TITLE: Proving the existence of good covers
QUESTION [14 upvotes]: Usually one proves the existence of good covers in compact manifolds by Riemannian methods: we pick an arbitrary Riemannian metric, prove that geodesically convex neighborhoods exist, that they are closed under finite intersections, and diffeomorphic to balls; this is, for example, the argument that Bott and Tu sketch in their book.

Is there a non-Riemannian approach to this?

While this is not necessary for most things, it is a nice fact that good covers can be found which realize the covering dimension bound. 

Is there a differential-topological way to find them?

REPLY [3 votes]: I think you can obtain a good cover of $C^2$ manifold (compact or not) from the charts/atlas definition and a little bit of topology (locally finite atlas and a relatively compact "shrinking" of it).
The very simple idea (akin to that in Vitali's answer) is that under a $C^2$ diffeomorphism between open subsets of euclidean $n$-space, the (pre-)image of a sufficiently small ball centered at a point will be convex, as soon as the curvature of its boundary "dominates" the second derivative of the diffeomorphism (or its inverse).
In formulas, if $\phi$ is the diffeomorphism, this boils down to the fact that the $C^2$ function $x\mapsto |\phi(x)-\phi(x_0)|^2$ has a positive definite hessian at $x_0$, hence is convex near $x_0$. 
With a little more care, I think you can still conclude if $\phi$ is only $C^{1+Lip}$.