Can speed be defined for a parametrized but irregular curve in a Riemannian manifold?

Here is a good reference that goes into some detail, of how to work with absolutely continuous curves on Riemannian manifolds: http://nyjm.albany.edu/j/2015/21-12v.pdf In other words, there is a reasonable extension of notions like the speed of a curve on a Riemannian manifold so that the answer to your question is negative.

Notably, a similar strategy sometimes allows you to work even with curves which are defined in an abstract metric space, with no manifold structure at all. For this, a good reference is the first half of the book by Ambrosio, Gigli, and Savaré.

$X\vee Y\cong (X\times\{y_0\})\cup (\{x_0\}\times Y)$

Is the Riemann integral of a strictly smaller function strictly smaller?

Order of Cyclic Subgroups

Does cardinality really have something to do with the number of elements in a infinite set?

Elementary proof of transformations of domino tilings.

Strictly associative coproducts

Is there a nice description of the field of fractions of the ring of polynomials with integer coefficients?

Lower bound for monochromatic triangles in $K_n$

Integers that satisfy $a^3= b^2 + 4$

Prove $\frac{1}{2} + \cos(x) + \cos(2x) + \dots+ \cos(nx) = \frac{\sin(n+\frac{1}{2})x}{2\sin(\frac{1}{2}x)}$ for $x \neq 0, \pm 2\pi, \pm 4\pi,\dots$

Why is it necessary to have convexity outside the feasible region?

Can two functions with different codomains be equal?