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In this talk I will focus on the critical points of some functionals defined on Riemannian and pseudo-Riemannian geometries in various dimensions.One of these functionals is the one defined on two surfaces in $R^3$. We know some special cases of such functionals in classical differential geometry where the minimal and Willmore surfaces are the critical points of these functionals. In the general case such geometrical functionals are constructed by the minimal and Gauss curvatures of the surfaces. In such cases it is very difficult to investigate the critical points of these functionals. For this purpose, I shall introduce soliton techniques to solve the corresponding Euler-Lagrange equations [M. Gürses and S. Tek, Korteweg de Vries Surfaces , Nonlinear Analysis: Theory, Method and Application, Vol. 95, 11-22 (2014)].

Geometrical functionals defined on higher dimensional pseudo-Riemannian geometries (space-times) are very important in general relativity. In this talk I shall only consider the case where the Lagrange function is constructed solely by the curvature tensor and its covariant tensors. Investigation of critical points (black holes, AdS Waves) of these functionals is so popular reaserch area in the last three decades. Some of the critical points are universal. The corresponding space-time metrics solve the most general field equations. I shall give some examples of such critical points [ M. Gürses, S. Hervik, T. Ç. Şişman and B. Tekin, Anti-de Sitter Wave Solutions of Higher Derivative Theories , Physical Review Letters, Vol. 111, 101101 (2013)].