Is there already name for the generalization of Clothoids to curves on smooth manifolds, i.e. where the curve's curvature depends linearly on the curve's length-parameter? 


In the euclidean plane Clothoids are a suitable idealized model for the trajectory of vehicles moving at constant speed while the steering wheel is also rotated at constant speed and I wonder if there are also idealized models for that kind of driving on smooth surfaces. 



Name for Curves from Driving on Smooth Manifolds
Since this is a reference request: in [this article](http://www.ams.org/mathscinet-getitem?mr=1464047), Arroyo, Barros, and Garay just call what you are describing a *Cornu spiral* without much fanfare. ("Cornu spiral" or "Euler spiral" are other common names for the clothoid.)


In [this report/blog post](http://www.thephysicsmill.com/2013/02/23/textagconference-report/) Jonah Miller speaks about the analogue of the Euler spiral on the sphere (he arrives at those spirals from a different point of view). 


