<s>
Differential	B-Language
geometry	I-Language
is	O
a	O
mathematical	O
discipline	O
that	O
studies	O
the	O
geometry	O
of	O
smooth	O
shapes	O
and	O
smooth	O
spaces	O
,	O
otherwise	O
known	O
as	O
smooth	O
manifolds	B-Architecture
.	O
</s>
<s>
It	O
uses	O
the	O
techniques	O
of	O
differential	O
calculus	O
,	O
integral	B-Algorithm
calculus	I-Algorithm
,	O
linear	B-Language
algebra	I-Language
and	O
multilinear	O
algebra	O
.	O
</s>
<s>
The	O
simplest	O
examples	O
of	O
smooth	O
spaces	O
are	O
the	O
plane	O
and	O
space	O
curves	O
and	O
surfaces	O
in	O
the	O
three-dimensional	O
Euclidean	O
space	O
,	O
and	O
the	O
study	O
of	O
these	O
shapes	O
formed	O
the	O
basis	O
for	O
development	O
of	O
modern	O
differential	B-Language
geometry	I-Language
during	O
the	O
18th	O
and	O
19th	O
centuries	O
.	O
</s>
<s>
Since	O
the	O
late	O
19th	O
century	O
,	O
differential	B-Language
geometry	I-Language
has	O
grown	O
into	O
a	O
field	O
concerned	O
more	O
generally	O
with	O
geometric	O
structures	O
on	O
differentiable	O
manifolds	B-Architecture
.	O
</s>
<s>
Differential	B-Language
geometry	I-Language
is	O
closely	O
related	O
to	O
,	O
and	O
is	O
sometimes	O
taken	O
to	O
include	O
,	O
differential	O
topology	B-Architecture
,	O
which	O
concerns	O
itself	O
with	O
properties	O
of	O
differentiable	O
manifolds	B-Architecture
which	O
do	O
not	O
rely	O
on	O
any	O
additional	O
geometric	O
structure	O
(	O
see	O
that	O
article	O
for	O
more	O
discussion	O
on	O
the	O
distinction	O
between	O
the	O
two	O
subjects	O
)	O
.	O
</s>
<s>
Differential	B-Language
geometry	I-Language
is	O
also	O
related	O
to	O
the	O
geometric	O
aspects	O
of	O
the	O
theory	O
of	O
differential	O
equations	O
,	O
otherwise	O
known	O
as	O
geometric	O
analysis	O
.	O
</s>
<s>
Differential	B-Language
geometry	I-Language
finds	O
applications	O
throughout	O
mathematics	O
and	O
the	O
natural	O
sciences	O
.	O
</s>
<s>
Most	O
prominently	O
the	O
language	O
of	O
differential	B-Language
geometry	I-Language
was	O
used	O
by	O
Albert	O
Einstein	O
in	O
his	O
theory	O
of	O
general	O
relativity	O
,	O
and	O
subsequently	O
by	O
physicists	O
in	O
the	O
development	O
of	O
quantum	O
field	O
theory	O
and	O
the	O
standard	O
model	O
of	O
particle	O
physics	O
.	O
</s>
<s>
Outside	O
of	O
physics	O
,	O
differential	B-Language
geometry	I-Language
finds	O
applications	O
in	O
chemistry	O
,	O
economics	O
,	O
engineering	O
,	O
control	O
theory	O
,	O
computer	O
graphics	O
and	O
computer	B-Application
vision	I-Application
,	O
and	O
recently	O
in	O
machine	O
learning	O
.	O
</s>
<s>
The	O
history	O
and	O
development	O
of	O
differential	B-Language
geometry	I-Language
as	O
a	O
subject	O
begins	O
at	O
least	O
as	O
far	O
back	O
as	O
classical	O
antiquity	O
.	O
</s>
<s>
It	O
is	O
intimately	O
linked	O
to	O
the	O
development	O
of	O
geometry	O
more	O
generally	O
,	O
of	O
the	O
notion	O
of	O
space	O
and	O
shape	O
,	O
and	O
of	O
topology	B-Architecture
,	O
especially	O
the	O
study	O
of	O
manifolds	B-Architecture
.	O
</s>
<s>
In	O
this	O
section	O
we	O
focus	O
primarily	O
on	O
the	O
history	O
of	O
the	O
application	O
of	O
infinitesimal	O
methods	O
to	O
geometry	O
,	O
and	O
later	O
to	O
the	O
ideas	O
of	O
tangent	O
spaces	O
,	O
and	O
eventually	O
the	O
development	O
of	O
the	O
modern	O
formalism	O
of	O
the	O
subject	O
in	O
terms	O
of	O
tensors	B-Device
and	O
tensor	B-Device
fields	O
.	O
</s>
<s>
The	O
study	O
of	O
differential	B-Language
geometry	I-Language
,	O
or	O
at	O
least	O
the	O
study	O
of	O
the	O
geometry	O
of	O
smooth	O
shapes	O
,	O
can	O
be	O
traced	O
back	O
at	O
least	O
to	O
classical	O
antiquity	O
.	O
</s>
<s>
Implicitly	O
throughout	O
this	O
time	O
principles	O
that	O
form	O
the	O
foundation	O
of	O
differential	B-Language
geometry	I-Language
and	O
calculus	O
were	O
used	O
in	O
geodesy	O
,	O
although	O
in	O
a	O
much	O
simplified	O
form	O
.	O
</s>
<s>
Namely	O
,	O
