Is there a smooth (noncompact) manifold $M$ such that for any Riemannian metric $g$ on $M$ there are $p\_i$ and unit tangent vectors $v\_i \in T\_{p\_i}M$ such that $Ricc(g)|\_{p\_i}(v\_i,v\_i) \leq -i$? This seems unlikely, but I'm not sure how to prove it.




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Alternatively, what is the simplest example of a fixed $(M,g)$ with no lower Ricci bounds in the sense above? It seems that some conformal change of the standard Euclidean metric could accomplish this, but I dont see a simple way to do this.



Do manifolds with no Ricci lower bounds for any metric exist?
Any smooth manifold $M$ admits a metric with quadratic curvature decay.
In fact there is a complete Riemannian metric $g$ on $M$ such that $$|K\_x|=o(|x\_0-x|\_g^{-2}),$$
here $|K\_x|$ is absolute bound for sectional curvature of $g$ at point $x$,
$x\_0$ is a fixed point and and $|x\_0-x|\_g$ denotes distance induced by $g$.


In particular there is no examples which you are looking for.


See p. 96 in Gromov M., Volume and bounded cohomology, Publ. Math. IHES 56 (1982) 5–99.


