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Guest
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Given two points p and q on a smooth, path connected manifold, there
is a way to produce a C1 (continuously differentiable) curve joining
them, namely by connecting p and q by finitely many charts (path
connectedness) and picking up points in the intersections and joining
them by C1 curves taking care that at each of those points, the
*tangent* to th previous curve matches with that of the second.
However, I can't answer whether the two points can be joined by an
infinitely differentiable (smooth) curve, because I can't keep track
of all the derivatives and make them match. Please shed light.
Regards,
Saurav |
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