Integrable Equations and Their Evolutions Based on Intrinsic Geometry of Riemann Spaces

Abstract

The intrinsic geometry of surfaces and Riemannian spaces will be investigated. It is shown that many nonlinear partial differential equations with physical applications and soliton solutions can be determined from the components of the relevant metric for the space. The manifolds of interest are surfaces and higher-dimensional Riemannian spaces. Methods for specifying integrable evolutions of surfaces by means of these equations will also be presented.

1. Introduction

Surfaces arise in the study of many classical systems, especially those in which interfaces or fronts appear. This occurs in the study of surface waves, deformations of membranes, and boundaries between regions of differing viscosity or densities. Surfaces are also finding applications at present in the areas of classical string theory and quantum field theory [1, 2]. Two cases which are particularly important for applications and have received much attention are the cases in which the surface can be ascribed a constant mean curvature or a constant Gaussian curvature. Each of these cases has a wide variety of nonlinear partial differential equations associated with it, and each has been studied with great interest recently [3, 4].

This is due to the fact that many partial differential equations, or systems of differential equations, can be generated as a consequence of formulas which result from the study of surfaces. Some equations such as those related to constant mean curvature are a consequence of the theory, and many related to constant Gaussian curvature can be obtained as a result of specifying quantities which are in the formulas in a particular form. Moreover, many of these equations are known to possess soliton solutions [5], and so this provides a link between equations, solitons, and surfaces. This has certainly been shown to be the case in the study of constant mean curvature surfaces [6]. In fact, many nonlinear equations which have been studied recently, such as the KdV and mKdV equations, are directly related to surfaces which have constant Gaussian curvature [7, 8]. These equations have soliton solutions, and conversely solutions from these equations can be used to determine the first and second fundamental forms of a surface. Historically, Bianchi and Bäcklund introduced symmetry transformations which can be regarded as transformations between surfaces and have come to be called Bäcklund transformations. This was initially done for the sine-Gordon equation which allows the construction of new pseudospherical surfaces from a given surface. In an approach that uses the inverse scattering transform method, one starts with the system of 1+1-dimensional linear problems 𝜓_𝑥=𝑃𝜓, 𝜓_𝑦=𝑄𝜓 and then constructs explicit formulas for the immersion of one-parameter families of surfaces [9].

The study of the general theory of three-dimensional Riemann spaces has not been completed as far as a theory of surfaces is concerned [10]. In fact, various partial differential equations do arise within the study of problems in three and higher dimensional Riemann spaces as will be seen here. Three-dimensional Riemann spaces and possibly even higher dimensional versions play a crucial and fundamental role in general relativity and the study of gravity in general. One of the reasons for this is that any metric in general relativity can be transformed into a synchronic system of coordinates, which is one in which the spatial part of the metric is Riemannian. In this event, an initial value problem can be formulated in terms of three-dimensional Riemannian space plus one time variable in a very natural way. The study of higher-dimensional Riemann spaces can very well be of importance in the study of Kaluza-Klein theories as well as in formulating string theories.

The intrinsic geometry of surfaces and Riemannian spaces will be introduced and described without any reference to the enveloping space. Results from the study of these spaces are used to obtain nonlinear differential equations. This gives an automatic connection between surfaces and equations. The procedure will be to define a metric in mathematical form with a particular structure and a curvature which will depend on the components of the metric and given by a specific formula, in particular the curvature equation. This will be used to establish a relationship between the intrinsic geometry of the spaces as specified by the functions in the metric and multidimensional integrable equations which result from the curvature equation [11]. Methods for specifying integrable evolutions or deformations of surfaces will also be examined as well. The study of the solutions to these equations is not the main objective here, but to formulate some new, relevant equations and propose some new ones for study.

2. Nonlinear Equations Related to Surfaces

Consider two-dimensional surfaces which have a first fundamental form given in terms of components as

𝑑𝑠²_𝐼=𝐸𝑑𝑥²+2𝐹𝑑𝑥𝑑𝑦+𝐺𝑑𝑦², (2.1) where 𝐸, 𝐹, and 𝐺 are functions of the local coordinates (𝑥,𝑦) of a surface. The basic invariant which is characteristic of the intrinsic geometry of a surface is the Gaussian curvature, which is determined by

𝑅𝐾=¹²¹²𝑔=𝑅₁₂₁₂𝐻². (2.2) In (2.2), 𝑅₁₂₁₂ is the nonzero component of the Riemann tensor 𝑅_{𝛼𝛽𝛾𝛿}, and 𝐻 is related to 𝑔 by means of the expression [12]

𝑔=𝐸𝐺−𝐹²=𝐻². (2.3) Given the components of the first fundamental form or metric, the Christoffel symbols Γ^𝑘_𝑖𝑗 are evaluated by means of

Γ^𝑘_𝑖𝑗=12𝑔^𝑘𝑠(𝜕𝑔_𝑖𝑠𝜕𝑥^𝑗+𝜕𝑔_𝑗𝑠𝜕𝑥^𝑖−𝜕𝑔_𝑖𝑗𝜕𝑥^𝑠). (2.4) The component 𝑅₁₂₁₂ can be calculated by means of

𝑅₁₂₁₂=𝜕Γ₁₂₂−𝜕𝑥𝜕Γ₁₂₁𝜕𝑦+Γ^ℎ₁₁Γ_2ℎ2−Γ^ℎ₁₂Γ_2ℎ1. (2.5) In terms of the components of the metric of a surface given by (2.1), the Gaussian curvature 𝐾(𝑥,𝑦) can be evaluated by first calculating (2.5) and then putting this result into (2.2). This procedure shows that the Gaussian curvature 𝐾(𝑥,𝑦) is related to the components of the metric of the surface by means of the Gauss equation

1𝐾=𝐹2𝐻((𝐸𝐸𝐻^𝑦−1𝐻𝐺_𝑥)_𝑥2+(𝐻𝐹_𝑥−1𝐻𝐸_𝑦−𝐹𝐸𝐸𝐻^𝑥)_𝑦). (2.6) By specifying the components of the metric (2.1) in a particular way, it is possible to use (2.6) to generate various types of nonlinear equations. Several examples will serve to illustrate the main idea as to exactly how this can be done.

