Differential of riemannian exponential, geodesic triangles and jacobi fields

Question

Let $(M,g)$ be a riemannian manifold, with exponential $exp$, assumed to be globally a diffeomorphism. Denote its inverse by $log$. Let $a,b,c\in M$ be three points on the manifold. I would like to prove that $d(exp_a)(log_a(b))(log_a(b)-log_a(c))=-log_b(c)$. This result holds in the special case that $a=c$, but I was wondering for the general result where $a,b,c$ are distinct and form a geodesic triangle. This result makes sense at least in the case of manifolds with non-positive curvature, since on the one hand $\parallel log_a(b)-log_a(c) \parallel\leq\parallel log_b(c) \parallel$, by comparison with a triangle in euclidean space and toponogov's theorem, while on the other hand $\parallel d(exp_a)(log_a(b))w\parallel\geq\parallel w\parallel$, for any tangent $w$ by Cartan-Hadamard theorem. A similar result which holds trivially is that $d(exp_a)(log_a(X(t)))(\frac{d}{dt}log_a(X(t)))=\dot X(t)$ for a curve $X:I \rightarrow M$ after differentiating the equation $exp_a(log_a(X(t)))=X(t)$. I suspect that the notion of Jacobi fields will be helpful but I do not know how to preceed. Maybe the result I want does not hold exactly but something similar holds. Any idea will be higly appreciated.

Answer 1

In general this is not true. Just take $S^2$ and $a,b,c$ such that all interior angles are $\frac{\pi}{2}$.