I'm looking for an example of a contractible Riemannian manifold which is not uniformly contractible.
Definition: Let $ X $ be a metric space. We say $ X $ is uniformly contractible, if for all $r>0$ we have a $R>r$ such that every $r$-ball $ B_r(x) $ around any point $ x \in X $ is contractible inside the bigger ball $ B_R(x) \supset B_r(x) $.
In general I'm quite unclear about the notion of uniform contractibility. Is there any references which I can look at?
Start with a sequence of round 2-spheres $S(x_i, i)$ of the radius $i$. Now, remove from each sphere an open spherical cap with the perimeter of the unit length. The result is a contractible surface $S_i$ with boundary $C_i$ with the path metric induced from the sphere. But the contractivity function diverges to infinity as $i\to \infty$ because you need the entire surface to contract the circle $C_i$ to a point. Next, you want to connect the surfaces $S_i$ to form a contractible space. The simplest thing to do is just to wedge them at one point and take a path-metric. That's your example. With a bit more thought, you can modify this example to get a contractible Riemannian surface which is not uniformly contractible. You drill countably many holes in the plane so that each hole has unit circumference. Attach $S_i$ along the boundary of each hole. Lastly, smooth out the metric along each $C_i$.