Surfaces of constant curvature and geometric interpretations of the Klein-Gordon, Sine-Gordon and Sinh-Gordon equations

Boris A. Rosenfeld and Nadezhda E. Maryukova

Abstract: It is well known that the Sine-Gordon equation (SGE) $u_{xx}-u_{yy} = \sin u$ admits a geometric interpretation as the differential equation which determines surfaces of constant negative curvature in the Euclidean space $R^3$. This result can be generalized to the elliptic space $S^3$ and the hyperbolic space $H^3$. These results are analogous to the results of Chern that SGE also admits a geometric interpretation as the differential equation which determines spacelike surfaces of constant negative curvature in pseudo-Riemannian spaces $V_1^3$ of constant curvature, that is in the pseudo-Euclidean space $R^3_1$, in the pseudoelliptic space $S^3_1$, and in the pseudohyperbolic space $H^3_1$, and that the Sinh-Gordon equation (SHGE) $u_{xx}-u_{yy} = \sinh u$ admits geometric interpretations as the differential equation which determines timelike surfaces of constant positive curvature in the same spaces. In this paper it is proved also that the Klein-Gordon equation (KGE) $u_{xx}-u_{yy} = m^2u$ admits analogous geometric interpretations in the {\it Galilean space} $\Gamma^3$, and in the {\it pseudo-Galilean space} $\Gamma^3_1$, that is, in the affine space $ E^3 $ whose plane at infinity is endowed with the geometry of the Euclidean plane $ R^2 $ and of the pseudo-Euclidean plane $R^2_1$, respectively, in the {\it quasielliptic space} $S^{1,3}$, in the {\it quasihyperbolic space} $H^{1,3}$, in the {\it quasipseudoelliptic space} $S_{01}^{1,3}$, and in the {\it quasipseudohyperbolic space} $H^{1,3}_{01}$, that is, in the projective space $P^3$ whose collineations preserve two conjugate imaginary planes and two conjugate imaginary points on the line of their intersection, two conjugate imaginary planes and two real points on the line of their intersection, two real planes and two conjugate imaginary points on the line of their intersection, and two conjugate imaginary planes and two real points on the line of their intersection, respectively.

Classification (MSC2000): 53C35

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