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Apr 11, 2017 at 11:59 comment added Henry.L @user-unknown There is another source of literature that discuss natural parameterization, please see Kurtek, Sebastian, et al. "Parameterization-invariant shape statistics and probabilistic classification of anatomical surfaces." Biennial International Conference on Information Processing in Medical Imaging. Springer Berlin Heidelberg, 2011.
Apr 2, 2017 at 15:09 comment added Romeo Thus I am sorry, but I cannot accept your answer.
Apr 2, 2017 at 14:42 comment added Henry.L I am not familiar with regularity in measurable sense, so I am sorry I cannot help in that direction.
Apr 2, 2017 at 14:12 comment added Romeo Thanks a lot for your interest and for your references. I admit I have not a solid background in differential geometry, thus I am a bit lost in the huge book you mention. I went through the third chapter and some of the others but I did not find anything specifically related to my problem. The same for the paper you mention and another paper by Michor I have seen: it seems they do not discuss the regularity (at the level of measurability at least, which is the one I need) of the reparameterization maps. Could you be so kind to indicate me a more precise reference therein, please? Thanks!
Apr 2, 2017 at 13:06 history answered Henry.L CC BY-SA 3.0