"Almost geodesics" in Riemannian manifolds which cannot be loops

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 $$(1-\varepsilon)|v|\leq |df_x(v)|\leq (1+\varepsilon)|v|.$$ Let $\gamma\colon[a,b]\to M$ be a geodesic. Denote $\tilde\gamma:=f\circ \gamma$.

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 $length(\tilde \gamma)$ 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.

Sorry if my question is somewhat vague, its precise formulation is a part of the question.

Answer 1

Assume $M$ is complete. (It is easy to construct counterexamples in the noncomplete case.)

The inverse function theorem implies that the map $f$ is locally invertible; that is, if $p\in f(x)$, then there is a neighborhood $U\ni p$ and a right inverse $g\colon U\to M$ of $f$ such that $g(p)=x$. (Here we have to assume that $\varepsilon <1$.)

Now assume $\tilde \gamma=f\circ\gamma$ is a short loop based at $p$. Consider the geodesic homotopy $\tilde h_t$ from $\tilde \gamma$ to the constant map with image $p$. This homotopy admits a local lifting $h_t$ to $M$; moreover, the lengths of lifted curves $t\mapsto h_t(x)$ can be controlled. Therefore the maximal interval of definition of $h_t$ is a closed subinterval of $[0,1]$. On the other hand, this subinterval must be open. Hence $\tilde h_t$ admits a global lifting. In particular $\gamma$ is a loop --- a contradiction.