---- - [ ] more general version where subscript and superscript are different (not hard) Let $M$ be a [[smooth manifold]] of dimension $n$ 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. ([[covariant derivative along a curve]]) > Suppose $D:\Omega_{M}^{0}(TM) \to \Omega_{M}^{1}(TM)$ is a [[connection on a manifold|connection]] ([[covariant derivative]]) on $M$. We know how to take the "[[partial covariant derivative]]" $D_{X}Y$ of a [[vector field]] $Y$ along a [[vector field]] $X$. > > If $\gamma:I \to M$ is a smooth [[parameterized curve]] on the base $M$, $\dot{\gamma}(0) \neq 0$, we may define the **covariant derivative along $\gamma$ at $t_{\star}$**, denoted $D_{\dot{\gamma}}\dot{\gamma} |_{t_{\star}}$ as follows. It is justified below that: > - Locally ($|t|<\delta$ for some $\delta=\delta(t_{\star})$) there exists a [[vector field]] $X$ on $M$ satisfying $\dot{\gamma}(t)=X\big( \gamma(t) \big)$ > - If $Y$ is another [[vector field]] satisfying the same property, then $D_{X}Y |_{\gamma(t)} = D_{Y}Y |_{\gamma(t)}=D_{Y} X |_{\gamma(t)}.$ > ![[Pasted image 20250518115652.png]] > This means that it is [[well-defined]] to take $D_{\dot{\gamma}}\dot{\gamma} |_{t}:=D_{Y}Y |_{\gamma(t)}$, and this is what we will do. > > As the proof below shall exhibit, in [[coordinate chart|local coordinates]] $\gamma(t)=\big( x^{i}(t) \big)_{i=1}^{n}$, we have $D_{\dot{\gamma}}\dot{\gamma}=(\ddot{x}^{i}+\Gamma^{i}_{jk}(x) \dot{x}^{j} \dot{x}^{k})_{i=1, \dots, n}.$ > In particular, note this expression has no dependence on the choice of extension, and that it is manifestly a second-order notion. > > Observe that if $M$ is a [[Riemannian manifold]] and $D$ the [[Levi-Civita connection]], then this expression matches the [[geodesic on a Riemannian manifold|geodesic equation]] for $\gamma$, i.e., $\gamma \text{ a geodesic } \iff D_{\dot{\gamma} } \dot{\gamma}=0.$ > "Geodesics have zero acceleration." > > ---- #### ---- #### 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 > ``` First to be on the same page re: notation. By definition, $\gamma:I \to M$ is viewed as a [[smooth maps between manifolds|smooth]] map between manifolds, $\dot{\gamma}:=d \gamma$ is a map $TI=\mathbb{R} \to TM$, and $\dot{\gamma}(t)$ is defined to be the evaluation of $(d \gamma_{t}:\mathbb{R} \to T_{\gamma(t)}M)$ at the coordinate basis vector $\frac{ \partial }{ \partial t }$ for $T_{t}I$, $\dot{\gamma}(t)=d \gamma_{t}\left( \frac{ \partial }{ \partial t } \right)$. Write $\|\dot{\gamma}(t)\|_{g}^{2}=g_{\gamma(t)}(\dot{\gamma}(t), \dot{\gamma}(t))$, where $\dot{\gamma}(t) \in T_{\gamma(t)}M$, $\dot{\gamma}(t)=\dot{x}^{i}(t) \frac{ \partial }{ \partial x^{i} } |_{\gamma(t)}$. This lets as view $f:=g(\dot{\gamma}, \dot{\gamma})$ as a element of $C^{\infty}(U) \subset M$, $f\big( \gamma(t) \big):= g_{\gamma(t)}(\dot{\gamma}(t), \dot{\gamma}(t))$. So $\|\dot{\gamma}(t)\|_{g}^{2}=f\big( \gamma(t) \big), f=g(\dot{\gamma}, \dot{\gamma}) \in C^{\infty}(U).$ Then the expression $\frac{d}{dt} f\big( \gamma(t)\big)$ may be understood as taking the [[differential of a smooth map between smooth manifolds|differential]] of a composition of [[smooth maps between manifolds|smooth maps]] $I \to U \to \mathbb{R}$, and the chain rule for differentials applies: $\begin{align} \frac{d}{dt}\big( f (\gamma(t)) \big) &= d(f \circ \gamma) |_{t} \left( \frac{ \partial }{ \partial t } \right) \ \ (\text{definition}) \\ &= \underbrace{ df_{\gamma(t)} }_{ T_{\gamma}(t) M \to \mathbb{R} } \circ \underbrace{ d \gamma_{t} }_{ T_{t}I \to T_{\gamma(t)} M } \left( \frac{ \partial }{ \partial t } \right) \text{ (chain rule for differentials)} \\ &= df_{\gamma(t)} \big( \dot{\gamma}(t) \big) \text{ (definition of } \dot{\gamma(t)}) \\ &= \dot{\gamma}(t) [ f] \ \ (df \text{ as a derivation}) \end{align}$ (yeah hard to be clear without being messy)