I have began to study Riemannian Geometry and there I encountered a statement that I am unable to feel/understand/visualize.
It is the following statement that is still bothering me:
Any geodesic for a linear connection on a smooth manifold is an integral curve of some vector field in $TM$.
The instructor defines a vector field $G$ on $TM$ globally as follows(i.e. $G\in \chi(TM)$):
$G$ is a vector field that acts on $F\in C^{\infty}(TM)$ as follows:
$G(F)(p,v)=\frac{d}{dt}|_{t=0}F(\gamma_{p,v}(t),\gamma'_{p,v}(t))$ where $p\in M$ and $v\in T_pM$.But I cannot see visually what this vector field looks like for certain geodesic $\gamma_{p,v}$ through $\gamma(0)=p$ and initial velocity $\gamma'(0)=v$ and for certain $F\in C^\infty(TM)$.I want to get a feel of what it exactly would look like but I am unable to feel it.Since it is essential in geometry and topology to develop a good intuition,so I think I should have an associated picture in my mind corresponding to this.I want to understand the theorem in such a way that it starts seeming natural to me.Is there anyone who can help me by providing necessary intuitions and also some picture that I can associate with this theorem?Also,if anyone has a better idea which will make understanding these things easy,then the person is encouraged to share his/her ideas also.