Let M be an n-dimensional Riemannian manifold without boundary, with sectional curvature $ \geqslant -1$ . For a point $ p\in M$ , suppose there exist $ l, \delta>0$ , $ x,y \in M$ with $ d(p,x),d(p,y)>l$ and a geodesic $ px$ and $ py$ with angle $ \angle xpy>\frac{\pi}{2}+\delta$ . Let $ q$ be a point on geodesic $ px$ or $ py$ , **Question**: is there $ r>0$ , which depends only on $ n,l,\delta$ such that $ B_q(r)$ is homeomorphic to $ B_p(r)$ ?

Equivalently, we can state the question in the following way:

Let $ M_i$ be a sequence of Riemannian manifolds with $ sec \geqslant -1$ and diameter $ \leqslant D$ . Suppose $ (M_i,p_i)$ Gromov-Hausdorff converge (possibly collapse) to $ (X,p)$ (we know it’s an Alexandrov space). Suppose there exist $ l>0, \delta>0$ , $ x,y\in X$ with $ \angle xpy> \frac{pi}{2}+\delta$ . lift $ x,y$ to $ M_i$ , we get $ x_i,y_i\in M_i$ . with$ \angle x_i p_i y_i >\frac{\pi}{2}+\delta$ . Let $ q_i$ be a point on geodesic $ p_ix_i$ or $ p_iy_i$ . Question: Is there $ r>0$ , such that such that $ B_{q_i}(r)$ is homeomorphic to $ B_{p_i}(r)$ ?