Surfaces of a Constant Negative Curvature

I study the geometric notion of a differential system describing surfaces of a constant negative curvature and describe a family of pseudospherical surfaces for the nonlinear partial differential equations with constant Gaussian curvature . 1. Introduction In recent decades, a class of transformations having their origin in the work by B？cklund in the late nineteenth century has provided a basis for remarkable advances in the study of nonlinear partial differential equations (NLPDEs) [1]. The importance of B？cklund transformations (BTs) and their generalizations is basically twofold. Thus, on one hand, invariance under a BT may be used to generate an infinite sequence of solutions for certain NLPDEs by purely algebraic superposition principles. On the other hand, BTs may also be used to link certain NLPDEs (particularly nonlinear evolution equations (NLEEs) modelling nonlinear waves) to canonical forms whose properties are well known [2, 3]. Nonlinear wave phenomena have attracted the attention of physicists for a long time. Investigation of a certain kind of NLPDEs has made great progress in the last decades. These equations have a wide range of physical applications and share several remarkable properties [4–6]: (i) the initial value problem can be solved exactly in terms of linear procedures, the so-called “inverse scattering method (ISM);” (ii) they have an infinite number of “conservation laws;” (iii) they have “BTs;” (iv) they describe pseudo-spherical surfaces (pss), and hence one may interpret the other properties (i)–(iii) from a geometrical point of view; (v) they are completely integrable [1, 3]. This geometrical interpretation is a natural generalization of a classical example given by Chern and Tenenblat [2] who introduced the notion of a differential equation (DE) for a function that describes a pss, and they obtained a classification for such equations of type ( ). These results provide a systematic procedure to obtain a linear eigenvalue problem associated to any NLPDE of this type [7]. Sasaki [6] gave a geometrical interpretation for inverse scattering problem (ISP), considered by Ablowitz et al. [4], in terms of pss. Based on this interpretation, one may consider the following definition. Let be a two-dimensional differentiable manifold with coordinates ( ). A DE for a real function describes a pss if it is a necessary and sufficient condition for the existence of differentiable functions: depending on and its derivatives such that the one-forms satisfy the structure equations of a pss, that is, This structure was considered for the