My question is somewhat vague, its precise formulation is a part of the question. 
Let $M,N$ be smooth Riemannian manifolds of the same dimension.  Let $0<\varepsilon<\frac{inj(N)}{100}$. Let $f\colon M\to N$ be a smooth map such that for any $x\in M$ and any $v\in T_xM$ one has
$$(1-\varepsilon)|v|\leq |df_x(v)|\leq (1+\varepsilon)|v|.$$
Denote $\tilde\gamma:=f\circ \gamma\colon [a,b]\to N$.

**Question.** What conditions, formulated in terms of geometry of $N$ only (rather than $M$),  are satisfied by the curve $\tilde\gamma$  which guarantee that if the $length(\tilde \gamma)$ is small enough then $\tilde\gamma$ cannot be a loop, i.e.
$$\tilde\gamma(a)\ne \tilde\gamma(b)?$$


For example, if $\varepsilon=0$ then $\tilde\gamma$ is also a geodesic, and if $length(\tilde \gamma)<inj(N)$ then $\tilde \gamma$ cannot be a loop.