Pullback tangent bundle ($f^{\ast }TN$)

For a smooth map between manifolds $f:M\to N$, the pullback vector bundle $f^{\ast }TN$ has fibre $T_{f(p)}N$ at $p\in M$. A smooth section is a vector field along a map. It allows one to differentiate maps and curves without requiring them to be invertible.

A vector field along a smooth map between manifolds $f:M\to N$ assigns $Y_p\in T_{f(p)}N$ smoothly to each $p$. It is a section of the pullback tangent bundle, not necessarily a vector field on $N$. Along a curve $\gamma$, the velocity $\dot{\gamma }$ is such a section.