Riemannian compact six-dimensional manifolds Ricci-flat

Question

Are there Real compact six-dimensional manifolds Real Ricci-flat?

It is known that Calabi-Yau manifolds exist, that is, Complex compact three-dimensional Ricci-flat, but I don't know if Real compact six-dimensional manifolds exist.

If yes, can you provide me with examples or alternatively references?

Can you provide me also with examples of non-compact six-dimensional manifolds Ricci-flat but not Riemann-flat?

Answer 1

As stated in the comments, any Calabi--Yau three-fold is a six-dimensional (real) Ricci flat manifold. Some examples are discussed by S. Roan and S.-T. Yau in their paper On Ricci flat 3-fold.

Here are two three "easy" examples in the compact case:

1. A six-dimensional torus $T^6$.
2. The Riemannian product $T^2\times K^4$ of a two-dimensional torus and a K3 surface.
3. A Kummer threefold (see Example I in Roan–Yau).

Note that a K3 surface and a Kummer threefold have positive Euler characteristic, and hence, by the Gauss–Bonnet–Chern theorem, are not flat. Therefore the last two examples are not flat.

It also follows that the Riemannian product $\mathbb{R}^2 \times K^4$ is a noncompact six-dimensional Ricci flat manifold which is not flat.