Confusion with the Tangent bundle of product manifolds 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:


*

*Which of the equations above is right (or are all of them right)?

*What exactly ist the pullback of the projection map? Is it just the invers of the projection map?

 A: 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.
