Coordinate-free definition of arc length I'm trying to write the definition of arc length on a Riemannian manifold using fancy differential geometry language, mostly as a way to become familiarized with this language. Of course, writing this definition using just calculus is trivial.
Let $(M,g)$ be a Riemannian manifold, and $\alpha : I \to M$ a regular curve on it. (Think of $I$ as an abstract interval with a smooth structure, but no preferred coordinate.) Given a coordinate $t$ on $I$, the pullback metric $\alpha^\star g$ is represented by the $1 \times 1$ matrix whose lone entry is the squared length of the tangent vector field $v : I \to TI$ with coordinate $dt(v) = 1$. If the endpoints of $I$ have coordinate $t = a$ and $t = b$, the arc length of $\alpha$ is
$$\mathcal L(\alpha) = \int_a^b \sqrt {(\alpha^\star g) (v, v)} \, dt = \int_a^b \sqrt {g_{\alpha(t)}(d\alpha_t(v), d\alpha_t(v))} \, dt$$
I find this definition unsatisfying, because I needed to introduce $t$ and $v$ just to be able to state it, even though the associated geometric concept is independent of $t$ and $v$. Is there a coordinate-free way to state this definition?
 A: Suppose you reparameterize the line using a coordinate $s$. There exists a scalar field $k$ such that $dt = k \, ds$, hence we must use the vector field $w = kv$ in the calculation. Substituting in the integrand, we have
$$\sqrt {g(d\alpha(w), d\alpha(w))} \, ds = \sqrt {k^2 \, g(d\alpha(v), d\alpha(v))} \, \frac 1k \, dt = \frac {|k|} k \sqrt {g(d\alpha(v), d\alpha(v))} \, dt$$
By definition of regular parameterization, $k$ is never zero. The interval $I$ is connected, hence we have two possibilities:


*

*$k > 0$ everywhere, in which case $|k|/k = 1$, and the integrand's sign stays the same.

*$k < 0$ everywhere, in which case $|k|/k = -1$, and the integrand's sign is flipped.
Hence the arc length (as defined in the question) is not a fully coordinate-free notion. It depends on the orientation of the line induced by the parameterization.

An even better take at it is that the definition given in the question is slightly wrong. The integral in the question actually defines an arc length parameterization of the curve, if we allow the upper limit of integration to vary:
$$s = \sigma(t) = \int_a^t \sqrt {(\alpha^\star g) (v, v)} \, dt$$
However, the arc length itself is actually a Lebesgue integral over the set $[a,b]$:
$$\mathcal L(\alpha) = \int_{[a,b]} \sqrt {(\alpha^\star g) (v, v)} \, |dt|$$
It is not an integral of a form over a chain.