First put 𝐸=𝐺=0 and 𝐻=𝐹 in (2.6), then we obtain

(ln𝐹)_𝑥𝑦−𝐹𝐾=0. (2.7) Setting 𝐹=exp(𝜑), this equation becomes

𝜑_𝑥𝑦−𝐾exp(𝜑)=0. (2.8) The coordinate curves are minimal lines, and for the case of constant Gaussian curvature 𝐾, it specifies a particular nonlinear equation, the Liouville equation.

As another example, suppose that we take 𝐹=0, 𝐸=cos²𝜃, and 𝐺=sin²𝜃, then 𝐻=cos𝜃sin𝜃, 𝐺_𝑥=2sin𝜃cos𝜃𝜃_𝑥, and 𝐸_𝑦=−2cos𝜃sin𝜃𝜃_𝑦, and one has the lines of curvature as the coordinate curves. The Gauss equation (2.6) takes the form

𝜃_𝑥𝑥−𝜃_𝑦𝑦=−𝐻𝐾. (2.9) Taking a constant negative curvature 𝐾=−2𝑎² in (2.9), it is found that 𝜃 satisfies

𝜃_𝑥𝑥−𝜃_𝑦𝑦=𝑎²sin(2𝜃). (2.10) Thus, for different metrics, or systems of coordinates, on a surface, the Gauss equation (2.6) reduces to a simple partial differential equation, as in (2.8) or (2.10). These equations have come up within the analysis of the pure geometry of surfaces. The well-known cases of nonlinear partial differential equations, such as the Liouville and sine-Gordon equations, correspond to the case of constant Gaussian curvature.

Finally, another equation which has many applications to physics can be developed in this way. Consider a geodesic system of coordinates for which 𝐸=1, 𝐹=0, and (2.1) reduces to the diagonal form 𝑑𝑠²=𝑑𝑥²+𝐺𝑑𝑦². Thus the curves 𝑦 equal constant are geodesics, and for (2.3) in this metric, this implies that 𝐺=𝐻². The Gauss equation takes the form

𝐻_𝑥𝑥+𝐾(𝑥,𝑦)𝐻=0. (2.11) This is a linear equation, and when 𝐾 is fixed to be constant, the corresponding surfaces are the spherical or pseudospherical surfaces of constant curvature. It is important to realize that (2.11) is closely connected with the one-dimensional stationary Schrödinger equation [13, 14]. This can be seen by identifying

𝐻(𝑥,𝑦)=Re𝜓(𝑥,𝑦,𝜆),𝐾(𝑥,𝑦)=−𝑈(𝑥,𝑦)+𝜆²₀, (2.12) where Re denotes the real part, and 𝜆₀ is an arbitrary real or pure imaginary value of the spectral parameter 𝜆 in the equation

−𝜓_𝑥𝑥+𝑈(𝑥,𝑦)𝜓=𝜆²𝜓. (2.13) Therefore, solvable cases of the Schrödinger equation (2.13) provide the explicit expressions for the Gauss curvature 𝐾(𝑥,𝑦) and the metric of surfaces referred to their geodesic coordinates. Here in (2.13), the quantity 𝑦 acts as a parameter. In fact, in general, one can prescribe any dependence of 𝐾(𝑥,𝑦) and 𝐻(𝑥,𝑦) on the variable 𝑦.

A way in which the dependence on 𝑦 can be fixed is to require that the function 𝐻 obeys the additional linear equation

𝐻_𝑦=𝐴(𝐾,𝐾_𝑥,𝜕_𝑥)𝐻, (2.14) where 𝐴 is some additional differential operator. Recalling (2.12), where 𝜓 obeys (2.13), we require that in addition to (2.13), the function 𝜓 satisfies an equation of the form

𝜓_𝑦=𝐴(𝑈,𝑈_𝑥,…,𝜕_𝑥)𝜓, (2.15) where 𝐴 is the linear differential operator in 𝜕_𝑥. The compatibility of (2.13) and (2.15) gives the preservation of (2.11) in the 𝑦 variable. Moreover, the compatibility condition for (2.13) and (2.15) is equivalent to a nonlinear partial differential equation for 𝑈. As a simple example, take 𝐴=𝑐𝜕_𝑥, where 𝑐 is a constant, then 𝜓 satisfies the two equations

𝜓_𝑥𝑥=(𝑈−𝜆²)𝜓,𝜓_𝑦=𝑐𝜕_𝑥𝜓. (2.16) Differentiating the second equation twice with respect to 𝑥 yields

𝜕²_𝑥𝜕_𝑦𝜓=𝑐(𝜕_𝑥𝑈)𝜓+𝑐𝑈𝜕_𝑥𝜓−𝑐𝜆²𝜕_𝑥𝜓. (2.17) Similarly, differentiating the first equation with respect to 𝑦 gives

𝜕_𝑦𝜕²_𝑥𝜓=(𝜕_𝑦𝑈)𝜓+𝑐𝑈𝜕_𝑥𝜓−𝑐𝜆²𝜕_𝑥𝜓. (2.18) Equating these two derivatives and simplifying the result, a first-order equation for 𝑈 is obtained, namely, 𝑐𝜕_𝑥𝑈−𝜕_𝑦𝑈=0. The general solution to this equation is 𝑈=𝑈(𝑥+𝑐𝑦) and hence 𝐾=𝐾(𝑥+𝑐𝑦). In fact, a nonlinear equation arises with the choice of 𝐴 as a third-order differential operator.

