Related vector fields

Given a smooth map $F:M\to N$ on smooth manifolds, we know that the differential maps $d{F}_p:T_{p}M⟶T_{F(p)}N.$ However, we don’t know that this is a “nice” mapping between tangent spaces. F is not guaranteed to be either injective or surjective. So if we have a vector space $X\in \mathfrak{X}(M)$ then we don’t know if we can get a vector field $Y\in \mathfrak{X}(N)$ using $F$ since maybe it would not map a vector to $T_{F(p)}Y$.

Definition

Two vector fields $X\in \mathfrak{X}(M)$ and $Y\in \mathfrak{X}(N)$ are called F-related if $d{F}_p(X_p)=Y_{F(p)}$

Conditions

$X$ and $Y$ are $F$-related if and only if for every smooth (real-valued) function $f$ defined on an open subset of $N$, $X(f\circ F)=Y(f)\circ F.$

Proof

todo

References

@lee2013 - Chapter 8