as	O
far	O
back	O
as	O
Euclid	B-Language
's	O
Elements	O
it	O
was	O
understood	O
that	O
a	O
straight	O
line	O
could	O
be	O
defined	O
by	O
its	O
property	O
of	O
providing	O
the	O
shortest	O
distance	O
between	O
two	O
points	O
,	O
and	O
applying	O
this	O
same	O
principle	O
to	O
the	O
surface	O
of	O
the	O
Earth	O
leads	O
to	O
the	O
conclusion	O
that	O
great	O
circles	O
,	O
which	O
are	O
only	O
locally	O
similar	O
to	O
straight	O
lines	O
in	O
a	O
flat	O
plane	O
,	O
provide	O
the	O
shortest	O
path	O
between	O
two	O
points	O
on	O
the	O
Earth	O
's	O
surface	O
.	O
</s>
<s>
In	O
Euclid	B-Language
's	O
Elements	O
the	O
notion	O
of	O
tangency	O
of	O
a	O
line	O
to	O
a	O
circle	O
is	O
discussed	O
,	O
and	O
Archimedes	B-Device
applied	O
the	O
method	O
of	O
exhaustion	O
to	O
compute	O
the	O
areas	O
of	O
smooth	O
shapes	O
such	O
as	O
the	O
circle	O
,	O
and	O
the	O
volumes	O
of	O
smooth	O
three-dimensional	O
solids	O
such	O
as	O
the	O
sphere	O
,	O
cones	O
,	O
and	O
cylinders	O
.	O
</s>
<s>
There	O
was	O
little	O
development	O
in	O
the	O
theory	O
of	O
differential	B-Language
geometry	I-Language
between	O
antiquity	O
and	O
the	O
beginning	O
of	O
the	O
Renaissance	B-Application
.	O
</s>
<s>
Before	O
the	O
development	O
of	O
calculus	O
by	O
Newton	O
and	O
Leibniz	O
,	O
the	O
most	O
significant	O
development	O
in	O
the	O
understanding	O
of	O
differential	B-Language
geometry	I-Language
came	O
from	O
Gerardus	O
Mercator	O
's	O
development	O
of	O
the	O
Mercator	O
projection	O
as	O
a	O
way	O
of	O
mapping	O
the	O
Earth	O
.	O
</s>
<s>
This	O
fact	O
reflects	O
the	O
lack	O
of	O
a	O
metric-preserving	O
map	O
of	O
the	O
Earth	O
's	O
surface	O
onto	O
a	O
flat	O
plane	O
,	O
a	O
consequence	O
of	O
the	O
later	O
Theorema	O
Egregium	O
of	O
Gauss	B-Algorithm
.	O
</s>
<s>
At	O
this	O
same	O
time	O
the	O
orthogonality	B-Application
between	O
the	O
osculating	O
circles	O
of	O
a	O
plane	O
curve	O
and	O
the	O
tangent	O
directions	O
is	O
realised	O
,	O
and	O
the	O
first	O
analytical	O
formula	O
for	O
the	O
radius	O
of	O
an	O
osculating	O
circle	O
,	O
essentially	O
the	O
first	O
analytical	O
formula	O
for	O
the	O
notion	O
of	O
curvature	O
,	O
is	O
written	O
down	O
.	O
</s>
<s>
Importantly	O
Clairaut	O
introduced	O
the	O
terminology	O
of	O
curvature	O
and	O
double	O
curvature	O
,	O
essentially	O
the	O
notion	O
of	O
principal	O
curvatures	O
later	O
studied	O
by	O
Gauss	B-Algorithm
and	O
others	O
.	O
</s>
<s>
In	O
regards	O
to	O
differential	B-Language
geometry	I-Language
,	O
Euler	O
studied	O
the	O
notion	O
of	O
a	O
geodesic	O
on	O
a	O
surface	O
deriving	O
the	O
first	O
analytical	O
geodesic	O
equation	O
,	O
and	O
later	O
introduced	O
the	O
first	O
set	O
of	O
intrinsic	O
coordinate	O
systems	O
on	O
a	O
surface	O
,	O
beginning	O
the	O
theory	O
of	O
intrinsic	O
geometry	O
upon	O
which	O
modern	O
geometric	O
ideas	O
are	O
based	O
.	O
</s>
<s>
Around	O
this	O
time	O
Euler	O
's	O
study	O
of	O
mechanics	O
in	O
the	O
Mechanica	O
lead	O
to	O
the	O
realization	O
that	O
a	O
mass	O
traveling	O
along	O
a	O
surface	O
not	O
under	O
the	O
effect	O
of	O
any	O
force	O
would	O
traverse	O
a	O
geodesic	O
path	O
,	O
an	O
early	O
precursor	O
to	O
the	O
important	O
foundational	O
ideas	O
of	O
Einstein	O
's	O
general	O
relativity	O
,	O
and	O
also	O
to	O
the	O
Euler	O
–	O
Lagrange	O
equations	O
and	O
the	O
first	O
theory	O
of	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
,	O
which	O
underpins	O
in	O
modern	O
differential	B-Language
geometry	I-Language
many	O
techniques	O
in	O
symplectic	O
geometry	O
and	O
geometric	O
analysis	O
.	O
</s>
<s>
This	O
theory	O
was	O
used	O
by	O
Lagrange	O
,	O
a	O
co-developer	O
of	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
,	O
to	O
derive	O
the	O
first	O
differential	O
equation	O
describing	O
a	O
minimal	O
surface	O
in	O
terms	O
of	O
the	O
Euler	O
–	O
Lagrange	O
equation	O
.	O
