Consider the following observation:

Let $c_1$ be a geodesic on the unit round sphere $S^2$ with length $2\pi-\epsilon$, where $\epsilon$ is sufficiently small. Then $c_1$ has Morse index one because it contains a pair of conjugate points. If we close the gap of $c_1$ and consider the great circle $c_2$, it can be shown that $c_2$ also has index one.

Is this a mere coincidence? Is there any situation that generalizes this phenomenon, say, to the case of minimal surfaces? e.g. I think that if you consider the totally geodesic $S^2$ in $S^3$, when you remove a small disc from it, its Morse index remains one. Is there any theorem that tells us when we can cut out a ball from an embedded closed minimal submanifold and have the Morse index unchanged?