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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
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 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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Category Archives: Analysis
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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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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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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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
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An extension of the First Fundamental Theorem of Calculus
In this note, we record a simple extension of the first fundamental theorem of calculus. Let us recall the first and second fundamental theorems of calculus: Theorem 1 (First fundamental theorem of calculus) If is continuous, then for Theorem … Continue reading
Posted in Analysis, Calculus
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Mean value properties for harmonic functions on Riemannian manifolds
It is wellknown 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
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