Integrable Curves and Surfaces
نویسنده
چکیده
The connection of curves and surfaces in R3 to some nonlinear partial differential equations is very well known in differential geometry [1], [2]. Motion of curves on two dimensional surfaces in differential geometry lead to some integrable nonlinear differential equations such as nonlinear Schrödinger (NLS) equation [3], Korteweg de Vries (KdV) and modified Korteweg de Vries (mKdV) equations [4],[5]. Surface theory in three dimensional Euclidean space (R3) is widely used in different branches of science, particularly mathematics (differential geometry, topology, Partial Differential Equations (PDEs)), theoretical physics (string theory, general theory of relativity), and biology [6]-[11]. There are some special subclasses of 2-surfaces which arise in the branches of science aforementioned. For the classification of surfaces in R3, particular conditions are imposed on the Gaussian and mean curvatures. These conditions are sometimes given as algebraic relations between curvatures and sometimes given as differential equations for these two curvatures. Here are some examples of some subclasses of 2-surfaces:
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