</s>
<s>
Later	O
in	O
the	O
1700s	O
,	O
the	O
new	O
French	O
school	O
led	O
by	O
Gaspard	O
Monge	O
began	O
to	O
make	O
contributions	O
to	O
differential	B-Language
geometry	I-Language
.	O
</s>
<s>
The	O
field	O
of	O
differential	B-Language
geometry	I-Language
became	O
an	O
area	O
of	O
study	O
considered	O
in	O
its	O
own	O
right	O
,	O
distinct	O
from	O
the	O
more	O
broad	O
idea	O
of	O
analytic	O
geometry	O
,	O
in	O
the	O
1800s	O
,	O
primarily	O
through	O
the	O
foundational	O
work	O
of	O
Carl	O
Friedrich	O
Gauss	B-Algorithm
and	O
Bernhard	O
Riemann	O
,	O
and	O
also	O
in	O
the	O
important	O
contributions	O
of	O
Nikolai	O
Lobachevsky	O
on	O
hyperbolic	O
geometry	O
and	O
non-Euclidean	O
geometry	O
and	O
throughout	O
the	O
same	O
period	O
the	O
development	O
of	O
projective	O
geometry	O
.	O
</s>
<s>
Dubbed	O
the	O
single	O
most	O
important	O
work	O
in	O
the	O
history	O
of	O
differential	B-Language
geometry	I-Language
,	O
in	O
1827	O
Gauss	B-Algorithm
produced	O
the	O
Disquisitiones	O
generales	O
circa	O
superficies	O
curvas	O
detailing	O
the	O
general	O
theory	O
of	O
curved	O
surfaces	O
.	O
</s>
<s>
In	O
this	O
work	O
and	O
his	O
subsequent	O
papers	O
and	O
unpublished	O
notes	O
on	O
the	O
theory	O
of	O
surfaces	O
,	O
Gauss	B-Algorithm
has	O
been	O
dubbed	O
the	O
inventor	O
of	O
non-Euclidean	O
geometry	O
and	O
the	O
inventor	O
of	O
intrinsic	O
differential	B-Language
geometry	I-Language
.	O
</s>
<s>
In	O
his	O
fundamental	O
paper	O
Gauss	B-Algorithm
introduced	O
the	O
Gauss	B-Algorithm
map	O
,	O
Gaussian	O
curvature	O
,	O
first	O
and	O
second	O
fundamental	O
forms	O
,	O
proved	O
the	O
Theorema	O
Egregium	O
showing	O
the	O
intrinsic	O
nature	O
of	O
the	O
Gaussian	O
curvature	O
,	O
and	O
studied	O
geodesics	O
,	O
computing	O
the	O
area	O
of	O
a	O
geodesic	O
triangle	O
in	O
various	O
non-Euclidean	O
geometries	O
on	O
surfaces	O
.	O
</s>
<s>
At	O
this	O
time	O
Gauss	B-Algorithm
was	O
already	O
of	O
the	O
opinion	O
that	O
the	O
standard	O
paradigm	O
of	O
Euclidean	O
geometry	O
should	O
be	O
discarded	O
,	O
and	O
was	O
in	O
possession	O
of	O
private	O
manuscripts	O
on	O
non-Euclidean	O
geometry	O
which	O
informed	O
his	O
study	O
of	O
geodesic	O
triangles	O
.	O
</s>
<s>
Around	O
this	O
same	O
time	O
János	O
Bolyai	O
and	O
Lobachevsky	O
independently	O
discovered	O
hyperbolic	O
geometry	O
and	O
thus	O
demonstrated	O
the	O
existence	O
of	O
consistent	O
geometries	O
outside	O
Euclid	B-Language
's	O
paradigm	O
.	O
</s>
<s>
The	O
development	O
of	O
intrinsic	O
differential	B-Language
geometry	I-Language
in	O
the	O
language	O
of	O
Gauss	B-Algorithm
was	O
spurred	O
on	O
by	O
his	O
student	O
,	O
Bernhard	O
Riemann	O
in	O
his	O
Habilitationsschrift	O
,	O
On	O
the	O
hypotheses	O
which	O
lie	O
at	O
the	O
foundation	O
of	O
geometry	O
.	O
</s>
<s>
In	O
this	O
work	O
Riemann	O
introduced	O
the	O
notion	O
of	O
a	O
Riemannian	O
metric	O
and	O
the	O
Riemannian	O
curvature	O
tensor	B-Device
for	O
the	O
first	O
time	O
,	O
and	O
began	O
the	O
systematic	O
study	O
of	O
differential	B-Language
geometry	I-Language
in	O
higher	O
dimensions	O
.	O
</s>
<s>
This	O
intrinsic	O
point	O
of	O
view	O
in	O
terms	O
of	O
the	O
Riemannian	O
metric	O
,	O
denoted	O
by	O
by	O
Riemann	O
,	O
was	O
the	O
development	O
of	O
an	O
idea	O
of	O
Gauss	B-Algorithm
 '	O
about	O
the	O
linear	O
element	O
of	O
a	O
surface	O
.	O
</s>
<s>
At	O
this	O
time	O
Riemann	O
began	O
to	O
introduce	O
the	O
systematic	O
use	O
of	O
linear	B-Language
algebra	I-Language
and	O
multilinear	O
algebra	O
into	O
the	O
subject	O
,	O
making	O
great	O
use	O
of	O
the	O
theory	O
of	O
quadratic	O
forms	O
in	O
his	O
investigation	O
of	O
metrics	O
and	O
curvature	O
.	O
</s>
<s>
At	O
this	O
time	O
Riemann	O
did	O
not	O
yet	O
develop	O
the	O
modern	O
notion	O
of	O
a	O
manifold	B-Architecture
,	O
as	O
even	O
the	O
notion	O
of	O
a	O
topological	O
space	O
