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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.
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    $\begingroup$ Is this notation due to Riemann? Or could it be due to Einstein, for gravity? $\endgroup$
    – ziggurism
    Jan 12, 2018 at 19:47
  • $\begingroup$ The use of $g$ goes at least as far back as Einstein. It's used in his 1915 paper. $\endgroup$
    – Winther
    Jan 12, 2018 at 19:52
  • $\begingroup$ @Winther: That is interesting. The symbol $g$ being used by Einstein might at least explain its universal acceptance nowadays. $\endgroup$
    – Jan Bohr
    Jan 12, 2018 at 20:05
  • $\begingroup$ Maybe because the notion of the metric tensor comes from the works of Gauss on invariant properties of a surface? $\endgroup$ Jan 12, 2018 at 21:20
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    $\begingroup$ there are several people to look at in the late 1800's to early 1900's, an early one who used the tensor as such was Beltrami, en.wikipedia.org/wiki/Eugenio_Beltrami The same article mentions his influence on Ricci and on his student Levi-Civita en.wikipedia.org/wiki/Gregorio_Ricci-Curbastro $\endgroup$
    – Will Jagy
    Jan 12, 2018 at 21:53

3 Answers 3

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The letter "g" was originally used by Albert Einstein for gravity.

See this answer and references therein.

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I always thought it was $g$ for the Gram matrix, which, after picking a basis, specifies the component of an inner product:

$g_{ij} = \langle \hat{e}_i, \hat{e}_j \rangle$

On a Riemannian manifold, the metric is just an inner product-valued field.

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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 $g$.

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    $\begingroup$ Writing "Zahlen", and not "zahlen", one should also write "Gestalt", and not "gestalt", ... $\endgroup$ Jan 12, 2018 at 20:06
  • $\begingroup$ But Gestalt is a vague as Geometrie, surely connected to the big picture, but not so much to the concept of a bilinear form at every point. $\endgroup$
    – Jan Bohr
    Jan 12, 2018 at 20:11

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