---- Let $M$ be a [[smooth manifold]] and $TM$ its [[tangent bundle]]. Recall that the notations $\Gamma(TM)=\Omega^{0}_{M}(TM)=\mathscr{V}(M)$ denote the space of [[vector field|vector fields]] on $M$. > [!definition] Definition. (Partial covariant derivative) > Let $M$ be a [[smooth manifold]] and $E \xrightarrow{\pi}M$ a [[vector bundle]]. Given a [[connection on a vector bundle|connection]] ([[covariant derivative on a vector bundle|covariant derivative]]) $\nabla^{E}$ on $E$ and $X \in \Gamma(TM)$ a [[vector field|smooth vector field]] on $M$, the composition $\Omega^{0}_{M}(E) \xrightarrow{\nabla^{E}}^{} \Omega_{M}^{1}(E) \xrightarrow{\text{contract with }X}\Omega^{0}_{M}(E)$ defines a differential operator on $\Gamma(E)$. With $U$ simultaneously a [[vector bundle|local trivialization]] $(U,\Phi)$ and [[coordinate chart|coordinate neighborhood]] $(U, (x^{k}))$, and $(e_{i} )$ the [[vector bundle|local frame of sections]] determined by $\Phi$, we have locally $\nabla_{X}^{E}(s ^{i} e_{i})=(\frac{ \partial s^{i} }{ \partial x^{k} } X^{k} + \Gamma^{i}_{jk}s ^{j} X^{k}) e_{i}.$ > Hereon in this note we focus on the case of [[connection on a manifold|connections on manifolds]] because it is more intuitive and some nice identifications arise. But pretty much everything holds for more general [[connection on a vector bundle|vector bundle connections]]. ^definition > [!definition] Definition. ([[partial covariant derivative]]) > Given a [[connection on a manifold|connection]] ([[covariant derivative]]) $D:\Omega_{M}^{0}(TM) \to \Omega^{1}_{M}(TM)$ on $M$ and a smooth [[vector field]] $X \in \Gamma(TM)$, the composition $\Omega^{0}_{M}(TM) \xrightarrow{D} \Omega^{1}_{M}(TM) \xrightarrow{ \text{eval. at }X } \Omega^{0}_{M}(TM) $ yields an $\mathbb{R}$-[[linear map|linear]] differential operator on $\mathscr{V}(M)$. > > There might be way to generalize the result in [[the space of differential 1-forms is dual to that of vector fields]] to get the second map. For now, I am thinking about it as follows. A section $\omega \in \Omega^{1}_{M}(TM)$ is a [[differential form with values in a vector bundle|section of]] $\Gamma(T^{*}M \otimes TM)$. For $p \in M$, $\omega(p) \in T^{*}_{p}M \otimes T_{p}M$. [[isomorphism for tensor product and homset for finite-dimensional spaces|Identifying]] this latter space with the [[vector space]] $\text{End}(T_{p}M)$, $\omega(p)$ is an operator on [[tangent vector to a smooth manifold|tangent vectors]] to $M$ at $p$. Hence $\omega$ is an [[linear operator|operator]] on [[vector field|vector fields]] of $M$. > > This is a sort of geometrically-invariant analogue to the [[partial derivative]], and we will hence call it the **partial, or directional, covariant derivative of $Y$ along $X$**. It is denoted $D_{X}Y=(DY)(X)$. > > **Local expression.** Recall that, in a local trivialization/[[coordinate chart|coordinate neighborhood]] $U$, the covariant derivative $D$ looks like $d_{A}$ for [[connection on a vector bundle|some]] [[connection on a manifold|connection]] $A$ on $M$, where $d_{A}s =(ds+As)$. Locally, then, $D_{X}Y$ has $i$th component $\begin{align} > (D_{X}Y |_{U})_{i} & = (d_{A}Y |_{U}) \ \left( X^{k} \frac{ \partial }{ \partial x^{k} } \right) \\ > &= \left(\left( \frac{ \partial Y^{i} }{ \partial x^{k} } + \Gamma^{i}_{jk} Y^{j} \right) dx^{k} \right)\ \left( X^{k} \frac{ \partial }{ \partial x^{k} } \right) \\ > &= \frac{ \partial Y^{i} }{ \partial x^{k} } \cancel{ dx^{k} } \left( X^{k} \cancel{ \frac{ \partial }{ \partial x^{k} } } \right) + \Gamma^{i}_{jk} Y^{j} \cancel{ dx^{k} }\left( X^{k} \cancel{ \frac{ \partial }{ \partial x^{k} } } \right) \\ > &= \frac{ \partial Y^{i} }{ \partial x^{k} } X^{k} + \Gamma^{i}_{jk} Y^{j} X^{k}. > \end{align}$ > That is, $D_{X}Y |_{U}=\left(\frac{ \partial Y^{i} }{ \partial x^{k} } X^{k} + \Gamma^{i}_{jk}Y^{j}X^{k}\right) \frac{ \partial }{ \partial