Sorry if this question seems too trivial for this forum.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Does a compact manifold always have bounded sectional curvature?

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