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What (who) is Gauss-Bonnet formula - definition

Chern-Gauss-Bonnet theorem; Generalized Gauss-Bonnet theorem; Gauss–Bonnet term; Gauss-Bonnet term; Chern–Gauss–Bonnet formula; Chern-Gauss-Bonnet formula; Chern theorem; Generalized Gauss–Bonnet theorem; Chern-Gauss–Bonnet theorem; Chern formula; Chen-Gauss-Bonnet theorem; Chern-Gauss-Bonnet Theorem; Gauss-Bonnet equation

Chern–Gauss–Bonnet theorem

In mathematics, the Chern theorem (or the Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler-Poincaré characteristic (a topological invariant defined as the alternating sum of the Betti numbers of a topological space) of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial (the Euler class) of its curvature form (an analytical invariant).

https://en.wikipedia.org/wiki/Chern–Gauss–Bonnet_theorem

Gauss–Bonnet gravity

In general relativity, Gauss–Bonnet gravity, also referred to as Einstein–Gauss–Bonnet gravity, is a modification of the Einstein–Hilbert action to include the Gauss–Bonnet term (named after Carl Friedrich Gauss and Pierre Ossian Bonnet)

https://en.wikipedia.org/wiki/Gauss–Bonnet_gravity

Shoelace formula

MATHEMATICAL ALGORITHM TO DETERMINE THE AREA OF A SIMPLE POLYGON

Surveyor's formula; Shoelace algorithm; Shoelace Method; Gauss' area formula; Gauss's area formula; Gauss area formula

The shoelace formula, shoelace algorithm, or shoelace method (also known as Gauss's area formula and the surveyor's formula) is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. It is called the shoelace formula because of the constant cross-multiplying for the coordinates making up the polygon, like threading shoelaces.

https://en.wikipedia.org/wiki/Shoelace_formula

Wikipedia

Chern–Gauss–Bonnet theorem

In mathematics, the Chern theorem (or the Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler–Poincaré characteristic (a topological invariant defined as the alternating sum of the Betti numbers of a topological space) of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial (the Euler class) of its curvature form (an analytical invariant).

It is a highly non-trivial generalization of the classic Gauss–Bonnet theorem (for 2-dimensional manifolds / surfaces) to higher even-dimensional Riemannian manifolds. In 1943, Carl B. Allendoerfer and André Weil proved a special case for extrinsic manifolds. In a classic paper published in 1944, Shiing-Shen Chern proved the theorem in full generality connecting global topology with local geometry.

Riemann–Roch and Atiyah–Singer are other generalizations of the Gauss–Bonnet theorem.

https://en.wikipedia.org/wiki/Chern–Gauss–Bonnet theorem