x^{i} } $ > as a linear combination of basis vector fields $\frac{ \partial }{ \partial x^{i} }$. > $D_{X} \partial_{_{\ell}}=$ > [!equivalence] >Given a [[connection on a vector bundle|connection]] $D$ on $M$, $D_{(\cdot)}(-)$ is the unique operator $\nabla:\Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)$, $(X,Y) \mapsto \nabla_{X}Y$ satisfying > > 1. *($C^{\infty}(M)$-linearity in the first argument)* $\nabla_{fX+gY}Z=f \nabla_{X}Z+g \nabla_{Y}Z$; > 2. *(Additivity in the second argument)* $\nabla_{X}(Y+Z)=\nabla_{X}Y+\nabla_{X}Z$; > 3. *(Leibniz)* $\nabla_{X}(fY)=f \nabla _XY + (Xf) \, Y.$ > > > for all $X,Y,Z \in \Gamma(TM)$ and all [[smooth maps between manifolds|smooth functions]] $f,g \in C^{\infty}(M)$. > > > > > > > [!proof]- Proof. > > To show uniqueness, assume we have an operator ${\nabla}$ satisfying $(1)$-$(3)$. Define a connection by the rule $\begin{align} > > D^{\nabla}:\Omega_{M}^{0}(TM) &\to \Omega_{M}^{1}(TM) \\ > > Y & \mapsto \nabla(-, Y). > > \end{align}$ > > The three conditions guarantee that $D^{\nabla}$ indeed is a [[covariant derivative on a vector bundle]], [[three views on connections|hence a connection]]. (More?) > > > > > > > > Given $Y \in \Gamma(TM)$, define a section $S \in \Gamma(\operatorname{End} TM)$ by $S(X):=\nabla_{X}Y$. This is well-defined because $(1)$ gives $C^{\infty}(M)$-linearity in $X$, so $X \mapsto \nabla_{X}Y$ is indeed a $TM$-valued 1-form, > > > > Thus ${\nabla}(X,Y)=(DY)(X)=D(X,Y)$. > > > > Let us now conversely show that $D$ indeed satisfies $(1)$-$(3)$. > > > > **1.** Since $DZ \in \Gamma(\operatorname{End}(TM))$, $(DZ)_{p} \in \operatorname{End}TM$ is linear, so that $(DZ)_{p}(f(p)X_{p}+ g(p)Y_{p})=f(p)(DZ)_{p}X_{p}+g(p)(DZ)_{p}Y_{p}.$ > > The RHS is precisely the definition of the vector field $f (DZ)(X)+g(DZ)(Y)$, as desired. > > > > > > **2.** The abstract definition of [[covariant derivative on a vector bundle]] stipulates $\mathbb{R}$-linearity of $D:\Omega^{0}_{M}(TM) \to \Omega_{M}^{1}(TM)$, thus $D(X+Y)=DX+DY$. > > > > **3.** By the defining Leibniz rule for [[covariant derivative on a vector bundle]], $D(fY)=df \otimes Y + fDY$. We see that $\begin{align} > > D_{X}(fY)&=\big(D(fY))\big(X) \\ > > &= (df \otimes Y)(X) + \underbrace{ (f DY )(X) }_{ fD_{X}Y }\text{ (using additivity)} . > > \end{align}$ > > All that remains to show is $(df \otimes Y)(X)=(Xf)Y$, but this is immediate. [[isomorphism for tensor product and homset for finite-dimensional spaces|Identifying]] $df \otimes Y$ with the element $\big(p \mapsto Y_{p}\, (df_{p})(\_)\big)$ of $\Gamma(\operatorname{End }TM)$, evaluation at $X$ gives the [[vector field]] $Y \, (df)(X)$, and $(df)(X)$ is precisely the coordinate-free definition of $Xf$. > > > > > > [!generalization] > ^generalization Given a [[connection on a manifold|connection]] ([[covariant derivative on a vector bundle|covariant derivative]]) $D:\Omega_{M}^{0}(E) \to \Omega^{1}_{M}(E)$ on $M$ and a smooth [[vector field]] $X \in \Gamma(TM)$, the composition $\Omega^{0}_{M}(E) \xrightarrow{D} \Omega^{1}_{M}(E) \xrightarrow{ \text{eval. at }X } \Omega^{0}_{M}(E) $ yields a differential operator on $\Gamma(E)$. $D_{X} s=(Ds)(X)$ $s \in \Gamma(T^{*}M \otimes E)\cong \Gamma(T^{*}M) \otimes \Gamma(E)$ $X \in \Gamma(TM)$ $s_{i}=s_{ki}\, dx^{k} \otimes e_{i}$ $(Ds)_{i}=\left( \frac{ \partial s^{i} }{ \partial x^{k} } + \Gamma^{i}_{jk}s ^{j}\right)dx^{k}$, evaluation at $X=X^{k}\frac{ \partial }{ \partial x^{k} }$ defines a new section (here viewed as a vector of smooth functions on $U \subset M$) as $(D_{s}X)_{i}=\frac{ \partial s^{i} }{ \partial x^{k} }X^{k} + \Gamma^{i}_{jk} X^{k} $ $D_{X} s= (\frac{ \partial s^{i} }{ \partial x^{k} } X^{k} + \Gamma^{i}_{jk}s ^{j} X^{k}) e_{i}$ for $(e_{i})_{i=1}^{m}$ a frame of coordinate sections on $E$, $s = s ^{i} e_{i}$. ---- #### ---- #### 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 > ```