A coordinate system without a metric doesn't tell you anything about spacetime curvature, do you mean the Schwarzschild metric expressed in Kruskal-Szekeres coordinates? And if so, do you mean the Schwarzschild metric for an eternal black hole, or just for a spherically symmetric mass like a planet? Not that I would know the answer either way myself, but if what you're asking is clarified then maybe someone else would. Where did you read/hear that spacetime was flat at the center, by the way?