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Consider two tiny plane(flat) surfaces A and B meeting at a straight line L.We have a preassigned vector on A at some point P on it.Is it always possible to have a vector parallel to the first one and lying on B at some point on it(say Q)?

[The first vector(that on A) is assumed as not parallel to L]

The two planes may be considered as the adjacent tangent planes on a curved surface over which we are working out parallel transport along some curve lying on it.

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You are not allowed to rotate the tangent planes wrt each other-----they are fixed on the curved surface at the two points concerned. –  Anamitra Palit Dec 4 '12 at 13:42
    
If you are given with a curve, and if a vector is specified at some point P on it, then its always possible to parallel transport this vector to any other point Q on the curve. However if some extra conditions are imposed on the vector to be parallely transported and/or on the curve along which it is to be parallely transported, then it may not be always possible. For example geodesic curves which parallely transport their own tangent vectors may end in singularities. –  user10001 Dec 4 '12 at 14:30
    
Well since both surfaces are flat, I think the statement holds. The tangent spaces are the same everywhere on each surface and can be trivially transformed to that of the other surface, vice versa. –  namehere Dec 4 '12 at 14:36
    
@dushya that could be most of an answer –  David Z Dec 5 '12 at 3:45
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