Let $\mathcal{M}$ an riemannian submanifold of euclidean space, i.e. $\mathcal{M} \subset \mathbb{R}^n$. Equipped with the Levi-Civita- Connection. Additionally let $\mathbf{\gamma}^1$ and $\mathbf{\gamma}^2$ be two different geodesics on $\mathcal{M}$.
Where
$\mathbf{\gamma}^1: s \rightarrow \mathcal{M}$ with $\frac{\partial \mathbf{\gamma}^1(s)}{s}\Bigg|_{s=0} = \mathbf{t}_1$
$\mathbf{\gamma}^2: t \rightarrow \mathcal{M}$ with $\frac{\partial \mathbf{\gamma}^2(t)}{t}\Bigg|_{t=0} = \mathbf{t}_2$
Introducing the "geodesic fields"
$\mathbf{\gamma}^{12}: (s,t) \rightarrow \mathcal{M}$ with $\frac{\partial \mathbf{\gamma}^{12}(s,t)}{s}\Bigg|_{s=0} = \mathbf{t}_{12}(t)$ where $ \mathbf{t}_{12}(t)$ is the parallel transported vector of $\mathbf{t}_1$ along $t$.
$\mathbf{\gamma}^{21}: (t,s) \rightarrow \mathcal{M}$ with $\frac{\partial \mathbf{\gamma}^{21}(t,s)}{t}\Bigg|_{t=0} = \mathbf{t}_{21}(s)$ where $ \mathbf{t}_{21}(s)$ is the parallel transported vector of $\mathbf{t}_2$ along $s$.
Where $\mathbf{\gamma}^1$ and $\mathbf{\gamma}^2$ doesn't have to need the same constant speed.
My question regarding this is when does the lie bracket vanish at ${s,t=0}$ :
$(\frac{\partial^2\gamma^{12}}{\partial s \partial t}-\frac{\partial^2\gamma^{21}}{\partial t \partial s})\Bigg|_{s,t=0}\stackrel{?}{=}0$
I tried it for the unit sphere $\mathcal{S}^2$ and i calculated that for this case the lie bracket vanishes. I put two figures for the two families of $\mathbf{t}_{12}(t), \mathbf{t}_{21}(s)$ at the end.
My interest lies i.e. at $\mathcal{S}\mathcal{O}(3)$ because here it seems to that the lie bracket does not vanish.
It seems to be that it has something to do with the corresponding jacobi fields of $\mathbf{\gamma}^1$ and $\mathbf{\gamma}^2$ but this is not completely clear to me how this can help here. But for me it seems that changing the order of the derivatives in
$(\frac{\partial^2\gamma^{12}}{\partial s \partial t}-\frac{\partial^2\gamma^{21}}{\partial t \partial s})\Bigg|_{s,t=0}\stackrel{?}{=}0$
would lead to
$(\frac{\partial^2\gamma^{12}}{\partial t \partial s}-\frac{\partial^2\gamma^{21}}{\partial s \partial t})\Bigg|_{s,t=0}\stackrel{?}{=}0$
and then this can be rewritten as
$(\frac{\partial J^{12}(s)}{ \partial s}\Bigg|_{s=0}-\frac{\partial J^{21}(t)}{ \partial t}\Bigg|_{t=0})\stackrel{?}{=}0$
with $ J^{12}$ is the jacobi field of $\mathbf{\gamma}^1$
$J^{12}(s)= \frac{\partial\gamma^{12}(s,t)}{\partial t}\Bigg|_{t=0}$
and similar
$J^{21}(t)= \frac{\partial\gamma^{21}(t,s)}{\partial s}\Bigg|_{s=0}$
Family of geodesics and tangent vectors 1
Family of geodesics and tangent vectors 2
Civil engineer here. My understanding of differential geometry is rather basic. Thanks for your help.
I edited the definition of the lie bracket cause in the first version derivatives didn't make sense