  We propose a criterion for finding the minimum distance at which an interior
solution of Einstein's equations can be matched with an exterior asymptotically
flat solution. It is based upon the analysis of the eigenvalues of the Riemann
curvature tensor and their first derivatives, implying $C^3$ differentiability
conditions. The matching itself is performed by demanding continuity of the
curvature eigenvalues across the matching surface. We apply the $C^3$ matching
approach to spherically symmetric perfect fluid spacetimes and obtain the
physically meaningful condition that density and pressure should vanish on the
matching surface. Several perfect fluid solutions in Newton and Einstein
gravity are tested.
