Sorry if this question seems too trivial for this forum. A grad student at my university told me that a compact Riemannian manifold always has lower and upper curvature bounds. Is this really true? The problem seems to be that I don't fully understand the curvature tensor's continuity etc. What makes me a little skeptical is that I already spent quite a lot of time trying to find a source where this is explicitly stated, without any success. Usually I would expect such a statement as basic as this to appear in lots of books or lecture notes.