# Differential of riemannian exponential, geodesic triangles and jacobi fields

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.

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