6
$\begingroup$

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.


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.

$\endgroup$

1 Answer 1

16
$\begingroup$

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.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.