A Riemannian metric on a manifold is universally denoted with the letter $g$, but unlike many other mathematical objects like a function $f$, a distance $d$, a manifold $M$ or a group $G$ there does not seem to be a connection to the name or meaning of the actual quantity, neither in English nor German.
Questions:
- When and where has this notation been introduced?
- Is there an evident reason why the letter $g$ was chosen and accepted by everyone?
Some Remarks:
Of course $g$ carries a lot of information about the geometry of a manifold, but this seems to be a little too unspecific to me.
In his book on Riemannian geometry, Do Carmo only uses $g_{ij}$ but never $g$ for the metric. On the other hand at some point he uses $g$ for the Gauss-map of a surface, but that also does not directly relate to the metric tensor.
- Do Carmo also points out that Riemann introduced the concept of a quadratic form assigned to each point of a manifold in his paper "Über die Hypothesen, welche der Geometrie zugrunde liegen." But in there I did not find the notation $g$ as well.
The symbol $\mathbb Z$ is used because the German word for number is Zahlen.
In German, "gestalt" means "shape", "form", "figure", etc.
Perhaps this is the reason behind the choice of $g$.