Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a $C^2$-function and let $H=\left(\frac{\partial^2f}{\partial x_i \partial x_j}\right)_{1\le i,j\le n}$ be its Hessian matrix. Suppose I know that $ \det H(x_1,\ldots,x_n)\ge 0$ for all $x=(x_1,\ldots,x_n)$.
Does this have any geometric meaning for $f$?
e.g., when $n=1$, this means that $f$ is convex. This no longer holds for $n\ge 2$, but when $f$ is convex the Hessian determinant is certainly positive, so perhaps one could wonder if a weaker property holds.