Let $(M,g)$ be a closed Riemannian manifold. I want to show that
there exists a sufficiently small $\delta > 0$ such that if $\gamma_1: S^1 \to M$ and $\gamma_2: S^1 \to M$ are simple closed geodesics with $d(\gamma_1, \gamma_2) = \sup\ \{d(\gamma_1(\theta), \gamma_2(\theta)) : \theta \in S^1\} < \delta$, then they are ambient isotopic.
There are easy counterexamples if you drop the simple condition. I have attempted a proof as follows: Since $M$ is closed, there is some $\epsilon$ such that every point in $M$ has a totally normal neighborhood of radius $\epsilon$. As long as $\delta < \epsilon$, there is a unique shortest geodesic connecting $\gamma_1(\theta)$ to $\gamma_2(\theta)$ for each $\theta$. Geodesic flow along these connecting segments takes $\gamma_1$ to $\gamma_2$. But a priori, two of those connecting segments might intersect such that the flow is not an isotopy. I suspect that something about being geodesics prevents this, since in a neighborhood nearby geodesics should look something like parallel lines oriented in the same direction, but I haven't managed to come up with an argument.
Can this proof be finished, possibly with much smaller $\delta$? Is this statement even true?
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.