M. Khalifa Saad - Department of Mathematics, Faculty of Science, Islamic University of Madinah, KSA. S. G. Elgendi - Department of Mathematics, Faculty of Science, Islamic University of Madinah, KSA. - Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt.
Based on the first fundamental form, we study the conformal deformation of surfaces and calculate the transformation of their Gaussian curvatures. We establish this transformation in special parameterizations, such as isothermal and orthogonal parameterizations. We find under what conditions, the conformal transformation is isometric and we study the case in which the Gaussian curvature is invariant under a conformal transformation. Also, we show that a Monge surface is conformal to a plane if and only if they are isometric and prove that the two Monge surfaces are conformal if and only if they are isometric. Moreover, we investigate the conformal change of the geodesic under conformal surface mapping and prove that a geodesic \(\gamma(t)\) remains a geodesic under the conformal change if and only if the change is homothetic. Finally, we provide some illustrative examples to strengthen our main results. These examples not only serve to illustrate our primary results but also feature graphical representations for clarity.
M. Khalifa Saad, S. G. Elgendi, On the conformal deformations of surfaces in Euclidean space, Journal of Nonlinear Sciences and Applications, 18 (2025), no. 2, 75--84
Khalifa Saad M., Elgendi S. G., On the conformal deformations of surfaces in Euclidean space. J. Nonlinear Sci. Appl. (2025); 18(2):75--84
Khalifa Saad, M., Elgendi, S. G.. "On the conformal deformations of surfaces in Euclidean space." Journal of Nonlinear Sciences and Applications, 18, no. 2 (2025): 75--84
