Category Archives: Geometry

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

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

Weighted Hsiung-Minkowski formulas and rigidity of umbilic hypersurfaces

1. Motivation and Main Results A. D. Alexandrov [Ale1956], [Ale1962] proved that the only closed hypersurfaces of constant (higher order) mean curvature embedded in are round hyperspheres. The embeddedness assumption is essential. For instance, admits immersed tori with constant mean … Continue reading

Posted in Calculus, General Relativity, Geometry, Inequalities | Leave a comment

The discrete Gauss-Bonnet theorem

This is a slight extension of my previous note on discrete Gauss-Bonnet theorem. As mentioned in that note, this is a generalization of the well-known fact that the sum of the exterior angles of a polygon is always , which … Continue reading

Posted in Calculus, Combinatorics, Discrete Mathematics, Geometry, Topology | Leave a comment

A functional inequality on the boundary of static manifolds

1. introduction and statement of results The research in this article is largely motivated by the following result concerning a functional inequality on the boundary of bounded domains in the Euclidean space , proved in [MTX] Corollary 3.1. Theorem 1 … Continue reading

Posted in Calculus, General Relativity, Geometry, Inequalities | Leave a comment

Mean value properties for harmonic functions on Riemannian manifolds

It is well-known that a continuous function on satisfies the mean value property over spheres if and only if it is harmonic i.e. . More generally, this also holds on a Riemannian space form. However, this property is generally not … Continue reading

Posted in Analysis, Calculus, Geometry | 2 Comments