Category Archives: Geometry

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

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

Posted in Analysis, Calculus, Complex analysis, Geometry | Leave a comment

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

Posted in Analysis, Geometry, Inequalities | Leave a comment

Weighted isoperimetric inequalities in warped product manifolds

1. Introduction The classical isoperimetric inequality on the plane states that for a simple closed curve on , we have , where is the length of the curve and is the area of the region enclosed by it. The equality … Continue reading

Posted in Analysis, Calculus, Differential equations, Functional analysis, Geometry, Inequalities | Leave a comment

Faber-Krahn inequality

I record a proof of the Faber-Krahn inequality here, mainly for my own benefit. Let be one of the standard space forms: the Euclidean space , the unit sphere , or the hyperbolic space . Suppose is a bounded domain … Continue reading

Posted in Analysis, Differential equations, Functional analysis, Geometry, Inequalities | Leave a comment

Why is a² + b² ≥ 2ab ?

This post can be regarded as a sequel to my previous (and very ancient) post on 1+2+3+…. Though these two posts are not quite logically related, they share the same spirit (I’m asking a dumb question again). How can one … Continue reading

Posted in Calculus, Discrete Mathematics, Geometry, Inequalities, Linear Algebra, Probability | 1 Comment

On the existence of a metric compatible with a given connection

Question: Suppose we are given a torsion-free (i.e. the torsion tensor vanishes) affine connection on a smooth connected manifold . Does there exist a Riemannian metric such that its Levi-Civita connection is ? If so, is it unique if we … Continue reading

Posted in Differential equations, Geometry | 2 Comments

A curious identity on the median triangle

I just came across a curious identity about the angles of the “median triangle” of a given triangle, while I was reviewing a paper from a team participating in the Hang Lung Mathematics Award. Of course, I am not going … Continue reading

Posted in Geometry | 4 Comments