Prove that $T_{(p,q)}(M_1\times M_2) \approx T_pM_1 \times T_q M_2$ as vector spaces

Solution 1:

Let $pi_1:M_1\times M_2\to M_1$ and $pi_2:M_1\times M_2\to M_2$ be the natural projections. The maps are smooth and we can consider the map $F:T_{(p,q)}(M_1\times M_2)\to T_pM_1\times T_qM_2$, given by $F(v)=(d\pi_1(v),d\pi_2(v))$.

To prove that $F$ is injective let $v$ be a vector such that $F(v)=0$. If we have local coordinates $x_i$ on $M_1$ and $y_i$ on $M_2$ then $x_1,...,x_m,y_1,...,y_n$ are local coordinates on $M_1\times M_2$. It now follows that $$v=\sum a_i\frac{\partial}{\partial x_i}+\sum b_i\frac{\partial}{\partial y_i}.$$ Moreover $$0=d\pi_1(v)=\sum a_i\frac{\partial}{\partial x_i}$$ and $$0=d\pi_2(v)=\sum b_i\frac{\partial}{\partial y_i}$$ so $v=0+0=0$.