had	O
not	O
been	O
encountered	O
,	O
but	O
he	O
did	O
propose	O
that	O
it	O
might	O
be	O
possible	O
to	O
investigate	O
or	O
measure	O
the	O
properties	O
of	O
the	O
metric	O
of	O
spacetime	B-Protocol
through	O
the	O
analysis	O
of	O
masses	O
within	O
spacetime	B-Protocol
,	O
linking	O
with	O
the	O
earlier	O
observation	O
of	O
Euler	O
that	O
masses	O
under	O
the	O
effect	O
of	O
no	O
forces	O
would	O
travel	O
along	O
geodesics	O
on	O
surfaces	O
,	O
and	O
predicting	O
Einstein	O
's	O
fundamental	O
observation	O
of	O
the	O
equivalence	O
principle	O
a	O
full	O
60	O
years	O
before	O
it	O
appeared	O
in	O
the	O
scientific	O
literature	O
.	O
</s>
<s>
In	O
the	O
wake	O
of	O
Riemann	O
's	O
new	O
description	O
,	O
the	O
focus	O
of	O
techniques	O
used	O
to	O
study	O
differential	B-Language
geometry	I-Language
shifted	O
from	O
the	O
ad	O
hoc	O
and	O
extrinsic	O
methods	O
of	O
the	O
study	O
of	O
curves	O
and	O
surfaces	O
to	O
a	O
more	O
systematic	O
approach	O
in	O
terms	O
of	O
tensor	B-Device
calculus	O
and	O
Klein	O
's	O
Erlangen	O
program	O
,	O
and	O
progress	O
increased	O
in	O
the	O
field	O
.	O
</s>
<s>
The	O
notion	O
of	O
differential	O
calculus	O
on	O
curved	O
spaces	O
was	O
studied	O
by	O
Elwin	O
Christoffel	O
,	O
who	O
introduced	O
the	O
Christoffel	O
symbols	O
which	O
describe	O
the	O
covariant	O
derivative	O
in	O
1868	O
,	O
and	O
by	O
others	O
including	O
Eugenio	O
Beltrami	O
who	O
studied	O
many	O
analytic	O
questions	O
on	O
manifolds	B-Architecture
.	O
</s>
<s>
In	O
1899	O
Luigi	O
Bianchi	O
produced	O
his	O
Lectures	O
on	O
differential	B-Language
geometry	I-Language
which	O
studied	O
differential	B-Language
geometry	I-Language
from	O
Riemann	O
's	O
perspective	O
,	O
and	O
a	O
year	O
later	O
Tullio	O
Levi-Civita	O
and	O
Gregorio	O
Ricci-Curbastro	O
produced	O
their	O
textbook	O
systematically	O
developing	O
the	O
theory	O
of	O
absolute	O
differential	O
calculus	O
and	O
tensor	B-Device
calculus	O
.	O
</s>
<s>
It	O
was	O
in	O
this	O
language	O
that	O
differential	B-Language
geometry	I-Language
was	O
used	O
by	O
Einstein	O
in	O
the	O
development	O
of	O
general	O
relativity	O
and	O
pseudo-Riemannian	O
geometry	O
.	O
</s>
<s>
The	O
subject	O
of	O
modern	O
differential	B-Language
geometry	I-Language
emerged	O
out	O
of	O
the	O
early	O
1900s	O
in	O
response	O
to	O
the	O
foundational	O
contributions	O
of	O
many	O
mathematicians	O
,	O
including	O
importantly	O
the	O
work	O
of	O
Henri	O
Poincaré	O
on	O
the	O
foundations	O
of	O
topology	B-Architecture
.	O
</s>
<s>
As	O
part	O
of	O
this	O
broader	O
movement	O
,	O
the	O
notion	O
of	O
a	O
topological	O
space	O
was	O
distilled	O
in	O
by	O
Felix	O
Hausdorff	O
in	O
1914	O
,	O
and	O
by	O
1942	O
there	O
were	O
many	O
different	O
notions	O
of	O
manifold	B-Architecture
of	O
a	O
combinatorial	O
and	O
differential-geometric	O
nature	O
.	O
</s>
<s>
Einstein	O
's	O
theory	O
popularised	O
the	O
tensor	B-Device
calculus	O
of	O
Ricci	O
and	O
Levi-Civita	O
and	O
introduced	O
the	O
notation	O
for	O
a	O
Riemannian	O
metric	O
,	O
and	O
for	O
the	O
Christoffel	O
symbols	O
,	O
both	O
coming	O
from	O
G	O
in	O
Gravitation	O
.	O
</s>
<s>
Élie	O
Cartan	O
helped	O
reformulate	O
the	O
foundations	O
of	O
the	O
differential	B-Language
geometry	I-Language
of	O
smooth	O
manifolds	B-Architecture
in	O
terms	O
of	O
exterior	O
calculus	O
and	O
the	O
theory	O
of	O
moving	O
frames	O
,	O
leading	O
in	O
the	O
world	O
of	O
physics	O
to	O
Einstein	O
–	O
Cartan	O
theory	O
.	O
</s>
<s>
Following	O
this	O
early	O
development	O
,	O
many	O
mathematicians	O
contributed	O
to	O
the	O
development	O
of	O
the	O
modern	O
theory	O
,	O
including	O
Jean-Louis	O
Koszul	O
who	O
introduced	O
connections	O
on	O
vector	O
bundles	O
,	O
Shiing-Shen	O
Chern	O
who	O
introduced	O
characteristic	O
classes	O
to	O
the	O
subject	O
and	O
began	O
the	O
study	O
of	O
complex	O
manifolds	B-Architecture
