Confusion with the Tangent bundle of product manifolds

Question

Let's suppose $M$ and $N$ are manifolds.

Here 1 one states that \begin{equation} T(M \times N) \cong \pi_{M}^{*}(T M) \times \pi_{N}^{*}(T N) \end{equation} But in my notes I find \begin{equation} T(M \times N) \cong (T M) \times(T N) \end{equation} and in this question 2 I find the equation \begin{equation} T(M \times N) \cong\pi_{M}^{*} (T M) \oplus \pi_{N}^{*} (T N) \end{equation}

Now I have got two questions:

1. Which of the equations above is right (or are all of them right)?
2. What exactly ist the pullback of the projection map? Is it just the invers of the projection map?

Answer 1

Given a smooth map $f:X\to Y$ of manifolds and a smooth vector bundle $p:E\to Y$, the pullback vector bundle is defined as $$f^*E=\{(x,e)\in X\times E:f(x)=p(e)\},$$ which is a submanifold of $X\times E$ and maps to $X$ by the first projection. The vector space structure on the fiber $(f^*E)_x$ over $x\in X$ is defined by observing that the second projection gives a bijection $(f^*E)_x\to E_{f(x)}$, and so you pull back the vector space structure on $E_{f(x)}$ along this bijection. So, you can think of $f^*E$ as a vector bundle over $X$ such that the fiber over a point $x\in X$ is given by the fiber of $E$ over $f(x)$.

Now $TM$ is a vector bundle on $M$ and $\pi_M:M\times N\to M$, so the pullback $\pi_M^*(TM)$ is a vector bundle on $M\times N$. Similarly, $\pi_N^*(TN)$ is also a vector bundle on $M\times N$. We can then form the fiberwise direct sum (or equivalently, direct product) of these two vector bundles to get another vector bundle $\pi_M^*(TM)\oplus \pi_N^*(TN)=\pi_M^*(TM)\times \pi_N^*(TN)$ on $M\times N$, which your first and third statements correctly say is isomorphic to $T(M\times N)$.

The second statement is also correct, but it has a different meaning. It is just talking about the tangent spaces as manifolds, not as vector bundles: it says that the manifold $T(M\times N)$ is diffeomorphic to the product of the manifolds $TM$ and $TN$. So the $\times$ symbol in $TM\times TN$ has a different meaning than in the first statement: it denotes the ordinary Cartesian product of manifolds, whereas in the first statement it denotes the fiberwise Cartesian product of vector bundles.