In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers...

now called de Branges's theorem. He claims to have proved several important conjectures in mathematics, including the generalized Riemann hypothesis....

completely traceless part of the Riemann tensor. In d {\displaystyle d} dimensions this is related to the Kretschmann invariant by R a b c d R a b c d = C a...

Cauchy–Riemann equations with a conformal map must itself solve the Cauchy–Riemann equations. Thus the Cauchy–Riemann equations are conformally invariant. Let...

field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the...

In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli...

genus is the same. Therefore, the genus is an important topological invariant of a Riemann surface. On the other hand, Hodge theory shows that the genus coincides...

differential geometry, Riemann's minimal surface is a one-parameter family of minimal surfaces described by Bernhard Riemann in a posthumous paper published...

is invariant under holomorphic changes of coordinate. Indeed, if z = x + iy is holomorphic as a function of w = u + iv, then by the Cauchy–Riemann equations...

Bézout's theorem Brill–Noether theory Genus (mathematics) Riemann surface Riemann–Hurwitz formula Riemann–Roch theorem Abelian integral Differential of the first...

known since Riemann that the algebraic variety condition imposes extra constraints on a complex torus. The following criterion by Riemann decides whether...

The cross-ratio is invariant under the projective transformations of the line. In the case of a complex projective line, or the Riemann sphere, these transformations...

space; for an even dimensional submanifold, this is an invariant only depending on the Riemann metric of the submanifold). Their result would be valid...

Grothendieck's theorem (Fredholm kernel) Grothendieck–Riemann–Roch theorem Grothendieck's Séminaire de géométrie algébrique Grothendieck's six operations...

orientability (a normal invariant, also detected by homology) and genus (a homological invariant). Smooth closed manifolds have no local invariants (other than dimension)...

finite invariant measure on X, but the dynamics of the system is constrained to the level sets of I on X, hence the system possesses invariant sets of...

exactly the bijective conformal maps from the Riemann sphere to itself, i.e., the automorphisms of the Riemann sphere as a complex manifold; alternatively...

diffeomorphism. If a flow on a manifold splits the tangent bundle into three invariant subbundles, with one subbundle that is exponentially contracting, and...

over the Riemann surface M. Then a Higgs bundle ( E , φ ) {\displaystyle (E,\varphi )} is stable if, for each φ {\displaystyle \varphi } invariant subbundle...

theory of physics, a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation. A Lorentz scalar may be generated from...

disk/upper halfplane. The first important invariant of a compact Riemann surface is its genus, a topological invariant given by half the rank of the Abelian...

entitled to do since the Euler characteristic is a topological invariant. What the Riemann–Hurwitz formula does is to add in a correction to allow for ramification...

d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of...

write these operators as E, B, L respectively, the principal invariants of the Riemann tensor are obtained as follows: K 1 / 4 {\displaystyle K_{1}/4}...

Bernhard Riemann. In 1857, Riemann introduced the concept of Riemann surfaces as part of a study of the process of analytic continuation; Riemann surfaces...

In mathematics, a CR manifold, or Cauchy–Riemann manifold, is a differentiable manifold together with a geometric structure modeled on that of a real...

states. In mathematics, it has been used to calculate knot invariants and three-manifold invariants such as the Jones polynomial. Particularly, Chern–Simons...

Bäcklund transform which relates solutions of the same equation is called an invariant Bäcklund transform or auto-Bäcklund transform. If such a transform can...

Simon Donaldson. The definition was inspired by a comparison to geometric invariant theory (GIT) stability. In the special case of Fano varieties, K-stability...

compact, semi-simple Lie groups equipped with a bi-invariant Riemannian metric. Every compact Riemann surface of genus greater than 1 (with its usual metric...