Problem 3-2 (The tangent space to a product manifold)

Linearity. First, we demonstrate that $\alpha$ is a linear map. Obviously, we use the fact that each component $d{({\pi }_j)}_p$ in itself is linear (cf. Proposition 3.6(a)). Let $v,w\in T_p(M_1\times \dots \times M_k)$ and $c\in ℝ$ be arbitrary. We then have

Bijectivity. Now we demonstrate that $\alpha$ is bijective. To do so, we explicitly provide an inverse map by reverse engineering $\alpha$. For each $j$, let ${\iota }_j:M_j\to M_1\times \cdots \times M_k$ be the embedding onto the $M_j$ factor of $p$, i.e., ${\iota }_j(x)=(p_1,\dots ,p_{j-1},x,p_{j+1},\dots ,p_k)$. For any point $(p_1,\dots ,p_k)\in M_1\times \cdots \times M_k$, the map

defined by

is the inverse for $\alpha$. To see why, consider some $v_1\oplus \cdots \oplus v_k\in T_{p_1}{M}_1\oplus \cdots \oplus T_{p_k}{M}_k$. Unwrapping the definitions and using various properties from Proposition 3.6, we obtain

Notice that the functions themselves satisfy $({\pi }_i\circ {\iota }_j)(p_j)={\pi }_i({\iota }_j(p_j))={\pi }_i(p)=p_i$. In other words, ${\pi }_i\circ {\iota }_j$ is the identity function for $i=j$ and the constant function for $i\ne j$. By Proposition 3.6, $d({\pi }_j\circ {\iota }_j)=d({\mathrm{Id}<!--\; FUNCTION\; APPLICATION\; -->}_{T_{p_j}{M}_j})$ and by Problem 3.1, $d({\pi }_i\circ {\iota }_j)=0$ for $i\ne j$. We get

In other words, $\alpha \circ \beta$ is the identity function on $T_{p_1}{M}_1\oplus \cdots \oplus T_{p_k}{M}_k$. Thus, $\alpha$ is surjective. Since $\alpha$ is a linear transformation and since $T_p\left(M_1\times \cdots \times M_k\right)$ and $T_{p_1}{M}_1\oplus \cdots \oplus T_{p_k}{M}_k$ have the same dimension¹, $\alpha$ must be a bijection.

Manifolds with boundary. Suppose that $M_j$ is a smooth manifold with boundary for some $j$. Then $M_1\times \cdots \times M_k$ is a smooth manifold with boundary as well. Therefore, it has a tangent space at each point with well-defined differentials. We can therefore apply the same proof to the case of a manifold with boundary.