# Proof of uniqueness part of the Fundamental Theorem of the local theory of curves

• May 20th 2012, 11:04 AM
x3bnm
Proof of uniqueness part of the Fundamental Theorem of the local theory of curves
In DoCarmo's Differential Geometry book there's an excerpt (on page 19-20) like this:

------------------------------Beginning of Excerpt------------------------------------------------------------------------
Fundamental theorem of the local theory of curves:

Given differentiable functions $k(s) > 0$ and $\tau (s),$ where $s \in I$, there exists a regular parametrized curve $\alpha \colon I \rightarrow \mathrm{R^3}$
such that $s$ is the arc length, $k(s)$ is the curvature, and $\tau(s)$ is the torsion of $\alpha$. Moreover, any other curve $\bar{\alpha}$, satisfying the
same conditions, differs from $\alpha$ by a rigid motion; that is, there exists an orthogonal linear map $\rho$ of $\mathrm{R^3}$, with positive determinant,
and a vector $c$ such that $\bar{\alpha} = \rho \circ \alpha + c$.

.........

Proof of the Uniqueness Part of the Fundamental Theorem:

We first remark that arc length, curvature, and torsion are invariant under rigid motions; that means, for instance, that if $M \colon \mathrm{R^3} \rightarrow \mathrm{R^3}$
is a rigid motion and $\alpha = \alpha(t)$ is a parametrized curve, then

$\int_a^b \left|\frac{d{\alpha}}{dt}\right| dt = \int_a^b \left|\frac{d(M\circ \alpha)}{dt}\right| dt$

That is plausible, since these concepts are defined by using inner or vector products of certain derivatives (the derivatives are invariant under translations, and the
inner and vector products are expressed by means of lengths and angles of vectors, and thus also invariant under rigid motions).
-------------------------------End of Excerpt------------------------------------------------------------------------

My question is:
why's that
$\int_a^b \left|\frac{d{\alpha}}{dt}\right| dt = \int_a^b \left|\frac{d(M\circ \alpha)}{dt}\right| dt$?

How did the author prove this? Is it possible to help me prove this? I'm completely lost at the last line of the excerpt given above.
• May 20th 2012, 12:52 PM
x3bnm
Re: Proof of uniqueness part of the Fundamental Theorem of the local theory of curves
I found the solution at this website on page 8:

http://www.math.uregina.ca/~mareal/cs2.pdf

The proof is something like this:

To prove the uniqueness part, we choose the linear orthogonal transformation $M$ which maps the Frenet fame of $\alpha$ at $c$ to the Frenet frame of $\beta$ at $c$.

Note that both frames are orthonormal systems of vectors with the last vector equal to the vector product of the previous two: thus the transformation $M$ mentioned above exists and moreover, has
$\text{det}(M) = 1 \neq 0$.

We have that $\beta'(s) = M\alpha'(s)$, has unique solution for all $s$, by using a uniqueness property of the solutions of the system.[Q.E.D]