Geometry
is
an
original
field
of
mathematics,
and
is
indeed
the
oldest
of
all
sciences,
going
back
at
least
to
the
times
of
Euclid,
Pythagoras,
and
other
“natural
philosophers”
of
ancient
Greece.
Initially,
geometry
was
studied
to
understand
the
physical
world
we
live
in,
and
the
tradition
continues
to
this
day.
Witness
for
example,
the
spectacular
success
of
Einstein's
theory
of
general
relativity,
a
purely
geometric
theory
that
describes
gravitation
in
terms
of
the
curvature
of
a
four-dimensional
“spacetime”.
However,
geometry
transcends
far
beyond
physical
applications,
and
it
is
not
unreasonable
to
say
that
geometric
ideas
and
methods
have
always
permeated
every
field
of
mathematics.
In
modern
language,
the
central
object
of
study
in
geometry
is
a
manifold,
which
is
an
object
that
may
have
a
complicated
overall
shape,
but
such
that
on
small
scales
it
“looks
like”
ordinary
space
of
a
certain
dimension.
For
example,
a
1-dimensional
manifold
is
an
object
such
that
small
pieces
of
it
look
like
a
line,
although
in
general
it
looks
like
a
curve
rather
than
a
straight
line.
A
2-dimensional
manifold,
on
small
scales,
looks
like
a
(curved)
piece
of
paper
–
there
are
two
independent
directions
in
which
we
can
move
at
any
point.
For
example,
the
surface
of
the
Earth
is
a
2-dimensional
manifold.
An
n-dimensional
manifold
likewise
looks
locally
like
an
ordinary
n-dimensional
space.
This
does
not
necessarily
correspond
to
any
notion
of
“physical
space”.
As
an
example,
the
data
of
the
position
and
velocity
of
N
particles
in
a
room
is
described
by
6N
independent
variables,
because
each
particle
needs
3
numbers
to
describe
its
position
and
3
more
numbers
to
describe
its
velocity.
Hence,
the
“configuration
space”
of
this
system
is
a
6N-dimensional
manifold.
If
for
some
reason
the
motion
of
these
particles
were
not
independent
but
rather
constrained
in
some
way,
then
the
configuration
space
would
be
a
manifold
of
smaller
dimension.
Usually,
the
set
of
solutions
of
a
system
of
partial
differential
equations
has
the
structure
of
some
high
dimensional
manifold.
Understanding
the
“geometry”
of
this
manifold
often
gives
new
insight
into
the
nature
of
these
solutions,
and
to
the
actual
phenomenon
that
is
modeled
by
the
differential
equations,
whether
it
comes
from
physics,
economics,
engineering,
or
any
other
quantitative
science.
A
typical
problem
in
geometry
is
to
“classify”
all
manifolds
of
a
certain
type.
That
is,
we
first
decide
which
kinds
of
manifolds
we
are
interested
in,
then
decide
when
two
such
manifolds
should
basically
be
considered
to
be
the
same,
or
“equivalent”,
and
finally
try
to
determine
how
many
inequivalent
types
of
such
manifolds
exist.
For
example,
we
might
be
interested
in
studying
surfaces
(2-dimensional
manifolds)
that
lie
inside
the
usual
3-dimensional
space
that
we
can
see,
and
we
might
decide
that
two
such
surfaces
are
equivalent
if
one
can
be
“transformed”
into
the
other
by
translations
or
rotations.
This
is
the
study
of
the
Riemannian
geometry
of
surfaces
immersed
in
3-space,
and
was
classically
the
first
subfield
of
“differential
geometry”,
pioneered
by
mathematical
giants
such
as
Gauss
and
Riemann
in
the
1800’s.
Today,
there
are
many
different
subfields
of
geometry
that
are
actively
studied.
Here
we
describe
only
a
few
of
them:
- Riemannian geometry. This is the study of manifolds equipped with the additional structure of a Riemannian metric, which is a rule for measuring lengths of curves and angles between tangent vectors. A Riemannian manifold has curvature, and it is precisely this curvature that makes the laws of classical Euclidean geometry, that we learn in elementary school, to be different. For example, the sum of the interior angles of a “triangle” on a curved Riemannian manifold can be more or less than πif the curvature is positive or negative, respectively.
- Algebraic geometry. This is the study of algebraic varieties, which are solution sets of systems of polynomial equations. They are sometimes manifolds but also often have “singular points” at which they are not “smooth”. Because they are defined algebraically, there are many more tools available from abstract algebra to study them, and conversely many questions in pure algebra can be understood better by reformulating the problem in terms of algebraic geometry. Moreover, one can study varieties over any field, not just the real or complex numbers.
- Symplectic geometry. This is the study of manifolds equipped with an additional structure called a symplectic form. A symplectic form is in some sense (that can be made precise) the opposite of a Riemannian metric, and symplectic manifolds exhibit very different behaviour from Riemannian manifolds. For example, a famous theorem of Darboux says that all symplectic manifolds are “locally” the same, although globally they can be extremely different. Such a theorem is far from true in Riemannian geometry. Symplectic manifolds arise naturally in physical systems from classical mechanics, and are called “phases spaces” in physics. This branch of geometry is very topological in nature.
- Complex geometry. This is the study of manifolds which locally “look like” ordinary n-dimensional spaces that are modeled on the complex numbers rather than the real numbers. Because the analysis of holomorphic (or complex-analytic) functions is much more rigid than the real case (for example not all real smooth functions are real-analytic) there are many fewer “types” of complex manifolds, and there has been more success in (at least partial) classifications. This field is also very closely related to algebraic geometry.
The
above
list
is
far
from
exhaustive.
For
example,
the
field
of
Kaehler
geometry
is
in
some
sense
the
study
of
manifolds
which
lie
in
the
intersection
of
the
above
four
subfields.
Finally,
another
very
important
area
of
geometry
is
the
study
of
connections
(and
their
curvature)
on
vector
bundles,
also
commonly
called
“gauge
theory”.
This
field
was
independently
developed
by
both
physicists
and
mathematicians
around
the
1950’s.
When
the
two
camps
finally
got
together
in
the
1970’s
to
communicate,
led
by
renowned
figures
such
as
Atiyah,
Bott,
Singer,
and
Witten,
there
resulted
a
spectacular
succession
of
important
new
advances
in
both
fields.
Some
of
these
accomplishments
include
the
existence
of
“exotic”
4-dimensional
manifolds
and
the
discovery
of
new
invariants
that
distinguish
different
types
of
spaces.
Geometry
is
more
active
and
exciting
than
ever,
even
after
3000
years.
And
there
is
no
sign
of
it
letting
up.