# Category Archives: Geometry

Elementary geometry, differential geometry, Riemannian geometry, algebraic geometry.

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

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

## Archimedes’ principle for hyperbolic plane

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

## Archimedes and the area of sphere

I record here a remarkable discovery of Archimedes about the formula for the surface area of a sphere. I think the derivation is elementary enough so that it can be taught in high school. (I think I was not taught … Continue reading

## Euler’s formula e^ix = cos x + i sin x: a geometric approach

Today I mentioned the famous Euler’s formula briefly in my calculus class (when discussing hyperbolic functions, lecture notes here): where is a solution to (usually denoted by “”, but indeed there is no single-valued square root for complex numbers, or … Continue reading

## An inequality for functions on the plane

I accidentally came across a curious inequality for functions of two variables. I would like to know if this inequality is a special case of a more general result but I was unable to find a reference. It would also … Continue reading