I'm a rookie in Riemann Geometry and try to understand the drawing scheme of parallel transport, i.e, $D_{\dot{\gamma}(t)}X = 0$.
My current understanding is shown as below:
Let $(M,g,D)$ be a Riemann manifold which equips a Levi-Civita connection, in my textbook the author tells me that one of property of Levi-Civita connection is equivalent to the parallel transport is isomorphism preserved the inner product. So that if I have a curve $\gamma(t) : [a, b] \rightarrow M$ such that $D_{\dot{\gamma}}\dot{\gamma} = 0$, which implies that the tangent vector of every point in curve is isomorphism. So $X$ is parallel along $\gamma$, i.e, $D_{\dot{\gamma}}X$, follows that $\langle X(\gamma(t_1)), \dot{\gamma}(t_1) \rangle = \langle X(\gamma(t_2)), \dot{\gamma}(t_2) \rangle$. It helps me to understand following Drawing scheme of parallel transport.
What confused me is that above analysis needs the curve is geodesic, whether this understanding also hold if I delete this condition?
Thanks for your help in advanced:)