0Introduction
III Differential Geometry
0 Introduction
In differential geometry, the main object of study is a manifold. The motivation
is as follows — from IA, we know well how to do calculus on
R
n
. We can talk
about continuity, differentiable functions, derivatives etc. happily ever after.
However, sometimes, we want to do calculus on things other than
R
n
. Say,
we live on a sphere, namely the Earth. Does it make sense to “do calculus” on a
sphere? Surely it does.
The key insight is that these notions of differentiability, derivatives etc. are
local properties. To know if a function is differentiable at a point
p
, we only need
to know how the function behaves near
p
, and similarly such local information
tells us how to compute derivatives. The reason we can do calculus on a sphere
is because the sphere looks locally like
R
n
. Therefore, we can make sense of
calculus on a sphere.
Thus, what we want to do is to study calculus on things that look locally like
R
n
, and these are known as manifolds. Most of the time, our definitions from
usual calculus on
R
n
transfer directly to manifolds. However, sometimes the
global properties of our manifold will give us some new exciting things.
In fact, we’ve already seen such things when we did IA Vector Calculus. If
we have a vector field
R
3
→ R
3
whose curl vanishes everywhere, then we know
it is the gradient of some function. However, if we consider such a vector field
on
R
3
\ {
0
}
instead, then this is no longer true! Here the global topology of the
space gives rise to interesting phenomena we do not see at a local level.
When doing differential geometry, it is important to keep in mind that
what we’ve learnt in vector calculus is actually a mess.
R
3
has a lot of special
properties. Apart from being a topological space, it is also canonically a vector
space, and in fact an inner product space. When we did vector calculus, these
extra structure allowed us conflate many different concepts together. However,
when we pass on to manifolds, we no longer have these identifications, and we
have to be more careful.