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If $S$ is a surface with a geodesic on it, can we find another surface $S'$ such that these surfaces are tangent on the geodesic with the additional condition that there is no other intersection?

Furthermore, to what extent can we loosen the assumption of "geodesic"?

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Possibly relevant: – Jesse Madnick Sep 13 '11 at 21:47

How about the following construction:

Since this is a local problem we may assume $S$ in the form

$$S:\quad (u,v)\mapsto\bigl(u,v,f(u,v)\bigr)$$

with $f(0,0)=f_u(0,0)=f_v(0,0)=0$, and the given geodesic as

$$\gamma:\quad t\mapsto\bigl(t, g(t),f(t,g(t))\bigr)$$

with $g(0)=g'(0)=0$. Now define the surface $S'$ by

$$S':\quad (u,v)\mapsto\bigl(u,v, f(u,v)+ (v-g(u))^r\bigr)$$

with a sufficiently large $r$.

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If S was a sphere, and S' was a torus which was placed around the equator of the sphere, then the circle where they touch would be a geodisic on both surfaces, unless I have misunderstood the question.

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I think the question is not 'do there exist surfaces $S$, $S'$' but 'for all $S$ does there exist a $S'$'; that seems a bit harder... – Steven Stadnicki Sep 18 '11 at 15:45

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