Theorem 2.1. Suppose (2.15) is given by 4𝜓_𝑥𝑥𝑥−6𝑈𝜓_𝑥−3𝑈_𝑥𝜓+𝜓_𝑦=0. (2.19) Then the compatibility condition for (2.13) and (2.19) is equivalent to the following equation for 𝑈: 𝑈_𝑥𝑥𝑥−6𝑈𝑈_𝑥+𝑈_𝑦=0. (2.20)

Proof. Substituting (2.13) into (2.19), we have the pair 𝜓_𝑥𝑥=(𝑈−𝜆)𝜓,𝜓_𝑦=−𝑈_𝑥𝜓+2𝑈𝜓_𝑥+4𝜆²𝜓_𝑥. (2.21) Calculating the compatibility condition 𝜓_𝑥𝑥𝑦−𝜓_𝑦𝑥𝑥=0 based on (2.21), in order to enforce this, it is found that (2.20) must hold. Putting 𝐾(𝑥,𝑦)=−𝑈(𝑥,𝑦)+𝜆²₀, (2.20) takes the form 𝐾_𝑥𝑥𝑥+6𝐾𝐾_𝑥−6𝜆²₀𝐾_𝑥+𝐾_𝑦=0. (2.22) Equations (2.20) and (2.22) are precisely the KdV equation, which is known to be integrable. The basic idea of the method is to generate a solvable nonlinear system from the compatibility condition of linear partial differential equations with variable coefficients. Of course, the KdV equation has exact solutions, and so surfaces may be generated with soliton curvature and associated solitonic metric. Therefore, as an example, it can be checked that 𝑈 given below is a solution to (2.20): 𝑈(𝑥,𝑦)=−2𝑎²cosh²(𝑎(𝑥−4𝑎²𝑦)−𝑥₀),𝐾(𝑥,𝑦)=2𝑎²cosh²(𝑎(𝑥−4𝑎²𝑦)−𝑥₀)+𝜆²₀. (2.23) The coefficient 𝐻(𝑥,𝑦) in the metric obeys a nonlinear equation such as (2.22) for 𝐾. This can be obtained by solving (2.13) for 𝑈 and substituting (2.19) to give 𝜓_𝑥𝑥𝑥−6𝜆²𝜓_𝑥𝜓−3^𝑥𝜓_𝑥𝑥𝜓+𝜓_𝑦=0. (2.24)

3. Evolutions of Surfaces

Consider now integrable evolutions or deformations of surfaces which are referred to geodesic coordinates [15]. Suppose that the coefficient 𝐺 or 𝐻 of the metric and the Gaussian curvature depend on an additional parameter which can be thought of as a time variable; so we write 𝐻=𝐻(𝑥,𝑦,𝑡), 𝐾=𝐾(𝑥,𝑦,𝑡). Evolutions of 𝐻 and 𝐾 in 𝑡 can be sought which preserve (2.11), or by virtue of (2.12), evolutions in 𝑡 which preserve (2.13). They are given by the compatibility condition of (2.13) and a linear equation of the form (2.15) with 𝑦 replaced by 𝑡; so there exists the pair

−𝜓_𝑥𝑥+𝑈(𝑥,𝑡)𝜓=𝜆²𝜓,𝜓_𝑥=𝐴(𝑈,𝑈_𝑥,…,𝜕_𝑥)𝜓, (3.1) where 𝐴 is a linear differential operator. The evolutions which can be taken for 𝐴 can come from the KdV hierarchy [16, 17].

Dynamics for a surface can also be introduced by fixing the dependence of 𝐾 and 𝐻 using one of the equations from the KdV hierarchy and then specifying the evolution in 𝑡 by another equation from the hierarchy. The common solution 𝐾(𝑥,𝑦,𝑡) of the KdV equation (2.22) and a higher KdV equation, for example, 𝐾_𝑡+𝜕_𝑥^2𝑛+1𝐾+⋯=0 generates the evolution of surfaces.

Another way for fixing the dependence of 𝐾 and 𝐻 on 𝑦 and determination of the evolution in 𝑡 consists in considering the compatibility of a system composed of (2.13) and an equation of the form

𝜓_𝑡+𝑓(𝜕²_𝑥+𝐾)𝜓_𝑦+^𝑀_𝑛=1𝑈_𝑛(𝑥,𝑦,𝑡)𝜕^𝑛_𝑥𝜓=0, (3.2) where 𝑓 is an arbitrary polynomial, and 𝑈_𝑛 are functions. The compatibility of (2.13) and (3.2) is equivalent to a 2+1-dimensional integrable equation for 𝐾. The following theorem gives a case in point.

