Let $X_i, Y_i$ be vector fields on the manifolds M and N. $X_i\oplus Y_j$ on $M\times N$. $[X_1\oplus Y_1,X_2\oplus Y_2]=[X_1,Y_1]\oplus [X_2,Y_2]$

Question

Let $M$ and $N$ be two differentiable manifolds and $X_1,X_2$ be two vector fields on $M$ and $Y_1, Y_2$ on $N$. Using the fact that $T_p(M)\oplus T_q(N)$ is naturally isomorphic to $T_{(p,q)}(M\times N)$, show that $$[X_1\oplus Y_1, X_2\oplus Y_2]=[X_1,X_2]\oplus [Y_1,Y_2]$$

Everything that I'm trying is leading me to prove the bilinearity of the Lie bracket. I can't see why this identity is true at all.

Answer 1

In coordinates, you can write the vector fields,

$$X_1 = f_1^i \frac{\partial}{\partial x^i}, Y_1= g_1^i \frac{\partial}{\partial y^i}$$

And similarly for $X_2$ and $Y_2$, where the $x^i$ are coordinates on $M$ and the $y^i$ on $N$. In these coordinates it's easy to see that

$$ X_j \oplus Y_j = f_j^i \frac{\partial}{\partial x^i} +g_j^i \frac{\partial}{\partial y^i}$$

Now, try computing $(X_1 \oplus Y_1)(X_2 \oplus Y_2)$. Are there any cross-terms (e.g. $X_1Y_2$)?