TITLE: Why is arc length not a differential form?
QUESTION [8 upvotes]: I read that the arc length is not a differential form. But I don't understand why it isn't. I understand that differential forms are integrands and arc length is an expression which is integrable. What property of differential form does it not satisfy?

REPLY [6 votes]: I understand that differential forms are integrands and arc length is an expression which is integrable.

On a given parametrized curve, any local expression that can be integrated can be written (locally in the parameter of the the curve) as $f(p) dp$ by differentiating the integral of the expression.  This is a differential $1$-form, and accordingly the integral is invariant under changes of parameter. That is not a surprise, considering that $f(p)$ was defined by differentiating an invariant quantity.

What property of differential form does it not satisfy?

Arc length is integrated only along specified curves, and is a differential form on any one curve.  However, the infinitesimal arc length (Riemannian metric line-element) $ds = \sqrt{\sum (dx_i)^2}$ is defined without reference to any curve and is a local expression in the coordinates of the space in which the curves are to be drawn.   This local expression is not a differential form on the space, and it is not possible to re-write it as a differential form. If it were equal to a form, it could be restricted to any particular curve to obtain the arclength form on that curve.  But there is no such form, because (through a given point, in a small neighborhood) there must pass curves along which integration of a form gives zero and this is not the case for arc length.