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A rather short proof can be found here. I assume you are interested in the case when $f$ is a scalar function. Otherwise the Hodge Laplacian differs from the Laplace–Beltrami operator not only by a s …

There is a short conceptual proof of Gauß-Bonnet due to Chern (see also "A panoramic view of Riemannian geometry" by Berger). The argument assumes a basic familiarity with differential forms though. …

L.P. Eisenhart claims in his Riemannian Geometry (Princeton University Press, 1926, p. 82) that Bianchi was the first who discovered the algebraic identity in 1902. It is also indicated in Italian Ma …

I like the presentation of the Theorema Egregium in A Comprehensive Introduction to Differential Geometry (Volume 2) by Michael Spivak. A translation of the original paper by Gauss and the historical …

A big classification result that I'm aware of is due to Gromov and Lawson. Theorem. Let $M$ be a compact simply connected manifold of dimension $\geq 5$, which is not a spin. Then $M$ admits a met …

According to Hans Samelson's historical note "Differential Forms, the Early Days", both notions were introduced in Les Méthodes nouvelles de la Mécanique Céleste by Poincaré (vol. 3, Gauthier-Villar …

Yau asked in 1982 if there is any compact simply connected manifold with nonnegative curvature for which one can prove that it does not admit a metric of positive curvature. This question opens his li …

An elementary 'coordinate' proof is given in Ordinary Differential Equations by Philip Hartman. It doesn't even use the contraction mapping argument. The main effort is spent to show the $C^1$-regular …

The rolling motion of a convex symmetric body on a horizontal plane is a classical problem. In the symmetric case, Chaplygin was the first who showed that the full equations of motion can be reduced t …

I haven't heard of any recent breakthroughs. The strongest result that I know is due to Misha Bialy: Theorem. If almost every phase point of the billiard ball map in a strictly convex billiard table …

You might be interested in the recent article by Böröczky and Csikós concerning polytopes $P_n$ of $n$ facets with minimal surface area. They studied best approximations of a convex body $K\subset \ …

It is well known how to construct a Laplacian on a fractal using the Dirichlet forms (see e.g. the survey article by Strichartz). This implies, in particular, that a fractal can be "heated", i.e. one …

The inequality $$\det(A+B)\geq \det A +\det B$$ is implied by the Minkowski determinant theorem $$(\det(A+B))^{1/n}\geq (\det A)^{1/n}+(\det B)^{1/n}$$ which holds true for any non-negative $n\times …