Does an estimate of the form $$Cg(X,X)\leq \operatorname{Ric}(X,X)\leq \frac{1}{C}g(X,X)$$ hold for all $X$ under appropriate assumptions? Thanks.
2026-04-12 13:30:34.1776000634
Is the Ricci tensor controlled by the metric?
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Your estimate would imply, that Ric$(X,X)\geq 0$. This isn’t always the case. However, the statement that $C_1 g(X,X) \leq Ric(X,X)\leq C_2 g(X,X,)$ for some (maybe negative) constants $C_1,C_2$ is true on any compact manifold.