Helices

Erscheinungsjahr 2025
ISBN: 978-3-032-09662-3
Verlag: Springer

Buch, Englisch, 230 Seiten, Format (B × H): 160 mm x 241 mm, Gewicht: 545 g

Reihe: RSME Springer Series

This book provides an overview of generalised helices, or Lancret helices, in both Riemannian and Lorentzian backgrounds. We would like to highlight, if 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.

The level of work has been structured so that it is suitable for a Master of Science, links with the expertise of a Bachelor in Mathematics or Physics, and leaves an open door to the 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.

Zielgruppe

Research

Autoren/Hrsg.

Ferrandez Izquierdo, Angel

Fachgebiete

• Naturwissenschaften Biowissenschaften Angewandte Biologie Bioinformatik
• Mathematik | Informatik EDV | Informatik Angewandte Informatik Bioinformatik
• Geowissenschaften Umweltwissenschaften Umweltwissenschaften
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Topologie
• Mathematik | Informatik Mathematik Geometrie

Inhaltsverzeichnis

Chapter 1. Lancret helices in R3.- Chapter 2. Lancret helices in S3 and H3.- Chapter 3. The conformal extension and protein chains.- Chapter 4. Lancret helices in homogeneous 3-spaces.- Chapter 5. Lancret helices in the Minkowski space L3.- Chapter 6. Lancret helices in the de Sitter S31 and anti de Sitter H31 spaces.- Chapter 7. Lightlike helices in Ln, Sn1
and Hn1.- Chapter 8. Lightlike Lancret helices in Ln.- Chapter 9. The Hasimoto transformation.- Chapter 10. Variational models.- Chapter 11. Bosons in spaces forms.

Über Autor(innen)

Angel Ferrandez Izquierdo  has a Bachelor's degree in Mathematics from the University of Granada and a PhD in Mathematics from the University of Valencia. He is an emeritus professor of Geometry and Topology at the University of Murcia. He has worked on both Riemannian and Lorentzian geometry, such as the classification of submanifolds satisfying certain differential equations, Willmore surfaces, and variational problems on curves, looking for applications in physics and biology related to helices and relativistic particle models.