Is the Meusnier's theorem true for geodesics torsion? 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?
 A: 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
$$
\Psi = h_{12}\,{\omega_1}^2 + (h_{22}{-} h_{11})\,\omega_1\omega_2 - h_{12}\,{\omega_2}^2.
$$
