In the paper that can be found in this link: https://arxiv.org/pdf/1712.00082.pdf
there is a definition of the geodesic curvature that I can't quite understand.
Here is how this article presents it:
Let $\vec{X}(s)$ be a curve on a 2D manifold in $R^3$ which is parametrized by
the arc length $s$ as shown in the following figure.
Choose a vector function $\vec{V}(s)$ living in the tangent space at the position $\vec{X}(s)$ and is paralely transported along the curve. Then $k_g=\frac{d\theta}{ds}$, where $\theta$ is the angle between the velocity vector $\vec{T}=\frac{d\vec{X}(s)}{ds}$ and $\vec{V}(s)$, is defined as the geodesic curvature.
The paper states that the geodesic curvature $k_g$ reflects the deviation of the curve from the local geodesics.
What I don't understand are the following:
$1)$ Since $\vec{T}=\frac{d\vec{X}(s)}{ds}$ and $\vec{V}(s)$ are both parallely transported along the curve(former is a velocity vector and later is given), then shouldn't the angle between them remain constant? In Riemannian geometry, when we parallel transport a vector along a curve, its angle with the velocity vector of the curve remain constant.
Is it that the author might not use an affine connection that is compatible with the metric, which is the property that an affine connection needs in order to preserve lengths of vectors and angles between vectors during parallel transport? (see for example Do Carmo's Riemannian geometry, p.53 on Riemannian connections)
$2)$ How does the geodesic curvature relate to the Riemannian curvature found in Riemannian geometry? Also, is it an intrinsic quantity or an extrinsic one? Finally, how does it generalize to higher-dimensional manifolds that are not embedded in higher-dimensional manifolds?