Are nearby simple closed geodesics ambient isotopic?

Fix a geodesic loop $\gamma$ Define a tubular neighborhood $U$ s.t. $$ U =\{ x| d(x,\gamma )<\delta \} $$ where $\delta < \frac{{\rm Inj}\ M}{3} $ and ${\rm Inj}\ M$ is an injectivity radius of $M$. Hence any geodesic of length $< 3\delta$ is minimizing.

Assume that another geodesic loop $\gamma_2$ is in $U$

And define $f$ to be a function on $U$ by $$f(x)=d(x,\gamma ) $$

Since $\gamma$ is a geodesic so gradient vector field $X:=\nabla f$ is well-defined. If $F$ is a local flow of $X$, then $F$ is an isotopy sending $\gamma_2$ onto $\gamma$ :

Assume that $$\gamma_2(s_1)=F(\varepsilon,\gamma_2(s_2)),\ s_1<s_2 $$

Here ${\rm length}\ \gamma_2|[s_1,s_2]\leq 2\delta$ so that it is minimizing. In further $c(t):=F(t,\gamma_2 (s_2))$ is geodesic so that $\gamma_2$ goes out $U$. So it is a contradiction.