Theorem 3.1. (i) Consider the following pair of evolution equations with 𝑈=−𝐾:
𝜓_𝑥𝑥𝜓=−𝐾(𝑥,𝑦,𝑡)𝜓,_𝑡=−2(𝜕²_𝑥𝜓_𝑦+𝐾𝜓_𝑦)−2[^𝑥_−∞𝑑𝑠𝐾_𝑦(𝑠,𝑦,𝑡)]𝜓_𝑥−𝐾_𝑦𝜓, (3.3)(3.4) where here and in what follows one defines 𝜕_𝑥⁻¹𝜅(𝑥,𝑦)=^𝑥_−∞𝑑𝑠𝜅(𝑠,𝑦). (3.5) Then the compatibility condition of (3.3) and (3.4) is equivalent to the following equation for 𝐾(𝑥,𝑦,𝑡): 𝐾_𝑡+𝐾_𝑥𝑦𝑦+4𝐾𝐾_𝑦+2𝐾_𝑥𝜕_𝑥⁻¹𝐾_𝑦=0. (3.6)
(ii) Consider the following pair of evolution equations:
𝜓_𝑥𝑥=−𝐾(𝑥,𝑦,𝑡)𝜓+𝜆²𝜓𝜓,_𝑡=−2(𝜕²_𝑥𝜓_𝑦+𝐾𝜓_𝑦)−2(^𝑥_−∞𝑑𝑠𝐾_𝑦(𝑠,𝑦,𝑡))𝜓_𝑥−𝐾_𝑦𝜓. (3.7)(3.8) The compatibility condition for these two equations implies that 𝐾 satisfies 𝐾_𝑡+𝐾_𝑥𝑥𝑦+4𝐾𝐾_𝑦+2𝐾_𝑥𝜕_𝑥⁻¹𝐾_𝑦−2𝜆²𝐾_𝑦=0. (3.9) The proof of this Theorem runs along exactly the same lines as Theorem 2.1. All of the lengthy calculations throughout have been done with Maple [18].

Equation (3.6), for example, has an infinite set of integrals of motion, which are

𝐶_𝑛=𝑑𝑥𝑑𝑦𝑃_𝑛(𝑥,𝑦), (3.10) where the densities 𝑃_𝑛 are given by the recursion

𝑃_𝑚+1=𝑃_𝑚,𝑥+^𝑚−1_𝑘=1𝑃_𝑘𝑃_𝑚−𝑘,𝑃₁=−𝐾(𝑥,𝑦,𝑡),𝑚=1,2,3,…. (3.11) The first such integrals of motion turn out to be

𝐶₁=𝑑𝑥𝑑𝑦𝐾(𝑥,𝑦,𝑡),𝐶₂=𝑑𝑥𝑑𝑦𝐾²(𝑥,𝑦,𝑡),𝐶₃=𝑑𝑥𝑑𝑦(𝐾²_𝑥−2𝐾³). (3.12) Thus, there exists an infinite set of global characteristics of a surface that are invariant under the evolution discussed in Theorem 3.1.

As a generalization of (3.3) and (3.7) in Theorem 3.1, we have the Theorem which follows.

Theorem 3.2. Consider the following pair of evolution equations: 𝜕²_𝑥𝜓+𝜎𝜕_𝑦𝜕𝜓+𝐾(𝑥,𝑦,𝑡)𝜓=0,_𝑡𝜓+4𝜕³_𝑥𝜓+6𝐾𝜕_𝑥𝜓+3𝐾_𝑥𝜓−3𝜎(𝜕_𝑥⁻¹𝐾_𝑦)𝜓+𝛼=0, (3.13) where 𝜎²=±1 and 𝛼 is a constant. The compatibility condition for these two equations implies that 𝐾 satisfies 𝐾_𝑡𝑥+𝐾_{𝑥𝑥𝑥𝑥}+6(𝐾𝐾_𝑥)_𝑥+3𝜎²𝐾_𝑦𝑦=0. (3.14) This 2+1-dimensional equation is integrable.

A more systematic method [19] for passing to higher dimensions involves considering the compatibility condition for a pair of equations of the form

𝐿𝜓=𝜆𝜓,_𝑘𝑓_𝑘(𝐿)𝜕𝜓𝜕𝑧_𝑘+𝐴𝜓=0, (3.15) where 𝑓_𝑘 are arbitrary polynomials in the operator 𝐿, and 𝑧_𝑘 are the variables. A slightly different reformulation of this system will be of particular interest here, namely,

𝐿𝜓=𝜆𝜓,(𝐷+𝐴)𝜓=_𝑘𝜆^𝑘𝜕𝜓𝜕𝑧_𝑘+𝐴𝜓=0. (3.16) In particular, if there are three variables and we take 𝐷𝜓=(𝜆𝜕_𝑦+𝜕_𝑡)𝜓, with 𝐿𝜓=(𝜕²_𝑥+𝐾)𝜓, then (3.16) takes the form

𝜕²_𝑥𝜓=−𝐾𝜓+𝜆𝜓,𝜕_𝑡𝜓=−𝜆𝜕_𝑦𝜓−𝐴𝜓. (3.17) Differentiating the first of the equations in (3.17) with respect to 𝑡 and the second with respect to 𝑥 twice, the compatibility condition for (3.17) takes the following form:

𝜕_𝑡(𝐿−𝐾)𝜓+𝜆𝜕_𝑦(𝐿−𝐾)𝜓+(𝐿−𝐾)𝐴𝜓=0. (3.18) Expanding this out, we have the explicit form

−𝐾_𝑡𝜓+𝐾(𝜆𝜕_𝑦𝜓+𝐴𝜓)−𝜆(𝜆𝜕_𝑦𝜓+𝐴𝜓)+𝜆𝜕_𝑦(−𝐾𝜓+𝜆𝜓)+𝜕²_𝑥(𝐴𝜓)=0. (3.19) Let us work out (3.19) explicitly for the case in which 𝐴 is the operator

𝐴=𝛼𝜕³_𝑥+𝑓𝜕_𝑥+𝑔, (3.20) where 𝛼 is an arbitrary constant, and 𝑓 and 𝑔 depend on 𝑥.

