Differential Geometry - Computation Help

Question

I'm trying to learn differential geometry through one of MIT's online courses (lecture notes found here: http://ocw.mit.edu/courses/mathematics/18-950-differential-geometry-fall-2008/lecture-notes/ch1_revised.pdf) and am stuck with what should be an easy question. The question asks to show the following: Suppose $c(s)$ is a regular curve ($c'(s) \ne 0$) in the plane with $|c(s)| \le 1$ and suppose there is a point t with $|c(t)| = 1$. Then $|\kappa(t)| \ge 1$. Here, $\kappa$ is defined as

$\frac{det(c', c'')}{||c'||^3}.$

It's should be straightforward if we write $c$ as the graph of a function and rotate so that $c(t) = (0,1)$. However, I feel as if there should be a more elegant solution using only what's found in the first day's notes (found in the link). I just don't see it.

Answer 1

Geometrically, you should understand that if the curve $c$ stays inside the unit circle and touches at a point, it must be at least as curved as the circle at that point.

Here's a hint: Use the fact that $t$ is a local maximum of the function $|c(s)|^2$, and put your calculus to work :)

Answer 2

Let me assume, without loss of generality, that $c$ is parametrized by arclength, so that $|c'|\equiv 1$, and let me also assume $t_0=0$ (where $c(t_0)$ has length one). Furthermore, upon reparametrizing $c$ backwards, we may assume that $\mathcal{B}=(c(0),c'(0))$ is a direct orthonormal basis of $\Bbb R^2$ (this frame is indeed orthonormal, as follows from the fact that $|c|^2$ attains a local maximum at $0$, and the the preceding hypothesis).

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The curvature then takes on the form $$\kappa(0)=\det(c'(0),c''(0))=\det_{\mathcal B}(c'(0),c''(0))=-c(0)\cdot c''(0)$$ Also, the Taylor expansion of $c$ near $0$ is $c(t)=c(0)+tc'(0)+\frac12t^2c''(0)+o(t^2)$, and thus $$\begin{array}{ccl} |c(t)|^2&=&\underbrace{c(0)\cdot c(0)}_{=1}+2t\underbrace{c(0)\cdot c'(0)}_{=0}+(\underbrace{2\times\frac12 c''(0)\cdot c(0)}_{=-\kappa(0)}+\underbrace{c'(0)\cdot c'(0)}_{=1})t^2+o(t^2)\\ &=&1+(1-\kappa(0))t^2+o(t^2) \end{array}$$ This expression has to stay $\leq 1$ near $0$, which in turn forces $1-\kappa(0)\leq 0$