Is there a way to know if this is always a non-positive (sectional) curvature manifold?
Note this is a parametrized manifold that is locally conformally flat. Following Einstein Manifolds [Arthur L. Besse], the Ricci tensor (in coordinates), can be shown to be:
$R = -(n-2)(H_f - \nabla f \cdot \nabla f^T ) - \frac{n-2}{n}(\Delta f + \|\nabla f\|^2)I_{n\times n}$
where $H_f$ is the Hessian of $f$.
Then $(\mathbb{R}^n, g)$ is of non-positive (sectional) curvature if $R$ is negative semi-definite.