I was reading the section dedicated to the Gauss-Bonnet Theorem of the book "Differential Geometry: Connections, curvature and Characteristic Classes" of Loring W. Tu, and there is a previous result to the mean theorem which is very interesting. The result says:
For any unit speed curve $\gamma$ on a 2 dimensional oriented Riemannian manifold we have the following characterization of the geodesic curvature (of the curve of course):
$$k_g=\frac{d\theta}{ds}-w_1^2(T),$$
where $T= \dot{\gamma},$ $\theta$ is the angle between $T$ a previous vector field $e_1$ which is an element of an orthonormal basis $\lbrace e_1, e_2 \rbrace,$ and $w_1^2 $ is the connection form of the affine connection of the Riemannian manifold.
My question: is there an analogous result for curve on 3 dimensional Riemannian manifolds?
The generalization of "geodic curvature", is just the second fundamental form. See https://en.wikipedia.org/wiki/Second_fundamental_form.
It is a quadratic form with value in the normal bundle.
If $u$ is a normal unit vector field to your curve, and if your curve is givn by arc length parametrization $c(s)$, then you look what happens at the order 2 for the length of the speed vector of the curve $c(s)+tu(s)$ wher means that tou take the exponential.
And more precisley you find something like $<\nabla _c'(s) u(s), u(s)>= \kappa _u$