I am a complete newbie Riemannian Geometry with a particular application in mind so please excuse a lack of rigor in the question.
Suppose I have a manifold with sectional curvature everywhere negative and also lower bounded by $\kappa < 0$ (I think these are called Hadamard Manifolds). Between two point $x,y$ I consider two curves $\gamma_1(t), \gamma_2(t)$. I parallel transport a vector $v \in T_xM$ to $y$ along the two curves $\gamma_1, \gamma_2$ giving me the vectors $v_1, v_2 \in T_yM$. I wish to say something about difference $v_1 - v_2$, in particular to bound $\|v_1 - v_2\|$. My bound can depend upon the lengths of the curves $\gamma_1, \gamma_2$.
Please provide a reference that could help me prove the above quantitative bound. From the little I have read it seems the quantity I care about is fundamentally related to the curvature tensor. In that case can I find an upper bound with respect to a bound on the curvature tensor? If yes can I relate the bound to the sectional curvature?
A general reference to understand these notions of curvature would be appreciated too.