Category Archives: Differential geometry

A simple proof of the Gauss-Bonnet theorem for geodesic ball

Posted on 26/02/2018 by KKK

In this short note, we will give a simple proof of the Gauss-Bonnet theorem for a geodesic ball on a surface. The only prerequisite is the first variation formula and some knowledge of Jacobi field (second variation formula), in particular … Continue reading →

Posted in Calculus, Differential equations, Differential geometry, Geometry | Leave a comment

Archimedes’ principle for hyperbolic plane

Posted on 06/01/2018 by KKK

After writing the previous post, I realized that there is an exact analogue of the Archimedes’ principle for the surface area of the hyperbolic disk inside the hyperbolic plane. Let us fix the notations. Let be the -dimensional Minkowski space … Continue reading →

Posted in Calculus, Differential geometry, Geometry | Leave a comment

Hopf fibration double covers circle bundle of sphere

Posted on 09/11/2017 by KKK

Two days ago, I gave a seminar talk on Chern‘s proof of the generalized Gauss-Bonnet theorem. Here I record the answer to a question asked by one of my colleague during the talk. Although not directly related to the proof … Continue reading →

Posted in Algebra, Differential geometry, Group theory | 3 Comments