Why is the covariant derivative denoted with $D_t V$ rather than $D_\gamma V$?

Question

I'm reading John M. Lee's book " Riemannian Manifolds". On page 57, the covariant derivative of $V$ along a curve $\gamma$ is defined, where $V$ is a vector field along $\gamma$. It is denoted by $D_tV$.

So why is the subscript is $t$? Why not $\gamma$?

When $D_tV$ is evaluated at $t_0$, it is written as $D_tV(t_0)$. So it is not a variable, right? I'd like to know what the subscript $t$ stands for.

Answer 1

@Thomas is right -- in my book, curves are always treated as parametrized curves, that is, smooth (or sometimes piecewise smooth) maps from an interval into the manifold. The $t$ in the notation $D_t\gamma$ refers to the independent variable, i.e., the standard coordinate on $\mathbb R$. It's parallel to the notations $d/dt$ or $\partial/\partial x$ for ordinary derivatives. (Some authors use $D\gamma/dt$ for the covariant derivative along a curve, but to my eye that looks ugly and cumbersome.) All of these notations are illogical, strictly speaking, because the name of the independent variable is not an intrinsic property of a function. But they're too convenient to abandon just because they don't happen to be logically consistent!

The situation in which the $D_t$ notation really shows its power is when considering a parametrized family of curves (see Chapter 6 in my book). Given a smooth map $\Gamma\colon (-\epsilon,\epsilon)\times[a,b]\to M$, we can consider the curves $s\mapsto \Gamma(s,t)$ for fixed $t$, or the curves $t\mapsto \Gamma(s,t)$ for fixed $s$, and denote the covariant derivatives along these curves by $D_s$ and $D_t$, respectively. Any other notation quickly becomes unwieldy.

I'm going to be working on a second edition of my Riemannian Manifolds book during the next couple of years, so I welcome suggestions.

Jack Lee

Answer 2

The derivative depends on the parametrization ("speed") of $\gamma$. If you change the parametrization this will also result in a change of the length of $\gamma'$. So $\gamma$ would be ambiguous. One could use $\gamma'$, this depends on how you introduce these concepts.
And if you consider vector fields along curves you are actually looking at maps $\mathbb{R} \mapsto TM|\gamma$ (that is $TM$ restricted to $\gamma$, assuming $\gamma$ is a curve $\mathbb{R}\mapsto M$), so you are taking the (covariant) derivative of a vector field along the curve w.r.t. a tangent vector to $\mathbb{R}$. Instead of $t$ you'll also see that $\frac{\partial}{\partial t}$ is used instead of $t$ is common as subscript for $D$, emphasizing the fact that you take the derivative in direction of a tangent to $\mathbb{R}$. You may also encounter $\nabla$ in place of $D$.