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.
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.