TITLE: Do cotangent bundles have "bounded geometry"?
QUESTION [6 upvotes]: I have often heard the phrase "a manifold $M$ has bounded geometry" thrown around without ever seeing a precise definition of what this means. Apparent examples are compact manifolds and $\mathbb{R}^n$. What is the definition and can I therefore assume that cotangent bundles of compact manifolds have bounded geometry?

REPLY [3 votes]: Alexander Grigor'yan ("Heat Kernel and Analysis on Manifolds", 2009) gives the following definition:

a Riemannian manifold $(M, g)$ (of $\dim M = n$) has bounded geometry if and only if there exist the "bounds" $0<c<C$ and the radius $\varepsilon > 0$ such that for every $x \in M$ the geodesic ball $B(x, \varepsilon)$ is diffeomorphic to the standard Euclidean ball $B(0, \varepsilon) \subset \Bbb R^n$ under the diffeomorphism $\varphi _x : B(0, \varepsilon) \to B(x, \varepsilon)$ and $c g_0 \le \varphi _x ^* g \le C g_0$ (where $g_0$ is the usual Riemannian structure on $\Bbb R^n$).

(The definition is assembled from pieces found at pages 93 and 312.)
Notice that in general $T^*M$ has no natural Riemannian structure, therefore asking whether it has bounded geometry is meaningless, regardless of whether $M$ is compact or not.