On the one hand the determinant (technically the absolute value of the determinant) represents the "volume distortion" experienced by a region after being transformed.
On the other hand the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space.
Is there a link between the two? Can the determinant of a matrix be interpreted as a scalar curvature? If I makes a difference, I am mostly interested in the context of general relativity.