Cotangent Vectors, Differentials, and the Cotangent Bundle
There’s another construct in differential topology and geometry that isn’t quite so obvious as a tangent vector, but which is every bit as useful: a cotangent vector. A cotangent vector 
at a point 
is just an element of the dual space to 
, which we write as 
.
We actually have a really nice example of cotangent vectors already: a gadget that takes a tangent vector at 
and gives back a number. It’s the differential, which when given a vector returns the directional derivative in that direction. And we can generalize that right away.
Indeed, if 
is a smooth germ at , then we have a linear functional 
defined for all tangent vectors 
. We will call this functional the differential of at , and write =v(f)](https://s0.wp.com/latex.php?latex=%5Cleft%5Bdf%28p%29%5Cright%5D%28v%29%3Dv%28f%29&bg=e6e6e6&fg=333333&s=0&c=20201002)
.
If we have local coordinates 
at , then each coordinate function 
is a smooth function, which has differential 
. These actually furnish the dual basis to the coordinate vectors 
. Indeed, we calculate
![\displaystyle\begin{aligned}\left[dx^i(p)\right]\left(\frac{\partial}{\partial x^j}(p)\right)&=\left[\frac{\partial}{\partial x^j}\right](x^i)\\&=\left[D_j(u^i\circ x\circ x^{-1})\right](x(p))\\&=\delta_j^i\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5Bdx%5Ei%28p%29%5Cright%5D%5Cleft%28%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%5Ej%7D%28p%29%5Cright%29%26%3D%5Cleft%5B%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%5Ej%7D%5Cright%5D%28x%5Ei%29%5C%5C%26%3D%5Cleft%5BD_j%28u%5Ei%5Ccirc+x%5Ccirc+x%5E%7B-1%7D%29%5Cright%5D%28x%28p%29%29%5C%5C%26%3D%5Cdelta_j%5Ei%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0&c=20201002)
That is, evaluating the coordinate differential on the coordinate vector 
gives the value 
if 
and 
otherwise.
Of course, the 
define a basis of at every point 
, just like the define a basis of at every point . This was exactly what we needed to compare vectors — at least to some extent — at points within a local coordinate patch, and let us define the tangent bundle as a 
-dimensional manifold.
In exactly the same way, we can define the cotangent bundle 
. Given the coordinate patch we define a coordinate patch covering all the cotangent spaces with . The coordinate map is defined on a cotangent vector 
by
Everything else in the construction of the cotangent bundle proceeds exactly as it did for the tangent bundle, but we’re missing one thing: how to translate from one basis of coordinate differentials to another.
So, let’s say 
and 
are two coordinate maps at , defining coordinate differentials and 
. How are these two bases related? We can calculate this by applying to :
![\displaystyle\begin{aligned}\left[dy^j(p)\right]\left(\frac{\partial}{\partial x^i}(p)\right)&=\left[\frac{\partial}{\partial x^i}\right](y^j)\\&=\left[D_i(u^j\circ y\circ x^{-1})\right](x(p))\\&=J_i^j(p)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5Bdy%5Ej%28p%29%5Cright%5D%5Cleft%28%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%5Ei%7D%28p%29%5Cright%29%26%3D%5Cleft%5B%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%5Ei%7D%5Cright%5D%28y%5Ej%29%5C%5C%26%3D%5Cleft%5BD_i%28u%5Ej%5Ccirc+y%5Ccirc+x%5E%7B-1%7D%29%5Cright%5D%28x%28p%29%29%5C%5C%26%3DJ_i%5Ej%28p%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0&c=20201002)
where 
are the components of the Jacobian matrix of the transition function 
. What does this mean? Well, consider the linear functional
This has the same values on each of the as 
does, and we conclude that they are, in fact, the same cotangent vector:
On the other hand, recall that
That is, we use the Jacobian of one transition function to transform from the basis to the basis of , but the transpose of the same Jacobian to transform from the basis to the 
basis of . And this is actually just as we expect, since the transpose is actually the adjoint transformation, which automatically connects the dual spaces.
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April 13, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
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