I was trying to prove the following statement(#9(a) in Guillemin & Pollack 1.2) but I couldn't make much progress.
"Show that for any manifolds $X$ and $Y$, $$T_{(x,y)}(X\times Y)=T_x(X)\times T_y(Y).$$"
My attempt so far: Parametrise X and Y locally with $U\overset{\phi}{\longrightarrow}X$ and $V\overset{\psi}{\longrightarrow}Y$ where $U\subset \mathbf R^m$ and $V\subset \mathbf R^n$.
Now we can parametrise $U\times V\overset{\phi\times \psi}{\longrightarrow}X\times Y$. By taking the derivative map, we have the tangent plane.
$\mathbf R^{m+n}\overset{d(\phi\times \psi)}\longrightarrow T_{(x,y)}(X\times Y)$. I don't know what to do after this… Apparently we are supposed to set up some relation between $T_{(x,y)}(X\times Y)$ and $T_x(X)\times T_y(Y)$…
Anyone would like to help me out? Thanks!
Best Answer
I don't understand what you are trying to do and here's what I would do. Consider the canonical projections $\pi_X,\pi_Y$ from $X \times Y$ to $X$ and $Y$ respectively. Let $(p,q) \in X \times Y$ and now consider the map
$$f : T_{(p,q)}(X \times Y) \to T_p X \times T_q Y$$
that sends a vector $v$ to elements $\Big(d(\pi_X)_{(p,q)}(v), d(\pi_Y)_{(p,q)}(v) \Big)$. This is linear since $d(\pi_X)_{(p,q)}$ and $d(\pi_Y)_{(p,q)}$ are both linear. On the other hand, define $$g : T_p X \times T_q Y \to T_{(p,q)} (X \times Y)$$ that sends a pair of vectors $(v,w)$ to $d(\iota_X)_p(v) +d(\iota_Y)_q(w)$, where $\iota_X : X \to X\times Y$ sends $X$ to the slice $X \times \{q\}$ and similarly for $\iota_Y$.
Using that \begin{align} \pi_X \circ \iota_X =\operatorname{id}_X, \ \ \pi_Y\circ \iota_Y=\operatorname{id}_Y, \end{align}
$\pi_Y \circ \iota_X, \ \ \pi_X\circ \iota_Y$ are constant maps and chain rule, we have
\begin{align}(f \circ g) (v, w) &= f ( d(\iota_X)_p(v) +d(\iota_Y)_q(w)) \\ &= \Big(d(\pi_X)_{(p,q)}( d(\iota_X)_p(v) +d(\iota_Y)_q(w)), d(\pi_Y)_{(p,q)}( d(\iota_X)_p(v) +d(\iota_Y)_q(w)) \Big) \\ &=\Big(d(\pi_X\circ \iota_X)_p(v) +d(\pi_X\circ\iota_Y)_q(w)), d(\pi_Y\circ\iota_X)_p(v) +d(\pi_Y\circ\iota_Y)_q(w)) \Big) \\ &= (v, w) \end{align}
Thus $f$ is surjective. Since $T_{(p,q)}(X \times Y)$ and $ T_p X \times T_q Y$ have the same dimension, $f$ is a linear isomorphism.