# Alexandrov angles in Riemannian manifolds

Dear all, I am teaching a course in Riemannian geometry, and I would like to prove some comparison theorems in the next lessons, building on the well-known theory of Jacobi fields, and of Rauch comparison Theorem for Jacobi fields. I would like to stress the fact that several arguments can work in the case of geodesic metric spaces, and I'd like to prove as soon as possible that the metric and differential notion of angle coincide on Riemannian manifolds.

Let me briefly recall the well-known notion of Alexandrov angle. If $X$ is a geodesic metric space and $\gamma_1,\gamma_2$ are geodesics (parameterized by arc-length) exiting from a point $p$, then the Alexandrov angle between $\gamma_1$ and $\gamma_2$ is the quantity $$\angle_p (\gamma_1,\gamma_2)=\limsup_{t,t'\to 0} \overline{\angle}_p (\gamma_1(t),\gamma_2(t'))\ ,$$ where $\overline{\angle}_p (\gamma_1(t),\gamma_2(t))$ is the Euclidean angle (in the point corresponding to $p$) of the Euclidean comparison triangle for the triple $(p,\gamma_1(t),\gamma_2(t'))$.

It is well-known that, in the case when $X$ is a Riemannian manifold, then the above $\limsup$ is in fact a genuine limit, and the Alexandrov angle between two geodesics exiting from the same point coincides with the Riemannian angle between them.

Here is my question: Is there a direct proof of this fact? Here, by direct proof I mean a proof that does not use too much of the theory of CAT(k)-spaces.

In the book by Bridson and Haefliger, the above statement is proved in Corollary II.1A.7, which in turn relies on Proposition II.1.7, where several tools form the preceding sections are used. On the contrary, if we want to prove the easier fact that, in a Riemannian manifold, the limit $$\lim_{t\to 0} \overline{\angle}_p (\gamma_1(t),\gamma_2(t))$$ exists and is equal to the Riemannian angle, then an easy computation using normal coordinates seems to suffice. I was wondering if some local estimates (for example, the expansion of the metric in normal coordinates with the subsequent estimates on the deviation of the exponential from being an isometry) could work for computing the $\limsup$ in the definition of Alexandrov angle (the difficult case being when $t,t'$ convergence to $0$ at a very different speed).

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To show the upper bound you can use the triangle inequality --- come closer to $p$ along the geodesic and apply the local estimates. (This is the "first variation inequality" it holds in any metric space where angles defined.)
The lower bound follows since the injectivity radius at $p$ is positive. Indeed, if the angle is smaller, the geodesic $[\gamma_1(t)\gamma_2(\tau)]$ converges as $\tau\to0$ to an other geodesic distinct from $\gamma_1$.