<s>
In	O
differential	B-Language
geometry	I-Language
,	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
or	O
Riemannian	O
space	O
,	O
so	O
called	O
after	O
the	O
German	O
mathematician	O
Bernhard	O
Riemann	O
,	O
is	O
a	O
real	B-Architecture
,	O
smooth	O
manifold	B-Architecture
M	O
equipped	O
with	O
a	O
positive-definite	O
inner	O
product	O
gp	O
on	O
the	O
tangent	O
space	O
TpM	O
at	O
each	O
point	O
p	O
.	O
</s>
<s>
Riemannian	O
geometry	O
is	O
the	O
study	O
of	O
Riemannian	B-Architecture
manifolds	I-Architecture
.	O
</s>
<s>
A	O
Riemannian	O
metric	O
(	O
tensor	O
)	O
makes	O
it	O
possible	O
to	O
define	O
several	O
geometric	O
notions	O
on	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
,	O
such	O
as	O
angle	O
at	O
an	O
intersection	O
,	O
length	O
of	O
a	O
curve	O
,	O
area	O
of	O
a	O
surface	O
and	O
higher-dimensional	O
analogues	O
(	O
volume	O
,	O
etc	O
.	O
</s>
<s>
)	O
,	O
extrinsic	O
curvature	O
of	O
submanifolds	O
,	O
and	O
intrinsic	O
curvature	O
of	O
the	O
manifold	B-Architecture
itself	O
.	O
</s>
<s>
See	O
Differential	B-Language
geometry	I-Language
of	O
surfaces	O
.	O
</s>
<s>
Bernhard	O
Riemann	O
extended	O
Gauss	O
's	O
theory	O
to	O
higher-dimensional	O
spaces	O
called	O
manifolds	B-Architecture
in	O
a	O
way	O
that	O
also	O
allows	O
distances	O
and	O
angles	O
to	O
be	O
measured	O
and	O
the	O
notion	O
of	O
curvature	O
to	O
be	O
defined	O
,	O
again	O
in	O
a	O
way	O
that	O
is	O
intrinsic	O
to	O
the	O
manifold	B-Architecture
and	O
not	O
dependent	O
upon	O
its	O
embedding	O
in	O
higher-dimensional	O
spaces	O
.	O
</s>
<s>
Albert	O
Einstein	O
used	O
the	O
theory	O
of	O
pseudo-Riemannian	O
manifolds	O
(	O
a	O
generalization	O
of	O
Riemannian	B-Architecture
manifolds	I-Architecture
)	O
to	O
develop	O
his	O
general	O
theory	O
of	O
relativity	O
.	O
</s>
<s>
The	O
tangent	O
bundle	O
of	O
a	O
smooth	O
manifold	B-Architecture
assigns	O
to	O
each	O
point	O
of	O
a	O
vector	O
space	O
called	O
the	O
tangent	O
space	O
of	O
at	O
A	O
Riemannian	O
metric	O
(	O
by	O
its	O
definition	O
)	O
assigns	O
to	O
each	O
a	O
positive-definite	O
inner	O
product	O
along	O
with	O
which	O
comes	O
a	O
norm	O
defined	O
by	O
The	O
smooth	O
manifold	B-Architecture
endowed	O
with	O
this	O
metric	O
is	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
,	O
denoted	O
.	O
</s>
<s>
If	O
and	O
are	O
two	O
Riemannian	B-Architecture
manifolds	I-Architecture
,	O
with	O
a	O
diffeomorphism	O
,	O
then	O
is	O
called	O
an	O
isometry	O
if	O
i.e.	O
</s>
<s>
Examples	O
of	O
Riemannian	B-Architecture
manifolds	I-Architecture
will	O
be	O
discussed	O
below	O
.	O
</s>
<s>
A	O
famous	O
theorem	O
of	O
John	O
Nash	O
states	O
that	O
,	O
given	O
