The Unapologetic Mathematician

Mathematics for the interested outsider

Maps Intertwining Vector Fields

Let f:M\to N be a smooth map between manifolds, with derivative f_*:\mathcal{T}M\to\mathcal{T}N, and let X:M\to\mathcal{T}M and Y:N\to\mathcal{T}N be smooth vector fields. We can compose them as f_*\circ X:M\to\mathcal{T}N and Y\circ f:M\to\mathcal{T}N, and it makes sense to ask if these are the same map.

To put it another way, Y\circ f:p\mapsto Y_{f(p)} is the vector Y specifies for the point f(p). On the other hand, f_*\circ X:p\mapsto f_*(X_p) is the image of the vector X specifies for the point p. If these two vectors are the same for every p\in M, then we say that f “intertwines” the two vector fields, or that X and Y are “f-related”. The latter term is a bit awkward, which is why I prefer the former, especially since it does have that same commutative-diagram feel as intertwinors between representations.

Anyway, in the case that f is a diffeomorphism we can actually use this to transfer vector fields from one manifold to the other. Given a point q\in N, which point should it be compared to? the inverse image f^{-1}(q), of course. This point gets the vector X\left(f^{-1}(q)\right), which then gets sent to f_*\left(X\left(f^{-1}(q)\right)\right). That is, if we define Y=f_*\circ X\circ f^{-1}, then f is guaranteed to intertwine X and Y.

Since f_* is a linear map on each stalk it’s clear that if f intertwines X_1 and Y_1, as well as X_2 and Y_2, then f intertwines c_1X_1+c_2X_2 and c_1Y_1+c_2Y_2. But we’ve just seen that vector fields form a Lie algebra, and it would be nice if we could say the same for [X_1,X_2] and [Y_1,Y_2]. The catch is that we don’t just compute these point-by-point.

Let’s pick a text function \phi\in\mathcal{O}N and a point p\in M. We first check that

\displaystyle\begin{aligned}{}[(Y_i\phi)\circ f](q)&=(Y_i\phi)(f(q))\\&=Y_{i,f(q)}\phi\\&=[f_*X_{i,q}]\phi\\&=X_{i,q}(\phi\circ f)\end{aligned}

Now we can calculate

\displaystyle\begin{aligned}{}[Y_1,Y_2]_{f(p)}\phi&=Y_{1,f(p)}(Y_2\phi)-Y_{2,f(p)}(Y_1\phi)\\&=f_*X_{1,p}(Y_2\phi)-f_*X_{2,p}(Y_1\phi)\\&=X_{1,p}((Y_2\phi)\circ f)-X_{2,p}((Y_1\phi)\circ f)\\&=X_{1,p}(X_2(\phi\circ f))-X_{2,p}(X_1(\phi\circ f))\\&=[X_1,X_2]_p(\phi\circ f)\\&=(f_*[X_1,X_2]_p)\phi\end{aligned}

So f intertwines [X_1,X_2] and [Y_1,Y_2], as we asserted. In the case where f is a diffeomorphism, this means that the construction above gives us a homomorphism of Lie algebras from \mathcal{T}M to \mathcal{T}N.

June 3, 2011 - Posted by | Differential Topology, Topology


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