Can anyone give an example of Ricci flat Riemannian or Lorentzian Manifold that is not flat?

Question

Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat?

I am especially interset in the case of Lorentzian Manifold whose sign signature is (- ,+ ,+ , + ). Of course, the example is not constricted in Lorentzian case.

I know there are many Ricci flat case in General Relativiy which is the vacuum solution to Einstein's equation. But what I know, such as Kerr solution, are all geodesic incomplete. So I want a geodesic complete example and reference. Thanks!

Answer 1

All riemannian manifolds with holonomy contained in $SU(n) \subset SO(2n)$, $Sp(n) \subset SO(4n)$, $G_2 \subset SO(7)$ and $Spin(7) \subset SO(8)$ are Ricci-flat. There are plenty of non-flat examples; e.g., those with holonomy precisely those groups.

In the Lorentzian setting, you could consider a subclass of lorentzian symmetric spaces with solvable transvection group, the so-called Cahen-Wallach spacetimes. They are pp-waves with metric given in local coordinates by $$ 2 dudv + \sum_{i=1}^{n-2} dx_i^2 + \sum_{i,j=1}^{n-2} A_{ij} x^i x^j du^2 $$ where $A_{ij}$ are the entries of a symmetric matrix. If $A$ is traceless, the metric is Ricci-flat, but if $A \neq 0$ then it is non-flat.

Added (in response to the comment)

The riemannian result is classical. I learnt this from a book by Simon Salamon, “Riemannian Geometry and Holonomy Groups”, but there are some more recent lecture notes you might find useful: http://arxiv.org/abs/1206.3170. Concerning the lorentzian result, it is a simple calculation to determine the Riemann and Ricci tensors of the metric I wrote down. The metrics were introduced in the paper “Lorentzian symmetric spaces” by Michel Cahen and Nolan Wallach, Bull. Am. Math. Soc. 76 (1970), 585–591.

Answer 2

Your claim that "all solutions from general relativity are geodesically incomplete" is not true. The classical Schwarzschild/Kerr black hole solutions are geodesically incomplete, and along with Minkowski space (which is flat), these are the most well known explicit metrics.

However, many solutions to the (matterless) Einstein equations are geodesically complete. This follows, for example, from the work of Christodoulou-Klainerman http://press.princeton.edu/titles/5159.html, which (very roughly) says that for a sufficiently small appropriate perturbation of $(\mathbb{R}^3,\delta)$, using this as initial data for the Einstein equations yields a Lorentzian Ricci flat manifold which is geodesically complete (and much more: they showed it was asymptotic to Minkowski space in some sense). A nice discussion is given here.

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Continuing the theme of physically related Ricci flat metrics, here is an interesting (explicit) Riemannian example:

The Lorentzian Schwarzschild metric (in $4$-dimensions) is incomplete, as you say. However, a strange observation is that formally setting $\tau = it$ (known to the physicists as "Wick rotation") yields a Riemannian manifold which is Ricci flat. The amazing thing is that this metric turns out to be complete, as long as $\tau$ is considered to be periodic with the appropriate period.

This gives a complete Ricci flat metric on $S^2\times\mathbb{R}^2$.

See, for example, section 2 of http://arxiv.org/pdf/hep-th/9112065v1.pdf. Or you can search for "Euclidean black hole" or "Schwarzschild instanton" for more physics literature.

Answer 3

For an example, let $N$ be a compact complex hypersurface of degree $m+1$ of the complex projective space $\mathbb{CP}^m$ with complex dimension $m\ge 3$ (for $m=3$ this is a complex $K3$ surface). The first Chern class of $N$ vanishes, and hence $N$ admits a Ricci-flat but nonflat Riemannian metric, by a theorem of Yau.

Answer 4

Here's another explicit reference, on top of Ben Crowell's more general comment. The following paper discusses explicit examples of "pp-wave" spacetimes (Lorentzian, solving vacuum Einstein equations) that are geodesically complete: Causal structures of pp-waves by Veronika E. Hubeny and Mukund Rangamani (arXiv:hep-th/0211195).

Answer 5

Eguchi - Hanson metric over $T^*S^2$ is complete Ricci flat but not flat, which can be written down explicitly. In fact, $T^*S^n$ admits Calabi-Yau structure for each $n$.

Ref: Stenzel, Ricci flat metrics on the complexification of a compact rank one symmetric space.