# $F$-related vector fields

I'm having a bit of difficulty understanding $F$-related vector fields. I think I understand conceptually what's going on (taking a vector field on a manifold $M$ and getting a smooth vector field on $N$ by applying the differential of a map), but I'm having trouble applying the concept.

For example, here's a problem (8.14 in Lee's Smooth Manifolds textbook) Let $M,N$ be smooth manifolds with or without boundary and let $f:M\to N$ be a smooth map. Define $F:M\to M\times N$ by $F(x)=(x,f(x))$. Show that for each vector field $X$ on $M$, there is a smooth vector field $Y$ on $M\times N$ which is $F$ related to $X$.

Now, I see that $F$ is the graph of $f$, and so $F(M)$ is closed in $M\times N$, and there's a proposition in Lee which says that if we have a smooth vector field on a closed subset, we can extend it to a smooth vector field on the whole manifold. However, I'm really not sure where to go. I see that $F(M)\subset M\times N$ is an embedded manifold, so we have slice charts, but I'm struggling to see how to construct this $Y$.

• Hint: $F(M)$ is not just closed, it's a smooth manifold. What is its relation to $M$? – Travis Mar 3 '15 at 12:25
• You might first consider trying a simple example where $M = \mathbb{R}^{2}$ and $N = \mathbb{R}$. What are the vector fields in $\mathbb{R}^{2} \times \mathbb{R} = \mathbb{R}^{3}$ that are $F$ related to $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$? How can you extend this relationship to a vector field $X = f(x, y) \frac{\partial}{\partial x} + g(x, y) \frac{\partial}{\partial y}$ to find a vector field on $\mathbb{R}^{2} \times \mathbb{R}$ that is $F$ related to $X$? – THW Mar 4 '15 at 15:18

Let $$\Gamma=F(M)$$. As mentioned, $$\Gamma$$ is closed in $$M\times N$$. So it suffices to show that the rough vector field on $$Y$$ along $$\Gamma$$, defined by the equation $$Y_{(p,f(p))}=dF_p(X_p),$$ is a smooth vector field along $$\Gamma$$(, that is, each point $$(p,f(p))\in\Gamma$$ has a neighborhood on which $$Y$$ admits a smooth extension).
Let $$p\in M$$ be arbitrary, and let $$(U,\varphi)$$ and $$(V,\psi)$$ be smooth charts about $$x$$ and $$f(x)$$, respectively, such that $$f(U)\subset V$$. Let $$X^i$$ denote the component functions of $$X$$ in $$U$$:$$X=X^i\frac{\partial}{\partial x^i}\text{ on } U.$$ Note that $$X^i:U\to \mathbb{R}$$ is smooth because $$X$$ is smooth. Using these functions, we define $$\tilde{Y}:U\times V\to T(M\times N)$$ by$$\tilde{Y}_{(p,q)}=X^i(p)\frac{\partial}{\partial x^i}|_{(p,q)}+ \frac{\partial \hat{f}^i}{\partial x^j}(\varphi (p))\frac{\partial}{\partial y^i}|_{(p,q)},$$ where $$\hat{f}=\psi\circ f\circ \varphi^{-1}:\varphi(U)\to\psi(V)$$ is the coordinate representation of $$f$$. Because the component functions of $$\tilde{Y}$$ are smooth, $$\tilde{Y}$$ is smooth, too. Clearly $$\tilde{Y}$$ and $$Y$$ agrees on their common domain. Thus $$\tilde{Y}$$ is a desired extension.