In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature...

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...

Gauss–Bonnet gravity, also referred to as Einstein–Gauss–Bonnet gravity, is a modification of the Einstein–Hilbert action to include the Gauss–Bonnet...

The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial...

aspects such as the Gauss–Bonnet theorem, the uniformization theorem, the von Mangoldt-Hadamard theorem, and the embeddability theorem. There are other important...

topological data). It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications...

hyperbolic geometry Gauss–Bonnet theorem, a theorem about curvature in differential geometry for 2d surfaces Chern–Gauss–Bonnet theorem in differential geometry...

to the differential geometry of surfaces, including the Gauss–Bonnet theorem. Pierre Bonnet attended the Collège in Montpellier. In 1838 he entered the...

_{T}K\,dA.} A more general result is the Gauss–Bonnet theorem. Gauss's Theorema egregium (Latin: "remarkable theorem") states that Gaussian curvature of a...

geodesics. In particular, Gauss proves the local Gauss–Bonnet theorem on geodesic triangles, and generalizes Legendre's theorem on spherical triangles to...

the Gauss–Bonnet theorem for the two-dimensional case and the generalized Gauss–Bonnet theorem for the general case. A discrete analog of the Gauss–Bonnet...

du\,dv=\iint _{S}K\ dA} The Gauss–Bonnet theorem links total curvature of a surface to its topological properties. The Gauss map reflects many properties...

billionaire hedge fund manager. Chern's work, most notably the Chern-Gauss-Bonnet Theorem, Chern–Simons theory, and Chern classes, are still highly influential...

theorem of curves Gauss–Bonnet theorem for an elementary application of curvature Gauss map for more geometric properties of Gauss curvature Gauss's principle...

vertex resembles a local maximum or minimum (positive curvature). The Gauss–Bonnet theorem gives the total curvature as 2 π {\displaystyle 2\pi } times the...

determined up to a rigid motion of R3. Bonnet's theorem is a corollary of the Frobenius theorem, upon viewing the Gauss–Codazzi equations as a system of first-order...

This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem. Nash embedding theorems. They...

doi:10.2307/1970563. JSTOR 1970563. 1965: Ono, Takashi (1965). "The Gauss-Bonnet theorem and the Tamagawa number". Bulletin of the American Mathematical Society...

Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was...

at each of its eight vertices. Descartes' theorem on total angular defect (a form of the Gauss–Bonnet theorem) states that the sum of the angular defects...

0^{+}}12{\frac {\pi r^{2}-A(r)}{\pi r^{4}}}.} The theorem is closely related to the Gauss–Bonnet theorem. Berger, Marcel (2004), A Panoramic View of Riemannian...

Osculating circle Curve Fenchel's theorem Theorema egregium Gauss–Bonnet theorem First fundamental form Second fundamental form Gauss–Codazzi–Mainardi equations...

the Gauss–Bonnet theorem then provides a logical contradiction to the negativity of mass. As such, they were able to prove the positive mass theorem in...

highly abstract theorems from geometry to be used to gain insight, ranging from the Chern–Gauss–Bonnet theorem and the Riemann–Roch theorem to the Atiyah–Singer...

\mathbb {R} ^{3}} given as: The Gauss–Bonnet theorem relates the topology of a surface and its geometry. The Gauss–Bonnet theorem— For each bounded surface...

(riemannian geometry) Gauss's Theorema Egregium (differential geometry) Gauss–Bonnet theorem (differential geometry) Geroch's splitting theorem (differential...

genus is at least 1 {\displaystyle 1} . The Uniformization theorem and the Gauss–Bonnet theorem can both be applied to orientable Riemann surfaces with boundary...

triangle's angular defect is understood as a special case of the Gauss-Bonnet theorem where the curvature of a closed curve is not a function, but a measure...

non-rotating black hole must be spherical. Because the proof uses the Gauss–Bonnet theorem, it does not generalize to higher dimensions. The discovery of black...

preserved by general diffeomorphisms of the surface. However, the famous Gauss–Bonnet theorem for closed surfaces states that the integral of the Gaussian curvature...