Tangent Space of Product Manifold

Question

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!

Answer 1

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.

Answer 2

I'm assuming you want to show that there is an isomorphism between the two spaces, if so, I would refer you Loring Tu's excellent book introduction to manifolds. Problem 8.7 is the question you are looking for and the solution is at the back. In a nutshell, he showed an isomorphism by sending basis elements to basis elements using the differential of the projection maps.

(Tu's book can be found using Google, just google "tu introduction to manifolds")

Hope this helps.

Answer 3

Abusing a little. Nowhere in the problem does it say exactly what to use and what not to so, to prove that $$T_{(x,y)}(X\times Y)=T_{x}X\times T_{y}Y$$ you can use a dimensional argument. Given that dim$(T_{(x,y)}(X\times Y))=$dim$(T_{x}X\times T_{y}Y)$ and that both vectorial spaces are of finite dimension, then both are isomorphic.

I know, I know, It's ugly... But it is correct.