Three-dimensional geometry and topology. Volume 1, by William P. Thurston, (edited by Sylvio Levy). Princeton Mathematical Series, 35. £29.95. 1997. ISBN 0-691-08304-5 (Princeton University Press).

Preface.- 1. 1+1 Dimensional Integrable Systems.- 1.1 KdV equation, MKdV equation and their Darboux transformations. 1.1.1 Original Darboux transformation. 1.1.2 Darboux transformation for KdV equation. 1.1.3 Darboux transformation for MKdV equation. 1.1.4 Examples: single and double soliton solutions. 1.1.5 Relation between Darboux transformations for KdV equation and MKdV equation. 1.2 AKNS system. 1.2.1 2 x 2 AKNS system. 1.2.2 N x N AKNS system. 1.3 Darboux transformation. 1.3.1 Darboux transformation for AKNS system. 1.3.2 Invariance of equations under Darboux transformations. 1.3.3 Darboux transformations of higher degree and the theorem of permutability. 1.3.4 More results on the Darboux matrices of degree one. 1.4 KdV hierarchy, MKdV-SG hierarchy, NLS hierarchy and AKNS system with u(N) reduction. 1.4.1 KdV hierarchy. 1.4.2 MKdV-SG hierarchy. 1.4.3 NLS hierarchy. 1.4.4 AKNS system with u(N) reduction. 1.5 Darboux transformation and scattering, inverse scattering theory. 1.5.1 Outline of the scattering and inverse scattering theory for the 2 x 2 AKNS system . 1.5.2 Change of scattering data under Darboux transformations for su(2) AKNS system. 2. 2+1 Dimensional Integrable Systems.- 2.1 KP equation and its Darboux transformation. 2.2 2+1 dimensional AKNS system and DS equation. 2.3 Darboux transformation. 2.3.1 General Lax pair. 2.3.2 Darboux transformation of degree one. 2.3.3 Darboux transformation of higher degree and the theorem of permutability. 2.4 Darboux transformation and binary Darboux transformation for DS equation. 2.4.1 Darboux transformation for DSII equation. 2.4.2 Darboux transformation and binary Darboux transformation for DSI equation. 2.5 Application to 1+1 dimensional Gelfand-Dickey system. 2.6 Nonlinear constraints and Darboux transformation in 2+1 dimensions. 3. N + 1 Dimensional Integrable Systems.- 3.1 n + 1 dimensional AKNS system. 3.1.1 n + 1 dimensional AKNS system. 3.1.2Examples. 3.2 Darboux transformation and soliton solutions. 3.2.1 Darboux transformation. 3.2.2 u(N) case. 3.2.3 Soliton solutions. 3.3 A reduced system on Rn. 4. Surfaces of Constant Curvature, Backlund Congruences.- 4.1 Theory of surfaces in the Euclidean space R3. 4.2 Surfaces of constant negative Gauss curvature, sine-Gordon equation and Backlund transformations. 4.2.1 Relation between sine-Gordon equation and surface of constant negative Gauss curvature in R3. 4.2.2 Pseudo-spherical congruence. 4.2.3 Backlund transformation. 4.2.4 Darboux transformation. 4.2.5 Example. 4.3 Surface of constant Gauss curvature in the Minkowski space R2,1 and pseudo-spherical congruence. 4.3.1 Theory of surfaces in the Minkowski space R2,1. 4.3.2 Chebyshev coordinates for surfaces of constant Gauss curvature. 4.3.3 Pseudo-spherical congruence in R2,1. 4.3.4 Backlund transformation and Darboux transformation for surfaces of constant Gauss curvature in R2,1. 4.4 Orthogonal frame and Lax pair. 4.5 Surface of constant mean curvature. 4.5.1 Parallel surface in Euclidean space. 4.5.2 Construction of surfaces. 4.5.3 The case in Minkowski space. 5. Darboux Transformation and Harmonic Map.- 5.1 Definition of harmonic map and basic equations. 5.2 Harmonic maps from R2 or R1,1 to S2, H2 or S1,1. 5.3 Harmonic maps from R1,1 to U(N). 5.3.1 Riemannian metric on U(N). 5.3.2 Harmonic maps from R1,1 to U(N). 5.3.3 Single soliton solutions. 5.3.4 Multi-soliton solutions. 5.4 Harmonic maps from R2 to U(N). 5.4.1 Harmonic maps from R2 to U(N) and their Darboux transformations. 5.4.2 Soliton solutions. 5.4.3 Uniton. 5.4.4 Darboux transformation and singular Darboux transformation for unitons. 6. Generalized Self-Dual Yang-Mills and Yang-Mills-Higgs Equations.- 6.1 Generalized self-dual Yang-Mills flow. 6.1.1 Generalized self-dual Yang-Mills flow. 6.1.2 Darboux transformation. 6.1.3 Example. 6.1.4 Relation with AKNS system. 6.2 Yang-Mills-Higgs

