I know that there is an isomorphism $$T(M\times N) \simeq \pi_M^* TM \oplus \pi_N^*TN. $$
However, I would be interested in equality, not in isomorphisms.
I've read somewhere that $$T(M\times N) = TM \times N \oplus M \times TN.$$ Is this really true? It does not look right to me to take products of manifolds and vector bundles. Maybe there is some abuse of notation going on?
Shouldn't the tangent bundle be something like $$T(M\times N) = \ker d\pi_M \oplus \ker d\pi_N, $$ with $\pi_M, \pi_N$ the projections?
