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 Apr 11 accepted Understanding a particular case of modifying curvature and torsion as opposed to modifying the curve Apr 11 comment Understanding a particular case of modifying curvature and torsion as opposed to modifying the curve Thanks a lot. The question seems really silly once you look bacak, in fact right after Anthony Carapetis' comments. Apr 11 comment Understanding a particular case of modifying curvature and torsion as opposed to modifying the curve @AnthonyCarapetis, On second thoughts, I think I get what you said, the curvature and tosion functions and space curves are in a 1-1 correspondence with each other, so both the ways yield the same curve. Is that right?? Apr 11 comment Understanding a particular case of modifying curvature and torsion as opposed to modifying the curve @AnthonyCarapetis well from what I understand about the fundamental theorem of space curves, any set of curvature torsion functions uniquely determines a space curve, but I am not able to gather how that helps me establish the relationship between the curves??Can you add something more?? Apr 11 asked Understanding a particular case of modifying curvature and torsion as opposed to modifying the curve Apr 2 accepted Proof for showing that a set of space curves form a manifold Apr 2 comment Proof for showing that a set of space curves form a manifold Thanks for that. I am sorry I guess I was pushing the nitpicking a bit too far. Yeah will look into all that you said. Cheers!! Apr 2 comment Proof for showing that a set of space curves form a manifold Aah yes, thats true. Is there some way to circumvent this and still get a meaningful group structure. I was also thinking of inducing a quotient topology on $M_{\alpha}$ using $(0,\infty)$ through the same kind of map $\psi$ or rather the inverse of $\psi$ Apr 2 revised Proof for showing that a set of space curves form a manifold rectified error Apr 2 comment Proof for showing that a set of space curves form a manifold I am sorry for making it all sound convoluted. So a group multiplication cannot be defined for this the way I did? Apr 2 comment Proof for showing that a set of space curves form a manifold Yes the single curve $\alpha$ with curvature $\kappa$ and torsion $\tau$ generates the family $M_{\alpha}$, but if $\alpha$ is defined on $I$, then so si every other curve $\dfrac{\alpha}{\mu}$, right??Thats how I defined multiplication. But I guess what you are saying is that at any given point $\dfrac{\alpha}{\mu_1}$ and $\dfrac{\alpha}{\mu_2}$ correspond to different points on the individual curves, so the multiplication is not well defined, am I right? Apr 2 comment Proof for showing that a set of space curves form a manifold @Jesse Madnick. I am sorry about the first mistake, you are right. As for the parameters being different, well this might seem a little shady, but I have assumed that the all curves in $M_{\alpha}$ have been defined on the same domain of definition as $\alpha$, so $s$ is the parameter for all curves in $M_{\alpha}$, though $s$ is not the arc length parameter for any other curve apart from $\alpha$. I hope that does not make this problem go kaput. Thanks for your patience. Apr 1 comment Proof for showing that a set of space curves form a manifold @Jesse Madnick. I have edited the question to fill the gaps which you pointed out. I hope it fits in logically. Thanks a lot for your response. Apr 1 revised Proof for showing that a set of space curves form a manifold added 621 characters in body Apr 1 asked Proof for showing that a set of space curves form a manifold Mar 31 comment About multiples of curvature and torsion of a space curve Thanks, from what I understood by going through your link is that I indeed have different space curves and there is no reparametrisation happening, as curvature is an invariant under reparametrisation. Reparametrisation involves redefining the parameter over a different interval. Am I right? Mar 31 asked About multiples of curvature and torsion of a space curve Mar 28 awarded Yearling Mar 20 accepted Affine connection, metric and parallel transport and mutual interdependence Mar 20 comment Affine connection, metric and parallel transport and mutual interdependence Thanks for your edit.It is much more accessible than before.