The Generalized Weierstrass System in Three-Dimensional Euclidean Space

Authors

Paul Bracken, The University of Texas Rio Grande ValleyFollow

Document Type

Book

Publication Date

1-25-2019

Abstract

In this chapter, some recent advances in the area of generalized Weierstrass representations will be given. This is an approach to the theory of surfaces in Euclidean three space. Weierstrass representations permit the explicit construction of surfaces in the designated space. The discussion proceeds in a novel and introductory manner. The inducing formulas for the coordinates of a surface are derived and important conservation laws are formulated. These lead to the inducing mechanism of a surface in terms of solutions to a system of two-dimensional Dirac equations. A set of fundamental forms as well as expressions for the mean and Gaussian curvatures are derived. The Cartan moving frame picture is also formulated to put everything in a broader perspective. A connection with the nonlinear sigma model is presented, which has important applications in physics. Some relationships are established between integrable systems and geometry by way of conclusion.

Comments

© 2019 The Author(s). Licensee IntechOpen.

Recommended Citation

Paul Bracken (January 25th 2019). The Generalized Weierstrass System in Three-Dimensional Euclidean Space, Manifolds II - Theory and Applications, Paul Bracken, IntechOpen, DOI: 10.5772/intechopen.82631. Available from: https://www.intechopen.com/chapters/65349

Creative Commons License

This work is licensed under a Creative Commons Attribution 3.0 License.

Publication Title

Manifolds II - Theory and Applications

DOI

10.5772/intechopen.82631