<s>
In	O
the	O
mathematical	O
field	O
of	O
differential	B-Language
geometry	I-Language
,	O
Ricci-flatness	O
is	O
a	O
condition	O
on	O
the	O
curvature	O
of	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
.	O
</s>
<s>
Ricci-flat	B-Algorithm
manifolds	I-Algorithm
are	O
a	O
special	O
kind	O
of	O
Einstein	O
manifold	O
.	O
</s>
<s>
In	O
theoretical	O
physics	O
,	O
Ricci-flat	B-Algorithm
Lorentzian	O
manifolds	O
are	O
of	O
fundamental	O
interest	O
,	O
as	O
they	O
are	O
the	O
solutions	O
of	O
Einstein	O
's	O
field	O
equations	O
in	O
vacuum	B-General_Concept
with	O
vanishing	O
cosmological	O
constant	O
.	O
</s>
<s>
In	O
Lorentzian	O
geometry	O
,	O
a	O
number	O
of	O
Ricci-flat	B-Algorithm
metrics	I-Algorithm
are	O
known	O
from	O
works	O
of	O
Karl	O
Schwarzschild	O
,	O
Roy	O
Kerr	O
,	O
and	O
Yvonne	O
Choquet-Bruhat	O
.	O
</s>
<s>
In	O
Riemannian	O
geometry	O
,	O
Shing-Tung	O
Yau	O
's	O
resolution	O
of	O
the	O
Calabi	O
conjecture	O
produced	O
a	O
number	O
of	O
Ricci-flat	B-Algorithm
metrics	I-Algorithm
on	O
Kähler	O
manifolds	O
.	O
</s>
<s>
A	O
pseudo-Riemannian	O
manifold	O
is	O
said	O
to	O
be	O
Ricci-flat	B-Algorithm
if	O
its	O
Ricci	O
curvature	O
is	O
zero	O
.	O
</s>
<s>
It	O
is	O
direct	O
to	O
verify	O
that	O
,	O
except	O
in	O
dimension	O
two	O
,	O
a	O
metric	O
is	O
Ricci-flat	B-Algorithm
if	O
and	O
only	O
if	O
its	O
Einstein	O
tensor	O
is	O
zero	O
.	O
</s>
<s>
Ricci-flat	B-Algorithm
manifolds	I-Algorithm
are	O
one	O
of	O
three	O
special	O
type	O
of	O
Einstein	O
manifold	O
,	O
arising	O
as	O
the	O
special	O
case	O
of	O
scalar	O
curvature	O
equaling	O
zero	O
.	O
</s>
<s>
From	O
the	O
definition	O
of	O
the	O
Weyl	O
curvature	O
tensor	O
,	O
it	O
is	O
direct	O
to	O
see	O
that	O
any	O
Ricci-flat	B-Algorithm
metric	I-Algorithm
has	O
Weyl	O
curvature	O
equal	O
to	O
Riemann	O
curvature	O
tensor	O
.	O
</s>
<s>
Since	O
the	O
Weyl	O
curvature	O
vanishes	O
in	O
two	O
or	O
three	O
dimensions	O
,	O
every	O
Ricci-flat	B-Algorithm
metric	I-Algorithm
in	O
these	O
dimensions	O
is	O
flat	B-Algorithm
.	O
</s>
<s>
Conversely	O
,	O
it	O
is	O
automatic	O
from	O
the	O
definitions	O
that	O
any	O
flat	B-Algorithm
metric	I-Algorithm
is	O
Ricci-flat	B-Algorithm
.	O
</s>
<s>
The	O
study	O
of	O
flat	B-Algorithm
metrics	I-Algorithm
is	O
usually	O
considered	O
as	O
a	O
topic	O
unto	O
itself	O
.	O
</s>
<s>
As	O
such	O
,	O
the	O
study	O
of	O
Ricci-flat	B-Algorithm
metrics	I-Algorithm
is	O
only	O
a	O
distinct	O
topic	O
in	O
dimension	O
four	O
and	O
above	O
.	O
</s>
<s>
As	O
noted	O
above	O
,	O
any	O
flat	B-Algorithm
metric	I-Algorithm
is	O
Ricci-flat	B-Algorithm
.	O
</s>
<s>
However	O
it	O
is	O
nontrivial	O
to	O
identify	O
Ricci-flat	B-Algorithm
manifolds	I-Algorithm
whose	O
full	O
curvature	O
is	O
nonzero	O
.	O
</s>
<s>
In	O
1916	O
,	O
Karl	O
Schwarzschild	O
found	O
the	O
Schwarzschild	O
metrics	O
,	O
which	O
are	O
Ricci-flat	B-Algorithm
Lorentzian	O
manifolds	O
of	O
nonzero	O
curvature	O
.	O
</s>
<s>
These	O
metrics	O
are	O
fully	O
explicit	O
and	O
are	O
of	O
fundamental	O
interest	O
in	O