Theorem 3.3. With operator 𝐴 given by (3.20), (3.19) is given explicitly as follows: (𝐾_𝑡+𝛼𝐾_𝑥𝑥𝑥+(−3𝛼𝐾+𝑓+3𝛼𝜆)𝐾_𝑥+𝜆𝐾_𝑦−2𝜆𝑓_𝑥−𝑔_𝑥𝑥+2𝑓_𝑥𝐾)𝜓+(3𝛼𝐾_𝑥𝑥−𝑓_𝑥𝑥−2𝑔_𝑥)𝜓_𝑥=0. (3.21) The last term in 𝜓_𝑥 is absent from (3.21) provided that 𝑓 and 𝑔 satisfy the equation 𝑓_𝑥𝑥+2𝑔_𝑥−3𝛼𝐾_𝑥𝑥=0. (3.22) Therefore when (3.22) is satisfied, compatibility condition (3.19) holds if the equation for 𝐾 multiplying 𝜓 in (3.21) vanishes, namely, 𝐾_𝑡+𝛼𝐾_𝑥𝑥𝑥+(−3𝛼𝐾+𝑓+3𝛼𝜆)𝐾_𝑥+𝜆𝐾_𝑦−2𝜆𝑓_𝑥−𝑔_𝑥𝑥+2𝑓_𝑥𝐾=0. (3.23)

To prove Theorem 3.3, the compatibility condition for (3.17) is worked out, and the higher derivatives of 𝜓 in 𝑥 are eliminated using the first equation of (3.17). This is a long calculation.

A solution to (3.22) can be determined. Let us set

𝑓=𝑓+^𝑥_−∞𝐾_𝑦(𝑠,𝑦,𝑡)𝑑𝑠,𝑔=𝑐(𝐾_𝑦+𝛼𝐾_𝑥), (3.24) where constant 𝑐 will be determined in the process. Substituting (3.24) into (3.22), the solution will work if 𝑐=−1/2 and 𝑓 is determined from

𝑓_𝑥𝑥=4𝛼𝐾_𝑥𝑥, (3.25) which implies that we can take 𝑓=4𝛼𝐾. Let us summarize these results in the following theorem.

Theorem 3.4. Let 𝐴 be given by (3.20) with 𝑓 and 𝑔 given by 𝑓=4𝛼𝐾+^𝑥_−∞𝐾_𝑦1𝑑𝑠,𝑔=−2(𝐾_𝑦+𝛼𝐾_𝑥). (3.26) Then (3.19) is satisfied identically provided that 𝐾 satisfies the partial differential equation 𝐾_𝑡+12(3𝛼𝐾_𝑥𝑥𝑥+𝐾_𝑥𝑥𝑦)+(^𝑥_−∞𝐾_𝑦𝑑𝑠+9𝛼𝐾−5𝛼𝜆)𝐾_𝑥+(2𝐾−𝜆)𝐾_𝑦=0. (3.27)

4. Scalar Curvature of Three and Four-Dimensional Riemann Spaces

In the previous section, it has been seen that the Gauss equation for the two-dimensional Riemann spaces connects the Gaussian curvature 𝐾 to the coefficients of the metric. In fact, two of the three components of the surface metric can always be transformed away by a change of variables.

For three-dimensional Riemann spaces, by means of a transformation of coordinates, the corresponding metric can be converted to diagonal form

𝑑𝑠²=𝑒₁𝐻²₁𝑑𝑥²₁+𝑒₂𝐻²₂𝑑𝑥²₂+𝑒₃𝐻²₃𝑑𝑥²₃, (4.1) where 𝑒_𝑖=±1 allow us to control the signs in the metric and 𝐻_𝑖 are functions. In terms of the components of the metric, the scalar curvature is given by

𝑅=𝑔^𝑖𝑙𝑔^𝑘𝑚𝑅_{𝑖𝑘𝑙𝑚}. (4.2) For the diagonal metric (4.1), 𝑅 can be evaluated explicitly, and the scalar curvature takes the following form:

−12𝑅=1𝐻₁𝐻₂[𝑒₂𝜕𝜕𝑥₂(1𝐻₂𝜕𝐻₁𝜕𝑥₂)+𝑒₁𝜕𝜕𝑥₁(1𝐻₁𝜕𝐻₂𝜕𝑥₁+1)]𝐻₁𝐻₃[𝑒₁𝜕𝜕𝑥₁(1𝐻₁𝜕𝐻₃𝜕𝑥₁)+𝑒₃𝜕𝜕𝑥₃(1𝐻₃𝜕𝐻₁𝜕𝑥₃+1)]𝐻₂𝐻₃[𝑒₂𝜕𝜕𝑥₂(1𝐻₂𝜕𝐻₃𝜕𝑥₂)+𝑒₃𝜕𝜕𝑥₃(1𝐻₃𝜕𝐻₂𝜕𝑥₃+1)]𝐻₁𝐻₂𝐻₃[𝑒₁𝐻₁𝜕𝐻₂𝜕𝑥₁𝜕𝐻₃𝜕𝑥₁+𝑒₂𝐻₂𝜕𝐻₁𝜕𝑥₂𝜕𝐻₃𝜕𝑥₂+𝑒₃𝐻₃𝜕𝐻₁𝜕𝑥₃𝜕𝐻₂𝜕𝑥₃]. (4.3) The factor −1/2 on the left side arises from a factor of −2 on the right-hand side divided out.

