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 Why is a² + b² ≥ 2ab ?
 A remark on the divergence theorem
 The CauchySchwarz inequality and the Lagrange identity
 On the existence of a metric compatible with a given connection
 A curious identity on the median triangle
 27 lines on a smooth cubic surface
 Weighted HsiungMinkowski formulas and rigidity of umbilic hypersurfaces
 The discrete GaussBonnet theorem
 Why a vector field rotates about its curl?
 A functional inequality on the boundary of static manifolds
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Author Archives: KKK
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
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A remark on the divergence theorem
The divergence theorem states that for a compact domain in with piecewise smooth boundary , then for a smooth vector field on , we have where is the unit outward normal and is the divergence of . In most textbooks, … Continue reading
Posted in Calculus
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The CauchySchwarz inequality and the Lagrange identity
The classical Lagrange identity is the following: This can be proven by expanding and separating the terms into the crossterms part and the non crossterms part. The Lagrange identity implies the CauchySchwarz inequality in . And when , this can … Continue reading
Posted in Algebra, Group theory, Inequalities, Linear Algebra
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On the existence of a metric compatible with a given connection
Question: Suppose we are given a torsionfree (i.e. the torsion tensor vanishes) affine connection on a smooth connected manifold . Does there exist a Riemannian metric such that its LeviCivita connection is ? If so, is it unique if we … Continue reading
Posted in Differential equations, Geometry
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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
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Weighted HsiungMinkowski 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
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The discrete GaussBonnet theorem
This is a slight extension of my previous note on discrete GaussBonnet theorem. As mentioned in that note, this is a generalization of the wellknown fact that the sum of the exterior angles of a polygon is always , which … Continue reading
Posted in Calculus, Combinatorics, Discrete Mathematics, Geometry, Topology
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