Darboux Transformations in Integrable Systems: Theory and their Applications to Geometry

Confluent Hypergeometric Functions By Dr L J Slater Pp ix + 247 (Cambridge: At the University Press, 1960) 65s net

Confluent Hypergeometric Functions

and the corresponding Euler-Lagrange equation is H = 0, where H is the mean curvature vector field. If φ : (M, g) → (N, h) is a Riemannian immersion, then it is a critical point of the bienergy in C∞(M,N) if and only if it is a minimal immersion [26]. Thus, in order to study minimal immersions one can look at harmonic Riemannian immersions. A natural generalization of harmonic maps and minimal immersions can be given by considering the functionals obtained integrating the square of the norm of the tension field or of the mean curvature vector field, respectively. More precisely: • biharmonic maps are the critical points of the bienergy functional E2 : C∞(M,N) → R, E2(φ) = 12 ∫

A short survey on biharmonic maps between Riemannian manifolds

We review part of the classical theory of curves and surfaces in 3-dimensional Lorentz-Minkowski space. We focus in spacelike surfaces with constant mean curvature pointing the differences and similarities with the Euclidean space.

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Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space

We give a differential geometric interpretation for the classification of biharmonic curves in semi-Euclidean 3-space due to Chen and Ishikawa (1991).

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Biharmonic curves in Minkowski 3-space

Classifications of all biharmonic isoparametric hypersurfaces in the unit sphere, and all biharmonic homogeneous real hypersurfaces in the complex or quaternionic projective spaces are shown. Answers in case of bounded geometry to Chen’s conjecture or Caddeo, Montaldo and Piu’s one on biharmonic maps into a space of non positive curvature are given. Gauge field analogue is shown, indeed, the isolation phenomena of bi-Yang-Mills fields are obtained.

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Classifications and Isolation Phenomena of Bi-Harmonic Maps and Bi-Yang-Mills Fields

We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati–Konno–Ichikawa elastic beam equation, the complex Dym equation and the short pulse equation. They are related to the modified KdV or the sine–Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler–Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.

Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves

We show that every proper biharmonic curve in a 3-dimensional Sasakian space form of constant holomorphic sectional curvature H is a helix (both of whose geodesic curvature and geodesic torsion are constants). In particular, if H ≠ 1, then it is a slant helix, that is, a helix which makes constant angle α with the Reeb vector field with the property \(\kappa^{2}+\tau^{2}=1+(H-1)\sin^{2}\alpha\). Moreover, we construct parametric equations of proper biharmonic herices in Bianchi–Cartan–Vranceanu model spaces of a Sasakian space form.

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Jun-ichi Inoguchi

Papers

Biharmonic curves in Minkowski 3-space

Classifications and Isolation Phenomena of Bi-Harmonic Maps and Bi-Yang-Mills Fields

Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves

Biharmonic curves in 3-dimensional Sasakian space forms

Magnetic Curves in Cosymplectic Manifolds