I think it would be more accurate to say that the real reason why
Calabi-Yau, hyperkähler, $G_2$ and $\mathrm{Spin}(7)$ manifolds are of
interest in string theory is not their Ricci-flatness, but the fact
that they admit parallel spinor fields. Of course, in
positive-definite signature, existence of parallel spinor fields
implies Ricci-flatness, but the converse is still open for compact
riemannian manifolds, as discussed in this
question, and known to fail for noncompact manifolds as pointed
out in an answer to that question.
The similar question for lorentzian manifolds has a bit of history.
First of all, the holonomy principle states that a spin manifold
admits parallel spinor fields if and only if (the spin lift of) its
holonomy group is contained in the stabilizer subgroup of a nonzero
spinor. Some low-dimensional (i.e., $\leq 11$, the cases relevant to
string and M-theories) investigations (by Robert Bryant and myself,
independently) suggested that these subgroups are either of two types:
subgroups $G < \mathrm{Spin}(n) < \mathrm{Spin}(1,n)$, whence $G$ is
the ones corresponding to the cases 5-8 in the question, or else $G =
H \ltimes \mathbb{R}^{n-1}$, where $H < \mathrm{Spin}(n-1)$ is one of
the groups in cases 5-8 in the question. Thomas Leistner showed that
this persisted in the general case and, as Igor pointed out in his
answer, arrived at a classification of possible lorentzian holonomy
groups. Anton Galaev then constructed metrics with all the possible
holonomy groups, showing that they all arise. Their work is reviewed
in their
paper (MR2436228).
The basic difficulty in the indefinite-signature case is that the de
Rham decomposition theorem is modified. Recall that the de Rham
decomposition theorem states that if $(M,g)$ is a complete, connected
and simply connected positive-definite riemannian manifold and if the
holonomy group acts reducibly, then the manifold is a riemannian
product, whence it is enough to restrict to irreducible holonomy
representations. This is by no means a trivial problem, but is
tractable.
In contrast, in the indefinite signature situation, there is a
modification of this theorem due to Wu, which says that it is not
enough for the holonomy representation to be reducible, it has to be
nondegenerately reducible. This means that it is fully reducible
and the direct sums in the decomposition are orthogonal with respect
to the metric. This means that it is therefore not enough to restrict
oneself to irreducible holonomy representations. For example,
Bérard-Bergery and Ikemakhen proved that the only lorentzian holonomy
group acting irreducibly is $\mathrm{SO}_0(1,n)$ itself: namely, the
generic holonomy group.
It should be pointed out that in indefinite signature, the
integrability condition for the existence of parallel spinor fields is
not Ricci-flatness. Instead, it's that the image of the Ricci
operator $S: TM \to TM$, defined by $g(S(X),Y) = r(X,Y)$, with $r$ the
Ricci curvature, be isotropic. Hence if one is interested in
supersymmetric solutions of supergravity theories (without fluxes) one
is interested in Ricci-flat lorentzian manifolds (of the relevant
dimension) admitting parallel spinor fields. It is now not enough to
reduce the holonomy to the isotropy of a spinor, but the
Ricci-flatness equation must be imposed additionally.
In probability, time is usually handled as a nested sequence of $\sigma$-algebras (say $B_t$, with $B_t \subset B_s$ if $t\leq s$), and to find the reality (call the reality $f$, and it includes the state at all times past and future) at time $t$, one takes the conditional expectation $f_t := E[f | B_t ]$. The sequence $(f_t)$ is then a martingale (a uniformly integrable martingale, more precisely), and this construction is the essence of what the big deal is about martingales.
Brownian motion is a martingale that you've probably heard of, but this also handles simpler situations. For example, consider the experiment: toss a coin repeatedly, and keep track of how many heads you've thrown, minus how many tails. We can capture this experiment in the following way: For $0\leq x <1$, let $f_n(x)$ be the number of 1's minus the number of 0's among the first $n$ digits of the binary expansion of $x$, and let $B_n$ be the $\sigma$-algebra (in this case, a boolean algebra) generated by the intervals $[i/2^n,(i+1)/2^n)$. Then $f_n$ is $B_n$-measurable, and $E[f_t | B_s]=f_s$ for any natural numbers $s < t$, and the sequence $(f_n)_{n=1}^\infty$ is a martingale (albeit different from the type mentioned above). If you want to play any fair game on the "coin tosses" as they come up (allowing use of knowledge of all previous-in-time tosses), then your fortune at time $t$ is still a martingale.
In other words, the passage of time is captured as un-conditional-expectating a function.
For a practical introduction to martingales, I recommend Williams' "Probability with martingales." It is a marvel of writing, and in my humble opinion should be taken as a model for how to write a monograph.
Best Answer
Curvature in Mathematics and Physics (2012), by Shlomo Sternberg, based on an earlier book Semi-Riemann Geometry and General Relativity [free download from the author's website] covers much of the same material as O'Neill but is much more recent.