TITLE: The "ds" which appears in an integral with respect to arclength is not a 1-form. What is it?
QUESTION [29 upvotes]: The only reasonable way to interpret "$ds$" as a functional on tangent vectors has to be that it takes a tangent vector and spits out its length, but this is not linear.  So $ds$ is not a 1-form.  It still seems like a nice sort of object to think about integrating.  Does $ds$ fit into a larger class of gadgets generalizing differential forms?  Or it there some compelling reason that I shouldn't care about $ds$?

REPLY [7 votes]: It is an example of an absolute differential form, as defined by Toby Bartels here: http://ncatlab.org/nlab/show/absolute+differential+form.