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The point of Jacobi fields is to study variations of geodesics through geodesics, but the Jacobi equation $D_t^2 J + R(J,\dot\gamma)\dot\gamma=0$ makes sense for any curve $\gamma$, not just for geodesics.

Are solutions to this equation still interesting in other situations? For instance, does a variation of a curve along a "Jacobi field" yield curves with related geodesic curvatures? Maybe this is related to e.g. curve shortening flow or some other already studied notions.

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    $\begingroup$ Interesting... One would expect the Jacobi equation to make no sense for non-geodesic $\gamma$ since it comes from the second variation of Energy and usually it doesn't make any sense to consider second derivatives at non-critical points. $\endgroup$
    – Carlos Esparza
    Apr 28, 2020 at 22:37
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    $\begingroup$ @CarlosEsparza: Of course there is a metric running around on the path space (say of paths of Sobolev class $H^1$), hence one can define the second derivative everywhere. $\endgroup$
    – Thomas Rot
    May 12, 2020 at 7:38

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