# What is the definition of the vector field $\pi_*(\xi)$?

Let $$\pi : M \rightarrow B$$ be a fiber bundle. If $$\alpha \in A^i(M)$$ is a differential form on M of degree $$i$$, we denote by $$\int_{M/B}\alpha \in A^{i-n}(B)$$ the integration of $$\alpha$$ along the fiber.

In the article " Orbites Coadjointes et Cohomologie équivariante" (page 23) the authors say that if $$\xi$$ is a vector field on $$M$$ which we can project to a vector field $$\pi_*(\xi)$$ on the base $$B$$, then $$\iota(\pi_*(\xi)) \int_{M/B} \alpha = \int_{M/B} \iota (\xi) \alpha.$$

My question is how is the vector field $$\pi_* (\xi)$$ defined ? and what does it mean that a vector field on $$M$$ admit a projection on $$B$$ ?

Given a smooth map $$F:M\to N$$ and a vector field $$X\in\mathfrak{X}M$$, we say a vector field $$Y\in\mathfrak{X}N$$ is $$F$$-related to $$X$$ if $$d_pF(X(p))=Y(F(p))$$ for all $$p\in M$$ where $$d_pF:T_pM\to T_{f(p)}N$$ denotes the differential of $$F$$ at $$p$$.
In the case of a fiber bundle $$\pi:M\to B$$, one can show that, given a vector field $$X\in\mathfrak{X}M$$, there may or may not exist a $$\pi$$-related vector field on $$B$$, but if it does exist, it is unique. This vector field is what the authors refer to by $$\pi_*(X)$$.
• hi @Kajelad! Thank you for your answer. I'm just wondering in which cases does the F-related vector field to $X$ exist ? I guess for example that if $F$ is invertible then it exists, am I right ?
• @asma If $F$ is a diffeomorphism (i.e. smoothly invertible) both existence and uniqueness are guaranteed. Commented Jul 29, 2021 at 1:22