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Meusnier's theorem states that all curves on a surface $S$ embedded in $\Bbb R^3$ passing through a given point $P$ and having the same tangent $v\in T_PS$ also have the same normal curvature.

I think the same is true for geodesics torsion. See here for my definition of geodesics torsion.

My proof goes as: Take a curve $\gamma : (-\varepsilon, \varepsilon)\to S$ with $\gamma(0)=P,\,\gamma'(0)=v$ and let $u=n\times\gamma',$ where $n$ is the Gauss map. Then $$\langle dn(v),n\times v\rangle =\langle (n\circ\gamma)',u \rangle(0)=-\langle n\circ\gamma, u' \rangle (0)=-\langle n\circ\gamma,-\kappa_g v-\tau_g n \rangle(0) =\tau_g.$$ However, the inputs of this formula are independent of the curve. Therefore geodesic curvature depends only on $P$ and $V.$

But this statement and almost one-line proof cannot be found anywhere in the literature. Have I done something wrong?

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I believe that your $\tau_g$ is what É. Cartan calls the 'geodesic torsion' in his 1945 book Les systèmes différentiels extérieurs et leurs applications géométriques. He denotes his geodesic torsion by $1/T_g$. He gives your formula for $1/T_g$ in Chapter 7 as part of equation $(17)$.

The point is that what the OP calls $\tau_g$ is simply the value of what Cartan calls the third fundamental form $\Psi$ of the surface $S$ evaluated on the tangent vector to the curve, which is why it depends only on the tangent vector to the curve and not on any higher derivatives.

Note: Cartan's third fundamental form $\Psi$ (see equation $(14)$ in Chapter 7) is not what is nowadays called the 'third fundamental form' and usually denoted $I\!I\!I$. (The modern $I\!I\!I$ is just the pullback via the Gauss map of the standard metric on the $2$-sphere.) Instead, if, in an orthonormal coframe field, the first fundamental form of $S$ is $I = {\omega_1}^2 + {\omega_2}^2$ and the (usual) second fundamental form is $I\!I = h_{11}\,{\omega_1}^2 + 2h_{12}\,\omega_1\omega_2 + h_{22}\,{\omega_2}^2$, Cartan's third fundamental form is $$ \Psi = h_{12}\,{\omega_1}^2 + (h_{22}{-} h_{11})\,\omega_1\omega_2 - h_{12}\,{\omega_2}^2. $$

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  • $\begingroup$ Thank you very much. Is there any English translation of this book? $\endgroup$
    – Bumblebee
    Commented Oct 1, 2022 at 10:16
  • $\begingroup$ I know that an English translation was made some years ago. I saw it because I was asked to comment on it. (It was a preliminary version and had many typos and translation errors.) As far as I know, it has never been published. The French is not that hard to read, and, in fact, for your purposes, you don't need to read the first 6 Chapters, which constitute Part I, an introduction to exterior differential systems. Part II starts with Chapter 7, the first 2 sections of which is a straightforward introduction to the moving frame (assuming only that you are familiar with differential forms). $\endgroup$
    – Robert Bryant
    Commented Oct 1, 2022 at 10:56
  • $\begingroup$ Do you know whether there is an analog of Gauss–Codazzi equations for Cartan's third fundamental form. A prior there is no reason to believe this is the case. It appears that this third fundamental form together with the first fundamental form is not enough to recover the second fundamental form. I'm trying to understand why this happens and what are the missing information. $\endgroup$
    – Bumblebee
    Commented Nov 8, 2023 at 18:06
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    $\begingroup$ @Bumblebee: The 'missing information' in $\Psi$ is the mean curvature $H=\tfrac12(h_{11}+h_{22})$ and a choice of orientation of the surface. However, given $I$ and $\Psi$, one can recover the mean curvature $H$ up to a sign since, by the definition of $\Psi$ and the Gauss equation $H^2 = K - \det_I(\Psi)$, and the orientation of the surface can also be recovered up to a sign. Thus, there are analogs of the Gauss-Codazzi equations for the pair $(I,\Psi)$, but they are more algebraically complicated than for the pair $(I,I\!I)$. $\endgroup$
    – Robert Bryant
    Commented Nov 9, 2023 at 10:30

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