If $\alpha : [0,1] \to M$ is a geodesic curve in a surface $M\subseteq \mathbb R^3$, then is the "geodesic curvature of $\alpha$" equal to the "curvature of $\alpha$" (defined using the Frenet apparatus)?
Or is "geodesic curvature" an unrelated thing from "curvature of a curve" (especially for geodesic) but only their names are chosen to be very similar?
Thank you in advance.
On
The total (Frenet), normal and geodesic curvatures satisfy $\kappa^2=\kappa_n^2+\kappa_g^2$. For a geodesic the geodesic curvature vanishes $\kappa_g=0$.
So, they are not completely unrelated, but they are indeed different things. You can find this in many books about differential geometry of curves and surfaces.
It has no meaning for curves in space defined on a single parameter.
On a surface 'geodesic curvature' symbol $k_g$ is the same thing as " tangential curvature of a curve ". It is the deviation from straightness. Zero for straight, positive for right turn , negative for left turn in the tangential plane. Curvature has two components resolved along the tangent and normal respectively as $(k_g, k_n)$
$k_g$ is an isometric invariant never changing during bending along with Gauss curvature, integral curvature etc. other first fundamental form (FFF) definable scalars.
I also at first felt unhappy about the nomenclature. A geodesic or straight line has no curvature. Sounds as a contradiction in its own terms. ( terms introduced by Bonnet?) Imho should be referred to as curvature departure/ difference etc. On the other hand geodesic torsion τg makes full sense ( e.g., for helices on a cylinder). But be cautioned this may be an opinion only. Please read what is in the text books of differential geometry and stick to its definition by Liouville etc. on surfaces using coefficients/derivatives of FFF.