# The Unapologetic Mathematician

## 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 $\lambda$ at a point $p\in M$ is just an element of the dual space to $\mathcal{T}_pM$, which we write as $\mathcal{T}^*_pM$.

We actually have a really nice example of cotangent vectors already: a gadget that takes a tangent vector at $p$ 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 $f$ is a smooth germ at $p$, then we have a linear functional $v\mapsto v(f)$ defined for all tangent vectors $v\in\mathcal{T}_pM$. We will call this functional the differential of $f$ at $p$, and write $\left[df(p)\right](v)=v(f)$.

If we have local coordinates $(U,x)$ at $p$, then each coordinate function $x^i$ is a smooth function, which has differential $dx^i(p)$. These actually furnish the dual basis to the coordinate vectors $\frac{\partial}{\partial x^i}(p)$. 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}

That is, evaluating the coordinate differential $dx^i(p)$ on the coordinate vector $\frac{\partial}{\partial x^j}(p)$ gives the value $1$ if $i=j$ and $0$ otherwise.

Of course, the $dx^j(p)$ define a basis of $\mathcal{T}^*_pM$ at every point $p\in U$, just like the $\frac{\partial}{\partial x^j}(p)$ define a basis of $\mathcal{T}_pM$ at every point $p\in U$. 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 $2n$-dimensional manifold.

In exactly the same way, we can define the cotangent bundle $\mathcal{T}^*M$. Given the coordinate patch $(U,x)$ we define a coordinate patch covering all the cotangent spaces $\mathcal{T}^*_pM$ with $p\in U$. The coordinate map is defined on a cotangent vector $\lambda\in\mathcal{T}^*_pM$ by $\displaystyle\tilde{x}(\lambda)=\left(x^1(p),\dots,x^n(p),\lambda\left(\frac{\partial}{\partial x^1}(p)\right),\dots,\lambda\left(\frac{\partial}{\partial x^n}(p)\right)\right)$

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 $x$ and $y$ are two coordinate maps at $p$, defining coordinate differentials $dx^i(p)$ and $dy^j(p)$. How are these two bases related? We can calculate this by applying $dy^j(p)$ to $\frac{\partial}{\partial x^j}(p)$: \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}

where $J_i^j(p)$ are the components of the Jacobian matrix of the transition function $y\circ x^{-1}$. What does this mean? Well, consider the linear functional $\displaystyle\sum\limits_iJ_i^j(p)dx^i(p)$

This has the same values on each of the $\frac{\partial}{\partial x^i}(p)$ as $dy^j$ does, and we conclude that they are, in fact, the same cotangent vector: $\displaystyle dy^j(p)=\sum\limits_iJ_i^j(p)dx^i(p)$

On the other hand, recall that $\displaystyle\frac{\partial}{\partial x^i}(p)=\sum\limits_jJ_i^j(p)\frac{\partial}{\partial y^j}(p)$

That is, we use the Jacobian of one transition function to transform from the $dx^i(p)$ basis to the $dy^j(p)$ basis of $\mathcal{T}^*_pM$, but the transpose of the same Jacobian to transform from the $\frac{\partial}{\partial x^i}(p)$ basis to the $\frac{\partial}{\partial y^j}(p)$ basis of $\mathcal{T}_pM$. And this is actually just as we expect, since the transpose is actually the adjoint transformation, which automatically connects the dual spaces.

April 13, 2011