Let us begin with the simplest case for which 𝐻₁=𝐻₂=1 and 𝐻₃=𝐻(𝑥,𝑦,𝑧) with 𝑒₁=𝑒₃=1, 𝑒₂=𝜎². In this case, (4.3) takes the form

𝐻_𝑥𝑥±𝐻_𝑦𝑦+12𝑅𝐻=0. (4.4) By setting

−12𝑅=𝑈−𝐸,𝐻=𝜓, (4.5) (4.4) can be put in the equivalent form

−(𝜕²_𝑥+𝜎²𝜕²_𝑦)𝜓+𝑈(𝑥,𝑦,𝑧)𝜓=𝐸𝜓. (4.6) For the case 𝜎²=1, (4.6) becomes the two-dimensional stationary Schrödinger equation, while for the case 𝜎²=−1 the perturbed string equation results.

As in the two-dimensional case, one can specify the dependence of the scalar curvature 𝑅 on another variable 𝑧 by requiring that 𝐻, and hence 𝜓, obeys an additional linear partial differential equation compatible with (4.6). Consider Riemann spaces such that the scalar curvature and metric obey the linear system of equations

−(𝜕²_𝑥+𝜎²𝜕²_𝑦1)𝐻−2𝑅𝐻=0,𝐻_𝑧+𝐴(𝑅,…,𝜕_𝑥,𝜕_𝑦)𝐻=0. (4.7) In (4.7), 𝐴 is a linear partial differential operator. Equivalently, in terms of 𝑈 and 𝑉, the equations in (4.7) can be written as

−(𝜕²_𝑥−𝜎²𝜕²_𝑦)𝜓+𝑈𝜓=𝐸𝜓,𝜓_𝑧+𝐴(𝑈,…,𝜕_𝑥,𝜕_𝑦)𝜓=0. (4.8) The compatibility condition for this system is equivalent to a nonlinear partial differential equation for the function 𝑈. It will be useful to write the first equation in (4.7) and (4.8) in a different form for the following theorem by introducing the variables 𝑢 and 𝑣 which are defined such that 𝜕_𝑢=𝜕_𝑥−𝜎𝜕_𝑦 and 𝜕_𝑣=𝜕_𝑥+𝜎𝜕_𝑦.

Theorem 4.1. Consider the following system of partial differential equations: 𝐻_𝑢𝑣=12𝐻𝑅𝐻,_𝑧=−𝛼𝐻_𝑢𝑢𝑢−𝛽𝐻_𝑣𝑣𝑣+3𝛼𝑊_2𝑢𝐻_𝑢+3𝛽𝑊_1𝑣𝐻_𝑣, (4.9) where 𝛼 and 𝛽 are constants with 𝑊₁ and 𝑊₂ given by 𝑊₁=12^𝑢_−∞1𝑅𝑑𝑢=−2𝜕_𝑢⁻¹𝑅,𝑊₂=12^𝑣_−∞1𝑅𝑑𝑣=2𝜕_𝑣⁻¹𝑅, (4.10) Then the function 𝑅 satisfies the following partial differential equation: 𝑅_𝑧+𝛼𝑅_𝑢𝑢𝑢+𝛽𝑅_𝑣𝑣𝑣−3𝛼(𝑅𝑊_2𝑢)_𝑢−3𝛽(𝑅𝑊_1𝑣)_𝑣=0,𝑊_1,𝑢=𝑊_2,𝑣=12𝑅. (4.11)

The proof of this goes along exactly the same lines as the previous theorems. It suffices to substitute the integrals (4.10) into (4.9) and compute the compatibility condition by calculating the derivatives 𝐻_𝑧𝑢𝑣, 𝐻_𝑢𝑣𝑧, which is some work, and then putting them in the equation 𝐻_𝑧𝑢𝑣−𝐻_𝑢𝑣𝑧=0. This generates (4.11).

The result in (4.11) is the simplest and lowest member of a hierarchy of nonlinear partial differential equations which is referred to as the Nizhnik-Veselov-Novikov equation. This equation is integrable by means of the IST method. Its properties are different for the cases 𝜎²=±1 and 𝐸>0, 𝐸<0 and 𝐸=0.

Continuing further, some nonlinear partial differential equations will be produced which are associated with the scalar curvature equation (4.3).

