MATH 573 Modern Geometry I

Groups of transformations of Euclidean and pseudo-Euclidean spaces. The theory of curves. The theory of surfaces in three-dimensional space. The Riemannian metric. The second fundamental form. The Poincare model of Lobachevsky’s geometry. The complex geometry. Surfaces in complex space. The conformal form of the metric on a surface. Isothermal co-ordinates. Gaussian curvature in terms of conformal co-ordinates. Surfaces of constant curvature. The Fundamental Theorem of Surfaces. Gauss-Weingarten equations. Theorema Egregium of Gauss. Surfaces of constant negative curvature and the “Sine-Gordon” equation. Minimal surfaces. The Concept of a Manifold and the simplest Examples.