difference between locally equivalent and uniformly equivalent for metrics on a manifold Suppose $M$ is a manifold, and two metrics $g_0$ and $g_1$ are given on $M$. What is the difference between the assertion "$g_0$ and $g_1$ are locally equivalent" and "$g_0$ and $g_1$ are uniformly equivalent"?
(the paper I'm reading uses these terms without defining them).
 A: Locally equivalent: for every $p  \in  M $ there is a neighbourhoud $U(p)$ such that the metrics are equivalent in $U(p)$.
Uniformly equivalent: the two metrics are equivalent on $M$
Equivalent as such: there are positive constants $\lambda, \mu$ such that $g_0 \le \lambda g_1, g_1 \le \mu g_0$ (in the sense of positive definite two forms)
Note: your notation $g$ kind of implies you (resp. the author of the paper you are referring to) are talking about Riemannian metrics, which are scalar products when restricted to the tangent bundle of $M$ at a given point $p\in M$. On a Riemannian manifold there is another kind of metric in use, usually denoted by $d(p,q)$, which is defined as the $\inf$ of the length of all curves joining $p$ and $q$. In case you are referring to the latter, equivalence is just equivalence in the context of metric spaces, since this $d$ is, in fact, a metric in the classical sense on $M$. The concepts are related, of course, since the length of a curve is calculated with respect to the Riemannian metric $g$.
