Table of Contents

What is RicciFlatness?
A metric on a manifold assigns a length to any curve connecting a pair of points. Two nearby points with separation $dx^i$ ($i=1,2,\ldots,d$) are assigned a distance $ds$ defined by
(1)where the Einstein summation convention is assumed. We usually refer to $g_{ij}$ as the metric. The matrix inverse of $g_{ij}$ is written as $g^{ij}$.
The Ricci tensor is a secondrank symmetric tensor obtained from the Riemann curvature tensor by contracting a pair of indices $R_{ij} = g^{kl} R_{ikjl}$. In 3 dimensions, the Ricci tensor is sufficient to reconstruct the Riemann curvature tensor. Thus the vanishing of the Ricci tensor in 3 dimensions implies the vanishing of the Riemann curvature tensor. This is not true in 4 and higher dimensions. A manifold is called Ricciflat if the Ricci tensor vanishes at all points on the manifold.
Uses of RicciFlat manifolds
In string theory, manifolds that admit metrics for which the Ricci tensor vanishes play a special role. Such metrics are called RicciFlat. Here are some applications of such manifolds.
 Superstring theory requires spacetime to be tendimensional. However, spacetime as we perceive it at low energies is fourdimensional. A simple and effective way to get around this is to assume that six of the dimensions are compact and small enough to be invisible at low energies. String compactification assumes that spacetime is assumed to be of the form $\mathbb{R}^{1,3}\times M^{(6)}$, where $M$ is a compact sixdimensional manifold. Consistency of string propagation (conformal invariance) requires $M$ to be RicciFlat to leading order.
 Noncompact RicciFlat manifolds make an appearance in the context of the AdSCFT correspondence (more generally, the gravitygauge correspondence). In its simplest form, this correspondence relates type IIB string theory on a spacetime $AdS_5\times X^{(5)}$ to a fourdimensional superconformal field theory (CFT). Let $M^{(6)}$ be a noncompact sixdimensional manifold obtained as a cone over $X^{(5)}$ i.e., consider a sixdimensional metric obtained from the fivedimensional metric on $X$ (which we write as $ds^2_X$):
Again, consistency of string propagation on $AdS_5\times X^{(5)}$ translates into the condition that $M$ be RicciFlat.
 RicciFlat metrics appear as the fixedpoints of the dynamical system, called RicciFlow
where $g_{ij}$ are the components of the metric, i.e., $ds^2 = g_{ij}dx^idx^j$. This is an area being actively pursued in mathematics. The proof of the Poincare conjecture by Perelman makes use of this dynamical system.
This page is devoted to explicit examples of RicciFlat metrics, typically but not necessarily in sixdimensions. The methods used fall into two categories:
 Analytical: Here one produces an analytical expression for the metric and one can verify that the Ricci tensor vanishes. This is expected to work only for noncompact examples.
 Numerical: Here one numerically solves the nonlinear partial differential equation implied by the vanishing of the Riccitensor — this is the only method that works for compact manifolds. It appears to be necessary for some noncompact examples as well.
Extensions to RicciFlatness
There are two different generalisations or extensions of Ricciflatness that we will describe below:
 In string theory, RicciFlatness of the manifold $M$ is only the condition for conformal invariance at leading order. It obtains corrections, the socalled $\alpha'$ corrections as well as stringloop corrections. For instance, it has been shown that given a RicciFlat metric on a Kahler manifold, it is possible to find a metric, order by order in $\alpha'$, that satisfies the condition for conformal invariance. The argument is a formal one and not too many explicit examples exist for which this has been carried out. However, see HigherOrder Corrections to NonCompact CalabiYau Manifolds in String Theory by Lu, Pope and Stelle for some explicit examples.
 In the context of string theory, the metric is one of several fields in theory. For instance, there is a secondrank antisymmetric tensor, called the $B$field, $B=\tfrac12 B_{ij}dx^i\wedge dx^j$, a scalar called the dilaton, $\Phi$, that is common to all string theories. There are also other $p$form gauge fields,$C^{(p)}=\tfrac1{p!}C_{i_1\ldots i_p}\ dx^{i_1}\wedge \cdots \wedge dx^{i_p}$, that appear. So the condition for conformal invariance gives rise to a much more complicated system of coupled partial differential equations involving all these fields. In the limit that these fields vanish or take constant values, on recovers the condition that the metric must be Ricciflat. The term manifolds is typically used for spaces that are Riemannian and have a metric. A generalised manifold can be defined to be a space with the various massless fields of string theory and a generalised geometry on them is given by imposing the conditions for conformal invariance of string theory. Clearly, a string theorist like me would be interested in understanding these generalised geometries.