<s>
In	O
mathematics	O
,	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
is	O
said	O
to	O
be	O
flat	O
if	O
its	O
Riemann	O
curvature	O
tensor	O
is	O
everywhere	O
zero	O
.	O
</s>
<s>
Intuitively	O
,	O
a	O
flat	B-Algorithm
manifold	I-Algorithm
is	O
one	O
that	O
"	O
locally	O
looks	O
like	O
"	O
Euclidean	O
space	O
in	O
terms	O
of	O
distances	O
and	O
angles	O
,	O
e.g.	O
</s>
<s>
The	O
universal	O
cover	O
of	O
a	O
complete	O
flat	B-Algorithm
manifold	I-Algorithm
is	O
Euclidean	O
space	O
.	O
</s>
<s>
that	O
all	O
compact	O
flat	B-Algorithm
manifolds	I-Algorithm
are	O
finitely	O
covered	O
by	O
tori	O
;	O
the	O
3-dimensional	O
case	O
was	O
proved	O
earlier	O
by	O
.	O
</s>
<s>
The	O
following	O
manifolds	O
can	O
be	O
endowed	O
with	O
a	O
flat	B-Algorithm
metric	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
this	O
may	O
not	O
be	O
their	O
'	O
standard	O
 '	O
metric	O
(	O
for	O
example	O
,	O
the	O
flat	B-Algorithm
metric	I-Algorithm
on	O
the	O
2-dimensional	O
torus	O
is	O
not	O
the	O
metric	O
induced	O
by	O
its	O
usual	O
embedding	O
into	O
)	O
.	O
</s>
<s>
Every	O
one-dimensional	O
Riemannian	B-Architecture
manifold	I-Architecture
is	O
flat	O
.	O
</s>
<s>
Conversely	O
,	O
given	O
that	O
every	O
connected	O
one-dimensional	O
smooth	O
manifold	O
is	O
diffeomorphic	O
to	O
either	O
or	O
it	O
is	O
straightforward	O
to	O
see	O
that	O
every	O
connected	O
one-dimensional	O
Riemannian	B-Architecture
manifold	I-Architecture
is	O
isometric	O
to	O
one	O
of	O
the	O
following	O
(	O
each	O
with	O
their	O
standard	O
Riemannian	O
structure	O
)	O
:	O
</s>
<s>
The	O
simplicity	O
of	O
a	O
complete	O
description	O
in	O
this	O
case	O
could	O
be	O
ascribed	O
to	O
the	O
fact	O
that	O
every	O
one-dimensional	O
Riemannian	B-Architecture
manifold	I-Architecture
has	O
a	O
smooth	O
unit-length	O
vector	O
field	O
,	O
and	O
that	O
an	O
isometry	O
from	O
one	O
of	O
the	O
above	O
model	O
examples	O
is	O
provided	O
by	O
considering	O
an	O
integral	O
curve	O
.	O
</s>
<s>
If	O
is	O
a	O
smooth	O
two-dimensional	O
connected	O
complete	O
flat	O
Riemannian	B-Architecture
manifold	I-Architecture
,	O
then	O
must	O
be	O
diffeomorphic	O
to	O
the	O
Möbius	O
strip	O
,	O
or	O
the	O
Klein	O
bottle	O
.	O
</s>
<s>
These	O
are	O
all	O
groups	O
acting	O
freely	O
and	O
properly	O
discontinuously	O
on	O
and	O
so	O
the	O
various	O
coset	O
spaces	O
all	O
naturally	O
have	O
the	O
structure	O
of	O
two-dimensional	O
complete	O
flat	O
Riemannian	B-Architecture
manifolds	I-Architecture
.	O
</s>
<s>
None	O
of	O
them	O
are	O
isometric	O
to	O
one	O
another	O
,	O
and	O
any	O
smooth	O
two-dimensional	O
complete	O
flat	O
connected	O
Riemannian	B-Architecture
manifold	I-Architecture
is	O
isometric	O
to	O
one	O
of	O
them	O
.	O
</s>
<s>
There	O
are	O
17	O
compact	O
2-dimensional	O
orbifolds	O
with	O
flat	B-Algorithm
metric	I-Algorithm
(	O
including	O
the	O
torus	O
and	O
Klein	O
bottle	O
)	O
,	O
listed	O
in	O
the	O
article	O
on	O
orbifolds	O
,	O
that	O
correspond	O
to	O
the	O
17	O
wallpaper	O
groups	O
.	O
</s>
<s>
Note	O
that	O
the	O
standard	O
'	O
picture	O
 '	O
of	O
the	O
torus	O
as	O
a	O
doughnut	O
does	O
not	O
present	O
it	O
with	O
a	O
flat	B-Algorithm
metric	I-Algorithm
,	O
since	O
the	O
points	O
furthest	O
from	O
the	O
center	O
have	O
positive	O
curvature	O
while	O
the	O
points	O
closest	O
to	O
the	O
center	O
have	O
negative	O
curvature	O
.	O
</s>
<s>
Since	O
is	O
presented	O
as	O
an	O
embedded	O
submanifold	O
of	O
any	O
of	O
the	O
(	O
flat	O
)	O
product	O
structures	O
on	O
are	O
naturally	O
presented	O
as	O
submanifolds	O
of	O
Likewise	O
,	O
the	O
standard	O
three-dimensional	O
visualizations	O
of	O
the	O
Klein	O
bottle	O
do	O
not	O
present	O
a	O
flat	B-Algorithm
metric	I-Algorithm
.	O
</s>
<s>
The	O
standard	O
construction	O
of	O
a	O
Möbius	O
strip	O
,	O
by	O
gluing	O
ends	O
of	O
a	O
strip	O
of	O
paper	O
together	O
,	O
does	O
indeed	O
give	O
it	O
a	O
flat	B-Algorithm
metric	I-Algorithm
,	O
but	O
it	O
is	O
not	O
complete	O
.	O
</s>
<s>
The	O
3-torus	O
,	O
made	O
by	O
gluing	O
opposite	O
faces	O
of	O
a	O
cube	B-Application
.	O
</s>
<s>
The	O
manifold	O
made	O
by	O
gluing	O
opposite	O
faces	O
of	O
a	O
cube	B-Application
with	O
a	O
1/2	O
twist	O
on	O
one	O
pair	O
.	O
</s>
<s>
The	O
manifold	O
made	O
by	O
gluing	O
opposite	O
faces	O
of	O
a	O
cube	B-Application
with	O
a	O
1/4	O
twist	O
on	O
one	O
pair	O
.	O
</s>
<s>
Quotients	O
of	O
flat	B-Algorithm
manifolds	I-Algorithm
by	O
groups	O
acting	O
freely	O
.	O
</s>
<s>
Among	O
all	O
closed	O
manifolds	O
with	O
non-positive	O
sectional	O
curvature	O
,	O
flat	B-Algorithm
manifolds	I-Algorithm
are	O
characterized	O
as	O
precisely	O
those	O
with	O
an	O
amenable	O
fundamental	O
group	O
.	O
</s>
<s>
This	O
is	O
a	O
consequence	O
of	O
the	O
Adams-Ballmann	O
theorem	O
(	O
1998	O
)	O
,	O
which	O
establishes	O
this	O
characterization	O
in	O
the	O
much	O
more	O
general	O
setting	O
of	O
discrete	O
cocompact	O
groups	O
of	O
isometries	O
of	O
Hadamard	B-Algorithm
spaces	I-Algorithm
.	O
</s>