any	O
smooth	O
Riemannian	B-Architecture
manifold	I-Architecture
there	O
is	O
a	O
(	O
usually	O
large	O
)	O
number	O
and	O
an	O
embedding	O
such	O
that	O
the	O
pullback	O
by	O
of	O
the	O
standard	O
Riemannian	O
metric	O
on	O
is	O
Informally	O
,	O
the	O
entire	O
structure	O
of	O
a	O
smooth	O
Riemannian	B-Architecture
manifold	I-Architecture
can	O
be	O
encoded	O
by	O
a	O
diffeomorphism	O
to	O
a	O
certain	O
embedded	O
submanifold	O
of	O
some	O
Euclidean	O
space	O
.	O
</s>
<s>
In	O
this	O
sense	O
,	O
it	O
is	O
arguable	O
that	O
nothing	O
can	O
be	O
gained	O
from	O
the	O
consideration	O
of	O
abstract	O
smooth	O
manifolds	B-Architecture
and	O
their	O
Riemannian	O
metrics	O
.	O
</s>
<s>
However	O
,	O
there	O
are	O
many	O
natural	O
smooth	O
Riemannian	B-Architecture
manifolds	I-Architecture
,	O
such	O
as	O
the	O
set	O
of	O
rotations	O
of	O
three-dimensional	O
space	O
and	O
the	O
hyperbolic	O
space	O
,	O
of	O
which	O
any	O
representation	O
as	O
a	O
submanifold	O
of	O
Euclidean	O
space	O
will	O
fail	O
to	O
represent	O
their	O
remarkable	O
symmetries	O
and	O
properties	O
as	O
clearly	O
as	O
their	O
abstract	O
presentations	O
do	O
.	O
</s>
<s>
Let	O
be	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
and	O
let	O
be	O
an	O
embedded	O
submanifold	O
of	O
which	O
is	O
at	O
least	O
Then	O
the	O
restriction	O
of	O
g	O
to	O
vectors	O
tangent	O
along	O
N	O
defines	O
a	O
Riemannian	O
metric	O
over	O
N	O
.	O
</s>
<s>
Let	O
be	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
and	O
let	O
be	O
a	O
differentiable	O
map	O
.	O
</s>
<s>
An	O
important	O
example	O
occurs	O
when	O
is	O
not	O
simply-connected	O
,	O
so	O
that	O
there	O
is	O
a	O
covering	O
map	O
This	O
is	O
an	O
immersion	O
,	O
and	O
so	O
the	O
universal	O
cover	O
of	O
any	O
Riemannian	B-Architecture
manifold	I-Architecture
automatically	O
inherits	O
a	O
Riemannian	O
metric	O
.	O
</s>
<s>
More	O
generally	O
,	O
but	O
by	O
the	O
same	O
principle	O
,	O
any	O
covering	O
space	O
of	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
inherits	O
a	O
Riemannian	O
metric	O
.	O
</s>
<s>
Also	O
,	O
an	O
immersed	O
submanifold	O
of	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
inherits	O
a	O
Riemannian	O
metric	O
.	O
</s>
<s>
Let	O
and	O
be	O
two	O
Riemannian	B-Architecture
manifolds	I-Architecture
,	O
and	O
consider	O
the	O
cartesian	O
product	O
with	O
the	O
usual	O
product	O
smooth	O
structure	O
.	O
</s>
<s>
Let	O
be	O
a	O
smooth	O
coordinate	O
chart	O
on	O
and	O
let	O
be	O
a	O
smooth	O
coordinate	O
chart	O
on	O
Then	O
is	O
a	O
smooth	O
coordinate	O
chart	O
on	O
For	O
convenience	O
let	O
denote	O
the	O
collection	O
of	O
positive-definite	O
symmetric	O
