---- > [!definition] Definition. ([[related vector fields]]) > Let $M, N$ be [[smooth manifold|smooth manifolds]], $F:M \to N$ a [[smooth maps between manifolds|smooth map]]. Let $X \in \Gamma(TM)$ be a [[vector field]] on $M$. The evaluation $dF |_{p}(X_{p})$ produces a [[tangent vector to a smooth manifold|vector]] in $T_{F(p)}N$. However, this does not in general define a [[vector field]] on $N$.[^1] If $Y \in \Gamma(TN)$ is a [[vector field]] on $N$ with the property that for each $p \in M$, $dF_{p}(X_{p})=Y_{F(p)}$, then we say $X$ and $Y$ are **$F$-related**. ^definition > [!equivalence] > Unwinding the definitions, we see that $X$ and $Y$ are $f$-related if and only if for all $f \in C^{\infty}(U \subset N)$, $X(f \circ F)=(Yf) \circ F,$ viewing $X,Y$ as [[derivation|derivations]] of $C^{\infty}(U)$. >> [!proof]- >> By definition(s): > >$X(f \circ F)(p)=X_{p}(f \circ F)=dF_{p}(X_{p})f$ > > > >Meanwhile, $(Yf) \circ F(p)=(Yf)\big( F(p) \big)=Y_{F(p)}f.$ > > [!proposition] > If $F:M \to N$ is a [[diffeomorphism]], then for every smooth $X \in \Gamma(TM)$ there is a unique smooth $Y \in \Gamma(TN)$ that is $F$-related to $X$, given in the obvious way by $Y_{q}=dF_{F^{-1}(q)}\big(X_{F ^{-1}(q)}\big).$ > Writing $Y=(dF)X$, the identity in the equivalence above becomes $\big((dF)X) f \big) \circ F=X(f \circ F),$ as appears as equation (1.10) in Kovalev's notes. ---- #### [^1]: It is easy to see why nuance arises if, say, $f$ is not [[surjection|surjective]], or if $f$ is not [[injection|injective]]. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```