The first example to be considered corresponds to selecting 𝐻₁=𝐻₂=𝐻 and 𝐻₃=1, which implies that the metric has the form

𝑑𝑠²=𝐻²𝑑𝑥²₁−𝜎²𝐻²𝑑𝑥²₂−𝑑𝑥²₃. (4.12) Equation (4.3) for the scalar curvature 𝑅 then simplifies to the form

(ln𝐻)_𝑥𝑥−𝜎²(ln𝐻)_𝑦𝑦−2𝐻𝐻_𝑧𝑧−𝐻²_𝑧+12𝑅𝐻²=0. (4.13) Suppose, for example, that we take 𝐻=exp𝜑 so that upon differentiation 𝐻_𝑧=𝜑_𝑧𝑒^𝜑 and 𝐻_𝑧𝑧=𝜑²_𝑧𝑒^𝜑+𝜑_𝑧𝑧𝑒^𝜑, then the equation takes the form

𝜑_𝑥𝑥−𝜎²𝜑_𝑦𝑦−(2𝜑_𝑧𝑧+3𝜑²_𝑧)𝑒^2𝜑+12Re^2𝜑=0. (4.14) If instead we take 𝑒₃=+1 in the metric, the equation obtained from (4.3) is given by

𝜑_𝑥𝑥−𝜎²𝜑_𝑦𝑦+(2𝜑_𝑧𝑧+3𝜑²_𝑧)𝑒^2𝜑+12Re^2𝜑=0. (4.15) When 𝜑_𝑧=0 and 𝑅 is put equal to a constant value, (4.15) reduces to the Liouville equation.

Another interesting case is the one in which trigonometric functions appear in the metric. To have an example of this case, take 𝑒₁=1, 𝑒₂=∓𝜎² and 𝑒₃=∓1 with 𝜃=𝜃(𝑥,𝑦,𝑧), and 𝐻_𝑗 are given by

𝐻₁𝜃=cos2,𝐻₂𝜃=sin2,𝐻₃=1. (4.16) Equation (4.3) can be simplified to the form

𝜃_𝑥𝑥±𝜎²𝜃_𝑦𝑦∓cos𝜃𝜃_𝑧𝑧±34sin𝜃𝜃²_𝑧−14𝑅sin𝜃=0. (4.17) Suppose that 𝑒₁=1, 𝑒₂=−𝜎², and 𝑒₃=−1 such that

𝐻₁=cos𝜃,𝐻₂=sin𝜃,𝐻₃=𝜃_𝑧. (4.18) The following equation results from (4.3):

sin𝜃𝜃cos𝜃^𝑥𝑥𝑧−𝜎²cos𝜃𝜃sin𝜃^𝑦𝑦𝑧+(𝜃_𝑥𝑥+𝜎²𝜃_𝑦𝑦)𝜃_𝑧+12cos²𝜃(𝜃²_𝑥)_𝑧+𝜎²2sin²𝜃(𝜃²_𝑦)_𝑧1+(3−4𝑅)sin𝜃cos𝜃𝜃_𝑧=0. (4.19) Equation (4.19) can be considered to be a type of generalized sine-Gordon equation.

Another interesting equation results if we take 𝑒₁=1, 𝑒₂=−𝜎², and 𝑒₃=−1 with

𝐻₁=𝑒^𝜑,𝐻₂=𝑒^𝜑,𝐻₃=𝜑_𝑧. (4.20) The equation for 𝜑 obtained from (4.3) using (4.20) is given by

(𝜑_𝑥𝑥−𝜎²𝜑_𝑦𝑦)_𝑧+(𝜑_𝑥𝑥−𝜎²𝜑_𝑦𝑦)𝜑_𝑧1+(3+4𝑅)𝜑_𝑧𝑒^2𝜑=0. (4.21) Integrating this equation with respect to 𝑧, the following system is obtained:

𝜑_𝑥𝑥−𝜎²𝜑_𝑦𝑦−𝜏𝑒^−𝜑+𝑒^2𝜑=0,𝜏_𝑧1=(6+4𝑅)𝜑_𝑧𝑒^3𝜑. (4.22) To prove (4.22), multiply the first equation by 𝑒^𝜑 and differentiate both sides with respect to 𝑧 to obtain

(𝜑_𝑥𝑥−𝜎²𝜑_𝑦𝑦)_𝑧𝑒^𝜑+(𝜑_𝑥𝑥−𝜎²𝜑_𝑦𝑦)𝜑_𝑧𝑒^𝜑−𝜏_𝑧+3𝜑_𝑧𝑒^3𝜑=0. (4.23) Substituting 𝜏_𝑧 from (4.22) and factoring 𝑒^𝜑, the original equation (4.21) is obtained.

For the case in which the scalar curvature 𝑅 is independent of 𝑧, the second equation in (4.22) can be written as

𝜏_𝑧=131(6+4𝑅)(𝑒^3𝜑)_𝑧. (4.24) By integration, this implies that 𝜏 is given by

1𝜏=(2+12𝑅)𝑒^3𝜑+𝜉(𝑥,𝑦), (4.25) where 𝜉(𝑥,𝑦) is an arbitrary function of (𝑥,𝑦). Substituting 𝜏 from (4.25) into the first equation of (4.22), we get

𝜑_𝑥𝑥−𝜎²𝜑_𝑦𝑦−𝜉(𝑥,𝑦)𝑒^−𝜑𝑅−(1+12)𝑒^2𝜑=0. (4.26) Another interesting case to examine is the one in which the metric is given by 𝑑𝑠²=𝐻²(𝑑𝑥²+𝑑𝑦²+𝑑𝑧²). For the metric in this form, (4.3) reduces to

𝐻_𝑥𝑥+𝐻_𝑦𝑦+𝐻_𝑧𝑧−12𝐻(𝐻²_𝑥+𝐻²_𝑦+𝐻²_𝑧1)−8𝑅𝐻³=0. (4.27) In terms of a function 𝜑 which is defined by 𝐻=𝜑², this equation reduces to the following form:

