Pushfoward of smooth vector field is smooth?

Solution 1:

Here is the main issue: how can one define the pushforward of a vector field? Rather – when? Take two copies of the real line, parametrized as $M=\mathbb{R}\times \{0\}\sqcup\mathbb{R}\times \{1\}\subseteq \mathbb{R}^2$ and define a map $$ \pi:M\to \mathbb{R}$$ by $\pi(x,n)=x$. Then take the vector field defined by $X\in \mathfrak{X}(M)$ with $X_{(x,0)}=-1$ and $X_{(x,1)}=1$ for all $x\in \mathbb{R}$. Then, if we calculate $\pi_{*,(x,0)}X_{(x,0)}=-1\in T_x\mathbb{R}$ and $\pi_{*,(x,1)} X_{(x,1)}=1\in T_x\mathbb{R}$. The sensible way to (try to) define the pushforward vector field here is to set $Y=\pi_{*}X$ to be $Y_p=\pi_{*,q}(X_q)$ for some $q\in M$ with $\pi(q)=p$.

Unfortunately, this won't work because if for instance we choose $p=1$, then its preimages under $\pi$ are $(1,0)$ and $(1,1)$. If we choose $q=(1,0)$ we get $Y_p=-1$ and if we choose $q=0$ we get $Y_p=1$. So, the pushforward vector field is not well-defined in general.

The only way this could be defined is if the following criterion is met:

Let $F:M\to N$ denote a map of $\mathscr{C}^\infty$ manifolds. Then given $X\in \mathfrak{X}(M)$, there exists a vector field $F_*X\in \mathfrak{X}(F(M))$ defined as above if and only if for each $p\in N$, $F_{*,q}(X_{q})=F_{*,q'}(X_{q'})$ for all $q,q'\in F^{-1}(p)$.

An immediate Corollary is that when each $F^{-1}(p)$ contains only one point, the pushforward vector field is defined. So, if we have a smooth embedding $F:M\to N$ then the pushforward vector field is defined. It suffices to have a smooth injective map, too. An immersion might not work in general, however.

Now, for the first question a criterion (found in Tu's Introduction to Manifolds) says that a vector field $X\in \mathfrak{X}(M)$ is $\mathscr{C}^\infty$ if and only if for any $f\in \mathscr{C}^\infty(M)$ the map $p\mapsto X_pf$ is a smooth function. For convenience, assume $F$ is aan embedding. Now, for $g\in \mathscr{C}^\infty(N)$, fix $p=F(q)\in N$. Then $(F_*X)_p(g)=X_q(g\circ F)$ where we note that $g\circ F\in \mathscr{C}^\infty(M)$. So, the association $p\mapsto (F_*X)_p(g)$ is given by the composition, $p\mapsto F^{-1}(p)=q\mapsto X_q(g\circ F)$. By $X\in \mathfrak{X}(M)$ smooth, the second map is smooth. By $F$ admitting a smooth inverse defined on $F(M)$, the first map is smooth. Hence, the pushforward of a smooth vector field (by an embedding) is again smooth.