real	B-Architecture
matrices	O
.	O
</s>
<s>
Although	O
much	O
of	O
the	O
basic	O
theory	O
of	O
Riemannian	O
metrics	O
can	O
be	O
developed	O
by	O
only	O
using	O
that	O
a	O
smooth	O
manifold	B-Architecture
is	O
locally	O
Euclidean	O
,	O
for	O
this	O
result	O
it	O
is	O
necessary	O
to	O
include	O
in	O
the	O
definition	O
of	O
"	O
smooth	O
manifold	B-Architecture
"	O
that	O
it	O
is	O
Hausdorff	O
and	O
paracompact	O
.	O
</s>
<s>
Informally	O
,	O
one	O
may	O
say	O
that	O
one	O
is	O
asking	O
for	O
to	O
locally	O
'	O
stretch	O
itself	O
out	O
 '	O
as	O
much	O
as	O
it	O
can	O
,	O
subject	O
to	O
the	O
(	O
informally	O
considered	O
)	O
unit-speed	O
constraint	B-Application
.	O
</s>
<s>
As	O
above	O
,	O
let	O
be	O
a	O
connected	O
and	O
continuous	O
Riemannian	B-Architecture
manifold	I-Architecture
.	O
</s>
<s>
Let	O
be	O
a	O
connected	O
and	O
continuous	O
Riemannian	B-Architecture
manifold	I-Architecture
.	O
</s>
<s>
A	O
Riemannian	B-Architecture
manifold	I-Architecture
M	O
is	O
geodesically	O
complete	O
if	O
for	O
all	O
,	O
the	O
exponential	O
map	O
expp	O
is	O
defined	O
for	O
all	O
,	O
i.e.	O
</s>
<s>
If	O
M	O
is	O
complete	O
,	O
then	O
M	O
is	O
non-extendable	O
in	O
the	O
sense	O
that	O
it	O
is	O
not	O
isometric	O
to	O
an	O
open	O
proper	O
submanifold	O
of	O
any	O
other	O
Riemannian	B-Architecture
manifold	I-Architecture
.	O
</s>
<s>
The	O
converse	O
is	O
not	O
true	O
,	O
however	O
:	O
there	O
exist	O
non-extendable	O
manifolds	B-Architecture
that	O
are	O
not	O
complete	O
.	O
</s>
<s>
The	O
statements	O
and	O
theorems	O
above	O
are	O
for	O
finite-dimensional	O
manifolds	B-Architecture
—	O
manifolds	B-Architecture
whose	O
charts	O
map	O
to	O
open	O
subsets	O
of	O
These	O
can	O
be	O
extended	O
,	O
to	O
a	O
certain	O
degree	O
,	O
to	O
infinite-dimensional	O
manifolds	B-Architecture
;	O
that	O
is	O
,	O
manifolds	B-Architecture
that	O
are	O
modeled	O
after	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
;	O
for	O
example	O
,	O
Fréchet	O
,	O
Banach	O
and	O
Hilbert	B-Algorithm
manifolds	I-Algorithm
.	O
</s>
<s>
A	O
strong	O
Riemannian	O
metric	O
on	O
is	O
a	O
weak	O
Riemannian	O
metric	O
,	O
such	O
that	O
induces	O
the	O
topology	O
on	O
Note	O
that	O
if	O
is	O
not	O
a	O
Hilbert	B-Algorithm
manifold	I-Algorithm
then	O
cannot	O
be	O
a	O
strong	O
metric	O
.	O
</s>
<s>
Let	O
be	O
a	O
compact	O
Riemannian	B-Architecture
manifold	I-Architecture
and	O
denote	O
by	O
its	O
diffeomorphism	O
group	O
.	O
</s>
<s>
It	O
is	O
a	O
smooth	O
manifold	B-Architecture
(	O
see	O
here	O
)	O
and	O
in	O
fact	O
,	O
a	O
Lie	O
group	O
.	O
</s>
<s>
Theorem	O
:	O
Let	O
be	O
a	O
strong	O
Riemannian	B-Architecture
manifold	I-Architecture
.	O
</s>