,	O
Sir	O
William	O
Vallance	O
Douglas	O
Hodge	O
and	O
Georges	O
de	O
Rham	O
who	O
expanded	O
understanding	O
of	O
differential	O
forms	O
,	O
Charles	O
Ehresmann	O
who	O
introduced	O
the	O
theory	O
of	O
fibre	O
bundles	O
and	O
Ehresmann	O
connections	O
,	O
and	O
others	O
.	O
</s>
<s>
Of	O
particular	O
importance	O
was	O
Hermann	O
Weyl	O
who	O
made	O
important	O
contributions	O
to	O
the	O
foundations	O
of	O
general	O
relativity	O
,	O
introduced	O
the	O
Weyl	O
tensor	B-Device
providing	O
insight	O
into	O
conformal	O
geometry	O
,	O
and	O
first	O
defined	O
the	O
notion	O
of	O
a	O
gauge	O
leading	O
to	O
the	O
development	O
of	O
gauge	O
theory	O
in	O
physics	O
and	O
mathematics	O
.	O
</s>
<s>
In	O
the	O
middle	O
and	O
late	O
20th	O
century	O
differential	B-Language
geometry	I-Language
as	O
a	O
subject	O
expanded	O
in	O
scope	O
and	O
developed	O
links	O
to	O
other	O
areas	O
of	O
mathematics	O
and	O
physics	O
.	O
</s>
<s>
The	O
development	O
of	O
complex	O
geometry	O
was	O
spurred	O
on	O
by	O
parallel	O
results	O
in	O
algebraic	O
geometry	O
,	O
and	O
results	O
in	O
the	O
geometry	O
and	O
global	O
analysis	O
of	O
complex	O
manifolds	B-Architecture
were	O
proven	O
by	O
Shing-Tung	O
Yau	O
and	O
others	O
.	O
</s>
<s>
During	O
this	O
same	O
period	O
primarily	O
due	O
to	O
the	O
influence	O
of	O
Michael	O
Atiyah	O
,	O
new	O
links	O
between	O
theoretical	O
physics	O
and	O
differential	B-Language
geometry	I-Language
were	O
formed	O
.	O
</s>
<s>
Techniques	O
from	O
the	O
study	O
of	O
the	O
Yang	O
–	O
Mills	O
equations	O
and	O
gauge	O
theory	O
were	O
used	O
by	O
mathematicians	O
to	O
develop	O
new	O
invariants	O
of	O
smooth	O
manifolds	B-Architecture
.	O
</s>
<s>
Riemannian	O
geometry	O
studies	O
Riemannian	B-Architecture
manifolds	I-Architecture
,	O
smooth	O
manifolds	B-Architecture
with	O
a	O
Riemannian	O
metric	O
.	O
</s>
<s>
The	O
notion	O
of	O
a	O
directional	O
derivative	O
of	O
a	O
function	O
from	O
multivariable	O
calculus	O
is	O
extended	O
to	O
the	O
notion	O
of	O
a	O
covariant	O
derivative	O
of	O
a	O
tensor	B-Device
.	O
</s>
<s>
Many	O
concepts	O
of	O
analysis	O
and	O
differential	O
equations	O
have	O
been	O
generalized	O
to	O
the	O
setting	O
of	O
Riemannian	B-Architecture
manifolds	I-Architecture
.	O
</s>
<s>
A	O
distance-preserving	O
diffeomorphism	O
between	O
Riemannian	B-Architecture
manifolds	I-Architecture
is	O
called	O
an	O
isometry	O
.	O
</s>
<s>
However	O
,	O
the	O
Theorema	O
Egregium	O
of	O
Carl	O
Friedrich	O
Gauss	B-Algorithm
showed	O
that	O
for	O
surfaces	O
,	O
the	O
existence	O
of	O
a	O
local	O
isometry	O
imposes	O
that	O
the	O
Gaussian	O
curvatures	O
at	O
the	O
corresponding	O
points	O
must	O
be	O
the	O
same	O
.	O
</s>
<s>
In	O
higher	O
dimensions	O
,	O
the	O
Riemann	O
curvature	O
tensor	B-Device
is	O
an	O
important	O
pointwise	O
invariant	O
associated	O
with	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
that	O
measures	O
how	O
close	O
it	O
is	O
to	O
being	O
flat	O
.	O
</s>
<s>
An	O
important	O
class	O
of	O
Riemannian	B-Architecture
manifolds	I-Architecture
is	O
the	O
Riemannian	O
symmetric	O
spaces	O
,	O
whose	O
curvature	O
is	O
not	O
necessarily	O
constant	O
.	O
</s>
<s>
Pseudo-Riemannian	O
geometry	O
generalizes	O
Riemannian	O
geometry	O
to	O
the	O
case	O
in	O
which	O
the	O
metric	O
tensor	B-Device
need	O
not	O
be	O
positive-definite	O
.	O
</s>
<s>
A	O
special	O
case	O
of	O
this	O
is	O
a	O
Lorentzian	O
manifold	B-Architecture
,	O
which	O
is	O
the	O
mathematical	O
basis	O
of	O
Einstein	O
's	O
general	O
relativity	O
theory	O
of	O
gravity	O
.	O
</s>
<s>
Finsler	O
geometry	O
has	O
Finsler	O
manifolds	B-Architecture
as	O
the	O
main	O
object	O
of	O
study	O
.	O
</s>
<s>
This	O
is	O
a	O
differential	O
manifold	B-Architecture
with	O
a	O
Finsler	O
metric	O
,	O
that	O
is	O
,	O
a	O
Banach	O
norm	O
defined	O
on	O
each	O
tangent	O
space	O
.	O
</s>
