I recently came upon this condition for a vector field X on M to be affine $$[L_{X},\nabla_{Y}]-\nabla_{[X,Y]}=0$$ Operating this on another vector Z gives me ${\nabla}^{2}_{Y,Z}X+R(X,Y)Z=0$

Where $\nabla^{2}_{Y,Z}X=\nabla_Y \nabla_Z X-\nabla_{\nabla_YZ}X$. Now if I choose $Y=Z=T$ some tangent vector to a geodesics in M then I get back the geodesics deviation or Jacobi equation. I know that affine transformations maps geodesics to geodesics hence the deviation vector is also an affine vector field. Is this observation correct?. Further I have seen this affine condition also written as $$(L_X\nabla)_{Y}Z=\lim_{s\to 0}{1\over s}\left((\phi^{*}_s\nabla)_Y Z-\nabla_Y Z\right)=0$$ If I break this down I get $$\lim_{s\to 0}\lim_{t\to 0}{1\over s}{1\over t}\left((\phi^{-1}_{*s}{\bar{\tau}}^{t}_{0}\phi_{*s})Z_{x_t}-\tau^{t}_{0}Z_{x_t}\right)=0$$ Where $t$ is the parameter along the curve whose tangent is $Y$ and similarly $s$ is along $X$. $\phi_s$ are the flow maps along $X$ and ${\tau}$ are the parallel transport map. The above condition is equivalent to the following condition $${\bar{\tau}}^{t}_{0}\phi_{*s}=\phi_{*s}\tau^{t}_{0}$$ i.e, they commute. Further I have noticed that the transformations $C_t=\tau^{t}_{0}\phi_{*t}$ is a local $1$ parameter group of linear transformations of $T_x(M)$ from this one can see that there is a linear endomorphism A of $T_x(M)$ such that $C_t=\exp{-t(A_X)_x}$. Using this one can again check that $A_X=L_X-\nabla_X$ and one can write the affine condition as $$(\nabla_Y A_X)Z=R(X,Y)Z$$ I have tried to summarize here what I have read but somehow I am not able to make a concise understanding of what is going on?. What is the connection between $(A_X)$ and affine transformations and how is this whole thing connected to geodesic deviations?. Can anyone summarize this in a proper clear order?. Most of this material can be found in "Foundations of Differential geometry volume one" by Kobayashi and Nomizu. Please forgive me if the arguments are not clear as I have just started studying this topic.


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