I'm reading a paper on harmonic maps and in it, it mentions the following. You can see the same notation on the wiki on Harmonic maps.
Let $ \phi : M \to N $ be a smooth map between two Riemannian Manifolds. Can someone briefly explain this notation: $ \phi^{-1}(TN) $?
The Wikipedia page Harmonic map uses $\phi^{-1}TN$ to denote the pullback bundle of $TN$ via $\phi,$ otherwise written as $\phi^* TN.$ The differential $d\phi$ can be viewed as a section of the bundle $TM^* \otimes \phi^{-1}TN,$ since $d\phi(p)$ is a linear map $T_pM \to T_{\phi(p)}N$ and thus an element of $$T_p M^* \otimes T_{\phi(p)}N=(TM^*\otimes \phi^{-1}TN)_p.$$