Parametrizations and coordinates in differential geometry - what's the difference?

Question

From what I've read one can introduce the notion of a tangent vector to a point on a manifold in terms of an equivalence class of curves passing through that point (the equivalence relation being that they have the same tangent at that point). Now my confusion arises from the fact that in the texts that I've read, the author introduces a curve $\gamma :(a,b)\subset\mathbb{R}\rightarrow M$ that is parametrized in terms of some real parameter $t\in (a,b)$, with $a<0<b$ and $\gamma (0)=p\in M$. From this a tangent vector at a point $p\in M$ can be defined in a coordinate independent manner, in terms of the directional derivative of function $f:M\rightarrow\mathbb{R}$, as $$\frac{df}{dt}\bigg\vert_{p}$$ I know that if one introduces a coordinate chart $(U,\phi)$ (where $p\in U$) then the curve can be represented in terms of the local coordinates, $$(\phi\circ\gamma)(t)=\gamma (x^{1}(t),\ldots ,x^{n}(t))$$ So what distinguishes a parameter $t$ (parametrizing the curve $\gamma$) from a coordinate? Is it that the parameter is defined in terms of an intrinsic property of the curve (such as arc-length) and thus is independent of any coordinate system, or am I completely misunderstanding things?

Answer 1

Generally in geometry, the term "coordinate (function)" near a point $p$ of a manifold $M$ refers to one component of a coordinate chart $\phi$ defined in some open neighborhood $U$ of $p$.

In the preceding sense, even if $\gamma$ is a simple smooth curve through $p$ (i.e., $\gamma$ is one-to-one, infinitely differentiable, defined on some neighborhood of $0$, and satisfies $\gamma(0) = p$), the parameter $t$ of $\gamma$ is not a coordinate on $M$ because $\gamma$ is not (of itself) a parametrization of some open neighborhood of $p$ in $M$ (unless $M$ is a curve).

That said, there might well exist a parametrization of some neighborhood $U$ of $p$ that extends $\gamma$, in which case $t$ would become a coordinate at $p$ in the resulting chart.

Incidentally, when one says "the definition of a tangent vector at $p$ is coordinate-independent", it means in a loose philosophical sense that there is a definition, possibly referring to a coordinate chart near $p$ or a parametrization of a neighborhood $U$ of $p$, that defines precisely the same criterion no matter which chart is used. (Physicists often express the same idea as "the object transforms correctly" under change of coordinates.)

This is not the meaning one might read from the non-mathematical semantics of the phrase. :)

For example Wikipedia's definition of a tangent vector decrees that two smooth curves $\gamma_{1}$ and $\gamma_{2}$ through $p$ are equivalent if, with respect to some coordinate chart $\phi$ defined near $p$, $$ \frac{d}{dt}\bigg|_{t = 0} (\phi \circ \gamma_{1})(t) = \frac{d}{dt}\bigg|_{t = 0} (\phi \circ \gamma_{2})(t). $$ In concrete terms, if the curves are expressed as $n$-tuples of smooth, real-valued functions, they have the same velocity vector at $0$ in the sense of elementary calculus. It's easy to see this definition doesn't depend on the chart, though it does refer to an arbitrary chart.