the	O
mathematics	O
and	O
physics	O
of	O
black	B-Application
holes	I-Application
.	O
</s>
<s>
More	O
generally	O
,	O
in	O
general	O
relativity	O
,	O
Ricci-flat	B-Algorithm
Lorentzian	O
manifolds	O
represent	O
the	O
vacuum	B-General_Concept
solutions	O
of	O
Einstein	O
's	O
field	O
equations	O
with	O
vanishing	O
cosmological	O
constant	O
.	O
</s>
<s>
However	O
,	O
these	O
constructions	O
are	O
not	O
directly	O
helpful	O
for	O
Ricci-flat	B-Algorithm
Riemannian	O
metrics	O
,	O
in	O
the	O
sense	O
that	O
any	O
homogeneous	O
Riemannian	B-Architecture
manifold	I-Architecture
which	O
is	O
Ricci-flat	B-Algorithm
must	O
be	O
flat	B-Algorithm
.	O
</s>
<s>
However	O
,	O
there	O
are	O
homogeneous	O
(	O
and	O
even	O
symmetric	O
)	O
Lorentzian	O
manifolds	O
which	O
are	O
Ricci-flat	B-Algorithm
but	O
not	O
flat	B-Algorithm
,	O
as	O
follows	O
from	O
an	O
explicit	O
construction	O
and	O
computation	O
of	O
Lie	O
algebras	O
.	O
</s>
<s>
Until	O
Shing-Tung	O
Yau	O
's	O
resolution	O
of	O
the	O
Calabi	O
conjecture	O
in	O
the	O
1970s	O
,	O
it	O
was	O
not	O
known	O
whether	O
every	O
Ricci-flat	B-Algorithm
Riemannian	O
metric	O
on	O
a	O
closed	O
manifold	O
is	O
flat	B-Algorithm
.	O
</s>
<s>
His	O
work	O
,	O
using	O
techniques	O
of	O
partial	O
differential	O
equations	O
,	O
established	O
a	O
comprehensive	O
existence	O
theory	O
for	O
Ricci-flat	B-Algorithm
metrics	I-Algorithm
in	O
the	O
special	O
case	O
of	O
Kähler	O
metrics	O
on	O
closed	O
complex	O
manifolds	O
.	O
</s>
<s>
Such	O
Riemannian	B-Architecture
manifolds	I-Architecture
are	O
often	O
called	O
Calabi	O
–	O
Yau	O
manifolds	O
,	O
although	O
various	O
authors	O
use	O
this	O
name	O
in	O
slightly	O
different	O
ways	O
.	O
</s>
<s>
It	O
is	O
a	O
straightforward	O
consequence	O
of	O
standard	O
elliptic	O
regularity	O
results	O
that	O
any	O
Ricci-flat	B-Algorithm
Riemannian	O
metric	O
on	O
a	O
smooth	O
manifold	O
is	O
analytic	O
,	O
in	O
the	O
sense	O
that	O
harmonic	O
coordinates	O
define	O
a	O
compatible	O
analytic	O
structure	O
,	O
and	O
the	O
local	O
representation	O
of	O
the	O
metric	O
is	O
real-analytic	B-Language
.	O
</s>
<s>
Based	O
on	O
this	O
perspective	O
,	O
Yvonne	O
Choquet-Bruhat	O
developed	O
the	O
well-posedness	B-Algorithm
of	O
the	O
Ricci-flatness	O
condition	O
.	O
</s>
<s>
She	O
reached	O
a	O
definitive	O
result	O
in	O
collaboration	O
with	O
Robert	O
Geroch	O
in	O
the	O
1960s	O
,	O
establishing	O
how	O
a	O
certain	O
class	O
of	O
maximally	O
extended	O
Ricci-flat	B-Algorithm
Lorentzian	O
metrics	O
are	O
prescribed	O
and	O
constructed	O
by	O
certain	O
Riemannian	O
data	O
.	O
</s>
<s>
Moreover	O
,	O
the	O
analyticity	O
and	O
corresponding	O
unique	O
continuation	O
of	O
a	O
Ricci-flat	B-Algorithm
Riemannian	O
metric	O
has	O
a	O
fundamentally	O
different	O
character	O
than	O
Ricci-flat	B-Algorithm
Lorentzian	O
metrics	O
,	O
which	O
have	O
finite	O
speeds	O
of	O
propagation	O
and	O
fully	O
localizable	O
phenomena	O
.	O
</s>
<s>
Yau	O
's	O
existence	O
theorem	O
for	O
Ricci-flat	B-Algorithm
Kähler	O
metrics	O
established	O
