The simplest answer is that the curvature of space-time is a smoothly changing function of distance up to, and through, the event horizon. There is no indication from the curvature ( Riemann's Curvature Tensor, or even Ricci's for that matter) that anything serious is happening just inside the event horizon. What happens to the metric is another thing entirely. Einstein's 'equations' work just fine so long as the local curvature of space-time ( the strength of the gravitational field) does not become singular. This does not happen just inside an event horizon, but only happens as you approach the 'r=0' singularity itself. Math is our only guide.