In "Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition" Di Manfredo P. do Carmo I found the following theorem:
When it comes to proving that $F=0$ the following argument is used:
It is assuming the convergence of $\frac{\mathrm{d} \alpha}{ \mathrm{d} \sigma}$ given the convergence of $\alpha$, which is not true in general, am I wrong? Do you know if this proof can be fixed or if a correct proof can be found somewhere else?
Thank you in advance