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.](https://encyclopediaofmath.org/wiki/Geodesic_torsion) See [here](https://en.wikipedia.org/wiki/Darboux_frame#Geodesic_curvature,_normal_curvature,_and_relative_torsion) 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?



Is the Meusnier's theorem true for geodesics torsion?
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
$$
\Psi = h\_{12}\,{\omega\_1}^2 + (h\_{22}{-} h\_{11})\,\omega\_1\omega\_2 - h\_{12}\,{\omega\_2}^2.
$$