<s>
Riemannian	B-Architecture
manifolds	I-Architecture
are	O
special	O
cases	O
of	O
the	O
more	O
general	O
Finsler	O
manifolds	B-Architecture
.	O
</s>
<s>
A	O
Finsler	O
structure	O
on	O
a	O
manifold	B-Architecture
is	O
a	O
function	O
such	O
that	O
:	O
</s>
<s>
Symplectic	O
geometry	O
is	O
the	O
study	O
of	O
symplectic	O
manifolds	B-Architecture
.	O
</s>
<s>
An	O
almost	O
symplectic	O
manifold	B-Architecture
is	O
a	O
differentiable	O
manifold	B-Architecture
equipped	O
with	O
a	O
smoothly	O
varying	O
non-degenerate	B-Algorithm
skew-symmetric	B-Algorithm
bilinear	O
form	O
on	O
each	O
tangent	O
space	O
,	O
i.e.	O
,	O
a	O
nondegenerate	O
2-form	O
ω	O
,	O
called	O
the	O
symplectic	O
form	O
.	O
</s>
<s>
A	O
symplectic	O
manifold	B-Architecture
is	O
an	O
almost	O
symplectic	O
manifold	B-Architecture
for	O
which	O
the	O
symplectic	O
form	O
ω	O
is	O
closed	O
:	O
.	O
</s>
<s>
A	O
diffeomorphism	O
between	O
two	O
symplectic	O
manifolds	B-Architecture
which	O
preserves	O
the	O
symplectic	O
form	O
is	O
called	O
a	O
symplectomorphism	O
.	O
</s>
<s>
Non-degenerate	B-Algorithm
skew-symmetric	B-Algorithm
bilinear	O
forms	O
can	O
only	O
exist	O
on	O
even-dimensional	O
vector	O
spaces	O
,	O
so	O
symplectic	O
manifolds	B-Architecture
necessarily	O
have	O
even	O
dimension	O
.	O
</s>
<s>
In	O
dimension	O
2	O
,	O
a	O
symplectic	O
manifold	B-Architecture
is	O
just	O
a	O
surface	O
endowed	O
with	O
an	O
area	O
form	O
and	O
a	O
symplectomorphism	O
is	O
an	O
area-preserving	O
diffeomorphism	O
.	O
</s>
<s>
The	O
phase	O
space	O
of	O
a	O
mechanical	O
system	O
is	O
a	O
symplectic	O
manifold	B-Architecture
and	O
they	O
made	O
an	O
implicit	O
appearance	O
already	O
in	O
the	O
work	O
of	O
Joseph	O
Louis	O
Lagrange	O
on	O
analytical	O
mechanics	O
and	O
later	O
in	O
Carl	O
Gustav	O
Jacobi	O
's	O
and	O
William	O
Rowan	O
Hamilton	O
's	O
formulations	O
of	O
classical	O
mechanics	O
.	O
</s>
<s>
By	O
contrast	O
with	O
Riemannian	O
geometry	O
,	O
where	O
the	O
curvature	O
provides	O
a	O
local	O
invariant	O
of	O
Riemannian	B-Architecture
manifolds	I-Architecture
,	O
Darboux	O
's	O
theorem	O
states	O
that	O
all	O
symplectic	O
manifolds	B-Architecture
are	O
locally	O
isomorphic	O
.	O
</s>
<s>
The	O
only	O
invariants	O
of	O
a	O
symplectic	O
manifold	B-Architecture
are	O
global	O
in	O
nature	O
and	O
topological	O
aspects	O
play	O
a	O
prominent	O
role	O
in	O
symplectic	O
geometry	O
.	O
</s>
<s>
The	O
first	O
result	O
in	O
symplectic	O
topology	B-Architecture
is	O
probably	O
the	O
Poincaré	O
–	O
Birkhoff	O
theorem	O
,	O
conjectured	O
by	O
Henri	O
Poincaré	O
and	O
then	O
proved	O
by	O
G.D.	O
Birkhoff	O
in	O
1912	O
.	O
</s>
<s>
Contact	O
geometry	O
deals	O
with	O
certain	O
manifolds	B-Architecture
of	O
odd	O
dimension	O
.	O
</s>
<s>
A	O
contact	O
structure	O
on	O
a	O
-dimensional	O
manifold	B-Architecture
M	O
is	O
given	O
by	O
a	O
smooth	O
hyperplane	O
field	O
H	O
in	O
the	O
tangent	O
bundle	O
that	O
is	O
as	O
far	O
as	O
possible	O
from	O
being	O
associated	O
with	O
the	O
level	O
sets	O
of	O
a	O
differentiable	O
function	O
on	O
M	O
(	O
the	O
technical	O
term	O
is	O
"	O
completely	O
nonintegrable	O
tangent	O
hyperplane	O
distribution	O
"	O
)	O
.	O
</s>
<s>
A	O
local	O
1-form	O
on	O
M	O
is	O
a	O
contact	O
form	O
if	O
the	O
restriction	O
of	O
its	O
exterior	O
derivative	O
to	O
H	O
is	O
a	O
non-degenerate	B-Algorithm
two-form	O
and	O
thus	O
induces	O
a	O
symplectic	O
structure	O
on	O
Hp	O
at	O
each	O
point	O
.	O
</s>
<s>
A	O
contact	O
analogue	O
of	O
the	O
Darboux	O
theorem	O
holds	O
:	O
all	O
contact	O
structures	O
on	O
an	O
odd-dimensional	O
manifold	B-Architecture
are	O
locally	O
isomorphic	O
and	O
can	O
be	O
brought	O
to	O
a	O
certain	O
local	O
normal	O
form	O
by	O
a	O
suitable	O
choice	O
of	O
the	O
coordinate	O
system	O
.	O
</s>
<s>
Complex	O
differential	B-Language
geometry	I-Language
is	O
the	O
study	O
