Product Manifold: Tangent Spaces

Question

Problem

Given a product manifold.

How to prove that its tangent spaces split into direct sums: $$T_{(p,q)}(M\times N)\cong T_pM\oplus T_qN$$

Attempts

One could try the geometric perspective: $$\Phi:T_{(p,q)}(M\times N)\to T_pM\oplus T_qN:[(\alpha,\beta)]\mapsto([\alpha],[\beta])$$ $$\Psi:T_pM\oplus T_qN\to T_{(p,q)}(M\times N):([\alpha],[\beta])\mapsto[(\alpha,\beta)]$$ Then bijectivity becomes pretty easy but linearity quite nasty.

(Besides it is well-defined.)

One could also try the algebraic perspective: $$\Phi:T_{(p,q)}(M\times N)\to T_pM\oplus T_qN:\delta\mapsto(d_{(p,q)}\pi_M\delta,d_{(p,q)}\pi_N\delta)$$ Then linearity becomes evident but bijectivity becomes pain.

(Besides there's an explicit but ugly inverse.)

Is there maybe some nice trick??

Answer 1

Take "coordinate" projections $\pi_X,\pi_Y$ from $M \times N$ to $M$ and $N$. Let $(p,q) \in M \times N$, so that you have a map $F : T_{(p,q)}(M \times N) \to T_p M \times T_q N$ sending $v$ to $\left(d(\pi_X)_{(p,q)}(v), d(\pi_Y)_{(p,q)}(v) \right)$. This map is a linear map that is an isomorphism with inverse given by the linear map $g : T_p M \times T_q N \to T_{(p,q)} (M \times N)$ that sends $(v,w)$ to $d(s_M)_p(v) +d(s_N)_q(w)$ where $s_M : M\to M\times N$ sends $M$ to $M \times \{q\}$ and where $s_N$...