"Approximate Isometry" in Riemannian Geometry I apologize if the notion I'm asking about is well known, I'm no expert in geometry (and I did not find an answer via google).
Suppose $(X,g_X)$ and $(Y,g_Y)$ are (smooth) Riemannian manifolds. I'm wondering if one may describe a diffeomorphism $f:X \to Y$ as being "close to an isometry." Is there a criteria to measure this, and what would one call such a map? A natural idea is to call $f$ an $\epsilon$-isometry if for any $p\in X$, $u,v \in T_pX$, $$g^X_p(u,v) - g^Y_{f(p)}(dfu,dfv) < \epsilon$$
(This idea is analogous to that of an approximate isometry on Banach spaces). Does this concept exist and if so in what context is it useful? Thanks. 
 A: As Arctic Char said, the condition $g^X_p(u,v) - g^Y_{f(p)}(dfu,dfv) < \epsilon$ lacks scale invariance; as a result, it is only satisfied by isometries. The scale invariant form is
$$|g^X_p(u,v) - g^Y_{f(p)}(dfu,dfv)| \le \epsilon\|u\|_{g^X}\|v\|_{g^X}$$ 
which is essentially equivalent, up to $\epsilon$ being different, to
$$\frac{1}{1+\epsilon}\le  \frac{\|dfu\|_{g^Y}}{\|u\|_{g^X}} \le 1+\epsilon $$ 
(Because the inner product can be recovered from the norm.) One sometimes puts $e^\epsilon$ instead of $1+\epsilon$ there, which is shorter and works nicely with composition of maps, but looks weird typographically.
These are called $\epsilon$-bi-Lipschitz maps, or $(1+\epsilon)$-bi-Lipschitz maps. They are useful when studying the convergence of manifolds. A sequence of manifolds $\{M_k\}$ may be called a Cauchy sequence if for every $\epsilon>0$ there is $N$ such that any two manifolds $M_j$, $M_k$ with $j,k\ge N$ can be mapped onto each other in an $\epsilon$-bi-Lipschitz way. Then one begins to ponder whether a Cauchy sequence will have a limit, and discovers the limits don't need to be manifolds. This is related, though not identical, to Gromov-Hausdorff convergence of manifolds.
