Helices

This book provides an overview of generalised helices, or Lancret helices, in both Riemannian and Lorentzian backgrounds. We would like to highlight, it that were possible, this mathematical structure with great preponderance in Nature and even in our daily lives. Moreover, we intend to collect many somewhat scattered results and put them in a unique context using helices as leitmotiv. Everybody is aware of the ubiquity of helices in Nature and admires their intrinsic beauty, but it seems that people are not interested in knowing the omnipresence of helices in the real world. Without getting into great speculations, we soon realize that Nature is governed by a principle of minimum energy, that is, any natural process occurs at minimum cost. With that idea in mind, and thinking in a supercoiled DNA molecule, we are predisposed to think that this phenomenon reflects the need to place the greatest amount of matter in the least possible space. Or even, that the arrangement in a double supercoiled helix is the optimal organization to facilitate the mechanism of replication.

That said, we should think of some kind of energy functional, defined on a suitable space of curves, and determine its critical points. Furthermore, the theory of helices falls into place in a broader scenario. For instance, helicoids or surfaces with helicoidal symmetry basically inherit all their properties from their generating curves, the helices, and also arise in many contexts. We intended deliberately that the level of this work is suitable for a Master of Science, linking with the expertise of a Bachelor in Mathematics or Physics and leaving an open door to research world. In addition, we have tried to make a self-contained book, for which we have included several appendices. And, wishing that our eyes see beyond, we have deployed a wide and unsuspected series of helices in geometries where light-like vectors play a key role.