Manifold with Ricci curvature not bounded below

Question

Could somebody please show an example (or give a reference to one) of a connected complete Riemannian manifold whose Ricci curvature is not bounded by below?

I guess there are standard examples to this, somewhere...

Answer 1

Consider $g=dr^2+ f(r)^2 d\theta^2$ on $\mathbb{R}^2$ So if we have suitable sequences $x_n,\ y_n$ and $f$ s.t. $$ x_n<y_n<x_{n+1},\ f(x_n)>f(y_n)<f(x_{n+1}) $$

then Gaussian curvatures around $r=y_{n}$ go to $-\infty$