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...

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

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

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...

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...

_{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...

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...

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...

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...

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...

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

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...

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

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

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

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

interior), and in the case of Levi-Civita connections on surfaces, of Gauss–Bonnet theorem. Schild's ladder requires not only geodesics but also relative distance...

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...

non-compact surface to the Euler characteristic. It is akin to the Gauss–Bonnet theorem for a compact surface. A divergent path within a Riemannian manifold...

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...

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

higher-dimensional Riemannian geometry such as the Gauss–Bonnet theorem. According to the Whitney–Graustein theorem, the total curvature is invariant under a regular...

{\displaystyle (-1)^{d}} . For surfaces, these statements follow from the Gauss–Bonnet theorem. For four-dimensional manifolds, this follows from the finiteness...

dimensions using Betti numbers. In the mid nineteenth century, the Gauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature. Investigations...

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...