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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: http://en.wikipedia.org/wiki/Tubular_neighborhood2011-09-13

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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...2011-09-18