The Hyperbolic Plane and its Immersions into R³

We study how to describe and compute geometric features of curved surfaces in Euclidean three space. We describe the Gauß and mean curvature of a surface. We shall discuss applications to cartography, soap films and bubbles. We shall touch opun mechanical and variational principles. We shall also mention non-Euclidean geometries. We shall use the MAPLE computer algebra system to illustrate text and lectures.

Topics include (depending on time):

• Curves in the plane and in three space.
• Isoperimetric Inequality.
• Differentiating functions and vector fields on a surface.
• Normal curvature of curves on a surface.
• Gauß curvature Surfaces of revolutiuon.
• Minimal surfaces. Conformal maps. Mercator projection.
• Constant Mean Curvature Surfaces.
• Isometric Immersions. Second fundamental form.
• Complete manifolds. Geodesics.
• Gauß Bonnet Theorem.
• Hilbert's theorems on immersing the sphere and the hyperbolc plane.

Last updated: 3 / 26 / 4