Loewner's "forgotten" theorem

Abstract

Let $f(t)$ be a smooth and periodic function of one real variable. Then the planar curves $t\mapsto (f^{\prime }(t),f(t))$ and $t\mapsto (f^{\prime \prime }(t)-f(t),f^{\prime }(t))$ both have non-negative rotation number around every point not on the curve. These are the two simplest cases of a beautiful Theorem by C. Loewner. This article is expository, we prove the two statements by elementary means following work by Bol [3]. After that, we present Loewner's Theorem and his proof from [7].

10 pages, 6 figures

References in corpus (1)

• Framed holonomic knots