Pullback vector bundle ($f^{\ast }E$)

For a smooth map $f:M\to B$ and a vector bundle $\pi :E\to B$, the pullback bundle isPulling back local trivializations of $E$ makes it a vector bundle over $M$ whose transition functions are the original transition functions composed with $f$.

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.

For an embedded submanifold $i:M\hookrightarrow N$, a smooth section of a vector bundle $X\in \Gamma (i^{\ast }TN)$ locally extends to an ambient vector field. In an adapted chart $M=\{y=0\}$, extend each component $X^a(x)$ independently of the transverse coordinates $y$. If $X$ takes values in $TM$, the extension criterion is $(\tilde{X}F){|}_M=X(F{|}_M)$ for every local smooth function $F$ on $N$. Testing coordinate functions suffices. For tangent fields $X,Y$, this criterion also proves $[\tilde{X},\tilde{Y}]{|}_M=[X,Y]$. General fields with normal components have no extension-independent ambient Lie bracket.

// Target: riemannian-geometry.bigb

A vector field along a curve $\gamma :I\to M$ is a smooth section of a vector bundle with $V(t)\in T_{\gamma (t)}M$. Its values may differ at distinct times mapping to the same base point. The covariant derivative along a curve differentiates its time-dependent coefficients using the pulled-back connection on a vector bundle; it does not require extending it to one ambient vector field.