the	O
precise	O
topological	O
condition	O
under	O
which	O
such	O
a	O
metric	O
exists	O
on	O
a	O
given	O
closed	O
complex	O
manifold	O
:	O
the	O
first	O
Chern	O
class	O
of	O
the	O
holomorphic	O
tangent	O
bundle	O
must	O
be	O
zero	O
.	O
</s>
<s>
As	O
particular	O
cases	O
of	O
well-known	O
theorems	O
on	O
Riemannian	B-Architecture
manifolds	I-Architecture
of	O
nonnegative	O
Ricci	O
curvature	O
,	O
any	O
manifold	O
with	O
a	O
complete	O
Ricci-flat	B-Algorithm
Riemannian	O
metric	O
must	O
:	O
</s>
<s>
Every	O
Ricci-flat	B-Algorithm
Riemannian	B-Architecture
manifold	I-Architecture
in	O
this	O
class	O
is	O
flat	B-Algorithm
,	O
which	O
is	O
a	O
corollary	O
of	O
Cheeger	O
and	O
Gromoll	O
's	O
splitting	O
theorem	O
.	O
</s>
<s>
On	O
a	O
simply-connected	O
Kähler	O
manifold	O
,	O
a	O
Kähler	O
metric	O
is	O
Ricci-flat	B-Algorithm
if	O
and	O
only	O
if	O
the	O
holonomy	O
group	O
is	O
contained	O
in	O
the	O
special	O
unitary	O
group	O
.	O
</s>
<s>
On	O
a	O
general	O
Kähler	O
manifold	O
,	O
the	O
if	O
direction	O
still	O
holds	O
,	O
but	O
only	O
the	O
restricted	O
holonomy	O
group	O
of	O
a	O
Ricci-flat	B-Algorithm
Kähler	O
metric	O
is	O
necessarily	O
contained	O
in	O
the	O
special	O
unitary	O
group	O
.	O
</s>
<s>
A	O
hyperkähler	O
manifold	O
is	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
whose	O
holonomy	O
group	O
is	O
contained	O
in	O
the	O
symplectic	O
group	O
.	O
</s>
<s>
This	O
condition	O
on	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
may	O
also	O
be	O
characterized	O
(	O
roughly	O
speaking	O
)	O
by	O
the	O
existence	O
of	O
a	O
2-sphere	O
of	O
complex	O
structures	O
which	O
are	O
all	O
parallel	O
.	O
</s>
<s>
This	O
says	O
in	O
particular	O
that	O
every	O
hyperkähler	O
metric	O
is	O
Kähler	O
;	O
furthermore	O
,	O
via	O
the	O
Ambrose	O
–	O
Singer	O
theorem	O
,	O
every	O
such	O
metric	O
is	O
Ricci-flat	B-Algorithm
.	O
</s>
<s>
A	O
quaternion-Kähler	O
manifold	O
is	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
whose	O
holonomy	O
group	O
is	O
contained	O
in	O
the	O
Lie	O
group	O
.	O
</s>
<s>
Furthermore	O
,	O
any	O
Ricci-flat	B-Algorithm
quaternion-Kähler	O
manifold	O
must	O
be	O
locally	O
hyperkähler	O
,	O
meaning	O
that	O
the	O
restricted	O
holonomy	O
group	O
is	O
contained	O
in	O
the	O
symplectic	O
group	O
.	O
</s>
<s>
A	O
manifold	O
or	O
manifold	O
is	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
whose	O
holonomy	O
group	O
is	O
contained	O
in	O
the	O
Lie	O
groups	O
or	O
.	O
</s>
<s>
The	O
Ambrose	O
–	O
Singer	O
theorem	O
implies	O
that	O
any	O
such	O
manifold	O
is	O
Ricci-flat	B-Algorithm
.	O
</s>
<s>
Marcel	O
Berger	O
commented	O
that	O
all	O
known	O
examples	O
of	O
irreducible	O
Ricci-flat	B-Algorithm
Riemannian	O
metrics	O
on	O
simply-connected	O
closed	O
manifolds	O
have	O
special	O
holonomy	O
groups	O
,	O
according	O
to	O
the	O
above	O
possibilities	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
Berger	O
considered	O
Ricci-flat	B-Algorithm
manifolds	I-Algorithm
to	O
be	O
"	O
extremely	O
mysterious.	O
"	O
</s>