of	O
complex	O
manifolds	B-Architecture
.	O
</s>
<s>
An	O
almost	O
complex	O
manifold	B-Architecture
is	O
a	O
real	B-Architecture
manifold	I-Architecture
,	O
endowed	O
with	O
a	O
tensor	B-Device
of	O
type	O
(	O
1	O
,	O
1	O
)	O
,	O
i.e.	O
</s>
<s>
It	O
follows	O
from	O
this	O
definition	O
that	O
an	O
almost	O
complex	O
manifold	B-Architecture
is	O
even-dimensional	O
.	O
</s>
<s>
An	O
almost	O
complex	O
manifold	B-Architecture
is	O
called	O
complex	O
if	O
,	O
where	O
is	O
a	O
tensor	B-Device
of	O
type	O
(	O
2	O
,	O
1	O
)	O
related	O
to	O
,	O
called	O
the	O
Nijenhuis	O
tensor	B-Device
(	O
or	O
sometimes	O
the	O
torsion	O
)	O
.	O
</s>
<s>
An	O
almost	O
complex	O
manifold	B-Architecture
is	O
complex	O
if	O
and	O
only	O
if	O
it	O
admits	O
a	O
holomorphic	O
coordinate	O
atlas	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
is	O
called	O
a	O
Kähler	O
structure	O
,	O
and	O
a	O
Kähler	O
manifold	B-Architecture
is	O
a	O
manifold	B-Architecture
endowed	O
with	O
a	O
Kähler	O
structure	O
.	O
</s>
<s>
In	O
particular	O
,	O
a	O
Kähler	O
manifold	B-Architecture
is	O
both	O
a	O
complex	O
and	O
a	O
symplectic	O
manifold	B-Architecture
.	O
</s>
<s>
A	O
large	O
class	O
of	O
Kähler	O
manifolds	B-Architecture
(	O
the	O
class	O
of	O
Hodge	O
manifolds	B-Architecture
)	O
is	O
given	O
by	O
all	O
the	O
smooth	O
complex	O
projective	O
varieties	O
.	O
</s>
<s>
CR	O
geometry	O
is	O
the	O
study	O
of	O
the	O
intrinsic	O
geometry	O
of	O
boundaries	O
of	O
domains	O
in	O
complex	O
manifolds	B-Architecture
.	O
</s>
<s>
Differential	O
topology	B-Architecture
is	O
the	O
study	O
of	O
global	O
geometric	O
invariants	O
without	O
a	O
metric	O
or	O
symplectic	O
form	O
.	O
</s>
<s>
Differential	O
topology	B-Architecture
starts	O
from	O
the	O
natural	O
operations	O
such	O
as	O
Lie	O
derivative	O
of	O
natural	O
vector	O
bundles	O
and	O
de	O
Rham	O
differential	O
of	O
forms	O
.	O
</s>
<s>
A	O
Lie	O
group	O
is	O
a	O
group	O
in	O
the	O
category	O
of	O
smooth	O
manifolds	B-Architecture
.	O
</s>
<s>
Geometric	O
analysis	O
is	O
a	O
mathematical	O
discipline	O
where	O
tools	O
from	O
differential	O
equations	O
,	O
especially	O
elliptic	O
partial	O
differential	O
equations	O
are	O
used	O
to	O
establish	O
new	O
results	O
in	O
differential	B-Language
geometry	I-Language
and	O
differential	O
topology	B-Architecture
.	O
</s>
<s>
The	O
apparatus	O
of	O
vector	O
bundles	O
,	O
principal	O
bundles	O
,	O
and	O
connections	O
on	O
bundles	O
plays	O
an	O
extraordinarily	O
important	O
role	O
in	O
modern	O
differential	B-Language
geometry	I-Language
.	O
</s>
<s>
A	O
smooth	O
manifold	B-Architecture
always	O
carries	O
a	O
natural	O
vector	O
bundle	O
,	O
the	O
tangent	O
bundle	O
.	O
</s>
<s>
Loosely	O
speaking	O
,	O
this	O
structure	O
by	O
itself	O
is	O
sufficient	O
only	O
for	O
developing	O
analysis	O
on	O
the	O
manifold	B-Architecture
,	O
while	O
doing	O
geometry	O
requires	O
,	O
in	O
addition	O
,	O
some	O
way	O
to	O
relate	O
the	O
tangent	O
spaces	O
at	O
different	O
points	O
,	O
i.e.	O
</s>
<s>
More	O
generally	O
,	O
differential	B-Language
geometers	I-Language
consider	O
spaces	O
with	O
a	O
vector	O
bundle	O
and	O
an	O
arbitrary	O
affine	O
connection	O
which	O
is	O
not	O
defined	O
in	O
terms	O
of	O
a	O
metric	O
.	O
</s>
<s>
In	O
physics	O
,	O
the	O
manifold	B-Architecture
may	O
be	O
spacetime	B-Protocol
and	O
the	O
bundles	O
and	O
connections	O
are	O
related	O
to	O
various	O
physical	O
fields	O
.	O
</s>
<s>
From	O
the	O
beginning	O
and	O
through	O
the	O
middle	O
of	O
the	O
19th	O
century	O
,	O
differential	B-Language
geometry	I-Language
was	O
studied	O
from	O
the	O
extrinsic	O
point	O
of	O
view	O
:	O
curves	O
and	O
surfaces	O
were	O
considered	O
as	O
lying	O
in	O
a	O
Euclidean	O
space	O
of	O
higher	O
dimension	O
(	O
for	O
example	O
a	O
surface	O
in	O
an	O
ambient	O
space	O
of	O
three	O
dimensions	O
)	O
.	O
</s>
<s>
