I am struggling to understand the following definition (Theorem $4.24$) in John Lee's Introduction to Riemannian Manifolds:
Let $M$ be a smooth manifold and let $\nabla$ be a connection in $TM$. For each smooth
curve $\gamma:I\to M$, the connection determines a unique $\mathbb R$-linear operator
$$D_t:\mathfrak X(\gamma)\to\mathfrak X(\gamma)$$
called the covariant derivative along $\gamma$, satisfying the following properties:
- PRODUCT RULE:
$$\forall f\in C^\infty(I):D_t(f V)=f'V+fD_tV$$- If $V\in\mathfrak X(\gamma)$ is extendible, then for every extension $\widetilde V$ of $V$,
$$D_tV(t)=\nabla_{\gamma'(t)}\widetilde V.$$
Are we really considering one linear operator, or an entire family $(D_t)_{t\in I}$? For me it is not clear from the context: The last equation suggests that we are considering a family of operators, but I think that we also get a sensible definition by replacing $D_t$ with $D$ everywhere in the definition.
Best Answer
There is a single map. For now, let me just write $D:\mathfrak{X}(\gamma)\to\mathfrak{X}(\gamma)$ (even though it would be more precise to write $D^{\gamma}$ for example to indicate the dependence on the base curve $\gamma$). If you take a vector field $V$ along $\gamma$, you get another vector field $DV$ along $\gamma$. This means of course that for each parameter value $t\in I$, the value of $DV$ at $t$, namely $(DV)(t)$, is then an element of the particular tangent space $T_{\gamma(t)}M$.
If you have a vector field $W$ on an open subset of $M$ which extends $V$, in the sense $W\circ \gamma=V$, then the second condition states that for each $t\in I$, $(DV)(t)=\nabla_{\gamma’(t)}W$.
The usual notation $D_t$ (other common notations include $\frac{D}{Dt}$ or $\frac{D}{dt}$) has of course the same unnecessary ambiguous information, by the symbol $t$, as the usual Leibniz’s notation $\frac{d}{dt}$ does.