PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 61(75), pp. 119132 (1997) 

Surfaces of constant curvature and geometric interpretations of the KleinGordon, SineGordon and SinhGordon equationsBoris A. Rosenfeld and Nadezhda E. MaryukovaAbstract: It is well known that the SineGordon 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 pseudoRiemannian spaces $V_1^3$ of constant curvature, that is in the pseudoEuclidean space $R^3_1$, in the pseudoelliptic space $S^3_1$, and in the pseudohyperbolic space $H^3_1$, and that the SinhGordon 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 KleinGordon equation (KGE) $u_{xx}u_{yy} = m^2u$ admits analogous geometric interpretations in the {\it Galilean space} $\Gamma^3$, and in the {\it pseudoGalilean 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 pseudoEuclidean 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 Full text of the article:
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