The	O
simplest	O
results	O
are	O
those	O
in	O
the	O
differential	B-Language
geometry	I-Language
of	O
curves	O
and	O
differential	B-Language
geometry	I-Language
of	O
surfaces	O
.	O
</s>
<s>
The	O
fundamental	O
result	O
here	O
is	O
Gauss	B-Algorithm
's	O
theorema	O
egregium	O
,	O
to	O
the	O
effect	O
that	O
Gaussian	O
curvature	O
is	O
an	O
intrinsic	O
invariant	O
.	O
</s>
<s>
In	O
the	O
formalism	O
of	O
geometric	O
calculus	O
both	O
extrinsic	O
and	O
intrinsic	O
geometry	O
of	O
a	O
manifold	B-Architecture
can	O
be	O
characterized	O
by	O
a	O
single	O
bivector-valued	O
one-form	O
called	O
the	O
shape	O
operator	O
.	O
</s>
<s>
Below	O
are	O
some	O
examples	O
of	O
how	O
differential	B-Language
geometry	I-Language
is	O
applied	O
to	O
other	O
fields	O
of	O
science	O
and	O
mathematics	O
.	O
</s>
<s>
In	O
physics	O
,	O
differential	B-Language
geometry	I-Language
has	O
many	O
applications	O
,	O
including	O
:	O
</s>
<s>
Differential	B-Language
geometry	I-Language
is	O
the	O
language	O
in	O
which	O
Albert	O
Einstein	O
's	O
general	O
theory	O
of	O
relativity	O
is	O
expressed	O
.	O
</s>
<s>
According	O
to	O
the	O
theory	O
,	O
the	O
universe	O
is	O
a	O
smooth	O
manifold	B-Architecture
equipped	O
with	O
a	O
pseudo-Riemannian	O
metric	O
,	O
which	O
describes	O
the	O
curvature	O
of	O
spacetime	B-Protocol
.	O
</s>
<s>
Understanding	O
this	O
curvature	O
is	O
essential	O
for	O
the	O
positioning	O
of	O
satellites	B-Application
into	O
orbit	O
around	O
the	O
earth	O
.	O
</s>
<s>
Differential	B-Language
geometry	I-Language
is	O
also	O
indispensable	O
in	O
the	O
study	O
of	O
gravitational	O
lensing	O
and	O
black	B-Application
holes	I-Application
.	O
</s>
<s>
Differential	B-Language
geometry	I-Language
has	O
applications	O
to	O
both	O
Lagrangian	O
mechanics	O
and	O
Hamiltonian	O
mechanics	O
.	O
</s>
<s>
Symplectic	O
manifolds	B-Architecture
in	O
particular	O
can	O
be	O
used	O
to	O
study	O
Hamiltonian	O
systems	O
.	O
</s>
<s>
In	O
economics	O
,	O
differential	B-Language
geometry	I-Language
has	O
applications	O
to	O
the	O
field	O
of	O
econometrics	O
.	O
</s>
<s>
Geometric	B-Application
modeling	I-Application
(	O
including	O
computer	O
graphics	O
)	O
and	O
computer-aided	B-Application
geometric	I-Application
design	I-Application
draw	O
on	O
ideas	O
from	O
differential	B-Language
geometry	I-Language
.	O
</s>
<s>
In	O
engineering	O
,	O
differential	B-Language
geometry	I-Language
can	O
be	O
applied	O
to	O
solve	O
problems	O
in	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
.	O
</s>
<s>
In	O
probability	O
,	O
statistics	O
,	O
and	O
information	O
theory	O
,	O
one	O
can	O
interpret	O
various	O
structures	O
as	O
Riemannian	B-Architecture
manifolds	I-Architecture
,	O
which	O
yields	O
the	O
field	O
of	O
information	O
geometry	O
,	O
particularly	O
via	O
the	O
Fisher	O
information	O
metric	O
.	O
</s>
<s>
In	O
structural	O
geology	O
,	O
differential	B-Language
geometry	I-Language
is	O
used	O
to	O
analyze	O
and	O
describe	O
geologic	O
structures	O
.	O
</s>
<s>
In	O
computer	B-Application
vision	I-Application
,	O
differential	B-Language
geometry	I-Language
is	O
used	O
to	O
analyze	O
shapes	O
.	O
</s>
<s>
In	O
image	B-Algorithm
processing	I-Algorithm
,	O
differential	B-Language
geometry	I-Language
is	O
used	O
to	O
process	O
and	O
analyse	O
data	O
on	O
non-flat	O
surfaces	O
.	O
</s>
<s>
Grigori	O
Perelman	O
's	O
proof	O
of	O
the	O
Poincaré	O
conjecture	O
using	O
the	O
techniques	O
of	O
Ricci	O
flows	O
demonstrated	O
the	O
power	O
of	O
the	O
differential-geometric	O
approach	O
to	O
questions	O
in	O
topology	B-Architecture
and	O
it	O
highlighted	O
the	O
important	O
role	O
played	O
by	O
its	O
analytic	O
methods	O
.	O
</s>
<s>
In	O
wireless	O
communications	O
,	O
Grassmannian	O
manifolds	B-Architecture
are	O
used	O
for	O
beamforming	B-Algorithm
techniques	O
in	O
multiple	O
antenna	O
systems	O
.	O
</s>
