Is a Ricci flat metric on $\mathbb {R}^n$ flat?

Question

We know that any Ricci flat metric on $T^n$ must be flat by Cheeger-Gromoll splitting theorem. I am curious whether it is true for $\mathbb R^n$. I feel like I may miss something and there should be a simple answer.

Answer 1

The answer is no in all dimensions $n\ge 4$, see for instance,

LeBrun, Claude, Complete Ricci-flat Kähler metrics on $\mathbb{C}^n$ need not be flat, Several complex variables and complex geometry, Proc. Summer Res. Inst., Santa Cruz/CA (USA) 1989, Proc. Symp. Pure Math. 52, Part 2, 297-304 (1991). ZBL0739.53053.

for the case of even $n$; the odd-dimensional case is obtained by multiplying by the real line.