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.

Sorry if 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 $$(1-\varepsilon)|v|\leq |df_x(v)|\leq (1+\varepsilon)|v|.$$ 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.

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.

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.

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