# Why are the symbols E, F, G, L, M, and N used for the coefficients of the fundamental forms?

In differential geometry, if $$e_1$$ and $$e_2$$ are bases for a tangent space $$T_pM$$, then the coefficients of the first fundamental form is:

\begin{align}E&:=\left\\F&:=\left\\G&:=\left\end{align}

And for the second fundamental form, similar -- $$L:=\mathrm{I\!I}(e_1,e_1)$$, $$M\ldots$$, $$N\ldots$$, etc.

This is... pretty strange! Not the definitions - the nomenclature. Some more modern textbooks use E/F/G for the first coefficients and e/f/g/ for the second, but everyone seems to be in agreement that at least the letters E/F/G are to be used, and L/M/N are quite common to see.

Why were these letters used? (Or, is it known to be an arbitrary choice?)

## 1 Answer

See the paper by Gauss, Disquisitiones generales circa superficies curvas, from 1828. On page 24, we find: $$E\,dp^2 + 2F\,dp\,dq + G\,dq^2 .$$ Since then everyone continued to use the letters $$E,F,G$$. That would have been long before anyone used terms like "tangent space".

Why did Gauss choose those letters? On page 22, where letters $$E,F,G$$ are first indroduced, he had already used letters $$A,B,C,D$$. [As well as various small letters, Greek letters, letters with primes.]

Thanks to Mark Yasuda for the reference.