𝜑_𝑥𝑥+𝜑_𝑦𝑦+𝜑_𝑧𝑧−𝑅𝜑16⁵=0. (4.28)

Finally, some equations relevant to a four-dimensional Riemann space with diagonal metric which is an extension of (4.1) will be determined. The metric is written as

𝑑𝑠²=𝑒₁𝐻²₁𝑑𝑥²₁+𝑒₂𝐻²₂𝑑𝑥²₂+𝑒₃𝐻²₃𝑑𝑥²₃+𝑒₄𝐻²₄𝑑𝑥²₄, (4.29) where 𝑒_𝑖=±1 as before, and 𝐻_𝑗 are functions to be specified in a given case. The scalar curvature can be calculated by means of (4.2) given the components of metric (4.29). Some equations will be determined based on metric (4.29) in the following.

(i)Suppose that we take 𝑒₁=1, 𝑒₂=𝜎², 𝑒₃=1, and 𝑒₄=𝜎² with the functions 𝐻₁=𝐻₂=𝐻₃=exp(𝜑), 𝐻₄=1. Equation (4.2) produces the following equation: 𝜑_𝑥𝑥+𝜑_𝑦𝑦+𝜑_𝑧𝑧+32𝜎²𝑒^2𝜑𝜑_𝑤𝑤+12(𝜑²_𝑥+𝜎²𝜑²_𝑦+𝜑²_𝑧+6𝜎²𝑒^2𝜑𝜑²_𝑤1)−8Re^2𝜑=0. (4.30)(ii)The case in which 𝑒₁=1, 𝑒₂=𝜎², 𝑒₃=1, and 𝑒₄=𝜎² with functions 𝐻₁=𝐻₂=𝑒^𝜑 and 𝐻₃=𝐻₄=1 yields 𝜑_𝑥𝑥+𝜎²𝜑_𝑦𝑦+2𝑒^2𝜑𝜑_𝑧𝑧+2𝜎²𝑒^2𝜑𝜑_𝑤𝑤+3𝑒^2𝜑𝜑²_𝑧+3𝑒^2𝜑𝜑²_𝑤−14Re^2𝜑=0. (4.31)(iii) The selection 𝑒_𝑖=1 with 𝐻₁=𝐻₂=𝐻₃=𝑒^𝜑 and 𝐻₄=𝜑_𝑤 produces the following equation: 12𝜑_𝑤(𝜑_𝑥𝑥𝑤+𝜑_𝑦𝑦𝑤+𝜑_𝑧𝑧𝑤)𝜑_𝑥𝑥+𝜑_𝑦𝑦+𝜑_𝑧𝑧+12(𝜑²_𝑥+𝜑²_𝑦)+12𝜑_𝑤(𝜑_𝑥𝜑_𝑥𝑤+𝜑_𝑦𝜑_𝑦𝑤+𝜑_𝑧𝜑_𝑧𝑤1)+(3−8𝑅)𝑒^2𝜑=0. (4.32)(iv) When the metric is diagonal of the form 𝑑𝑠²=𝐻²(𝑑𝑥²+𝑑𝑦²+𝑑𝑧²+𝑑𝑤²) and all 𝑒_𝑖=1, the equation generated by (4.3) is given by 𝐻_𝑥𝑥+𝐻_𝑦𝑦+𝐻_𝑧𝑧+𝐻_𝑤𝑤−124𝑅𝐻³=0. (4.33) If we take 𝐻=𝜑² in (4.33), we obtain the equation 𝜑_𝑥𝑥+𝜑_𝑦𝑦+𝜑_𝑧𝑧+𝜑_𝑤𝑤+1𝜑(𝜑²_𝑥+𝜑²_𝑦+𝜑²_𝑧+𝜑²_𝑤1)−24𝑅𝜑⁵=0. (4.34)(v)As a final example, suppose that 𝑒₁=𝑒₃=1, 𝑒₂=𝑒₄=−1 and trigonometric functions are used for the 𝐻_𝑗 in the metric, 𝐻₁=𝐻₃=cos𝜃, 𝐻₂=𝐻₄=sin𝜃, then the equation for 𝜃 can be put in the form sin³𝜃cos𝜃(sin²𝜃−2cos²𝜃)(𝜃_𝑥𝑥+𝜃_𝑧𝑧)+cos³𝜃sin𝜃(cos²𝜃−2sin²𝜃)(𝜃_𝑦𝑦+𝜃_𝑤𝑤)+(4sin⁶𝜃−cos⁶𝜃)(𝜃²_𝑥+𝜃²_𝑧)−(4cos⁶𝜃−sin⁶𝜃)(𝜃²_𝑦+𝜃²_𝑤1)+4𝑅sin⁴𝜃cos⁴𝜃=0. (4.35)

5. Conclusions

It has been shown that several equations of physical interest arise within the analysis of the geometry of surfaces, such as the Liouville and Schrödinger equations, or as a result of the compatibility condition of two linear equations. Moreover, these equations can be combined in such a way as to produce integrable evolutions or deformations of surfaces. An introduction to a more systematic approach to going to higher dimensional partial differential equations has been mentioned. Finally, by calculating the scalar curvature based on the Riemannian metric for three- and four-dimensional Riemann spaces, a large group of nonlinear partial differential equations has been determined in both three and four dimensions. It remains as a separate task to begin to study the various kinds of solutions and their characteristics for some of these equations.