In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions...

philosophical papers. As a mathematician Kähler is known for a number of contributions: the Cartan–Kähler theorem on solutions of non-linear analytic differential...

Kuranishi provided a proof of Cartan's conjecture. This theorem is used in infinite-dimensional Lie theory. Cartan-Kähler theorem Bryant, Robert L.; Chern...

groups Hypercomplex numbers, division algebras Systems of PDEs, Cartan–Kähler theorem Theory of equivalence Integrable systems, theory of prolongation...

coframes must be real analytic in order for this to hold, because the Cartan-Kähler theorem requires analyticity.) Prolongation. This is the most intricate...

Cartan–Kähler theorem (partial differential equations) Cartan–Kuranishi prolongation theorem (partial differential equations) Cauchy–Kowalevski theorem (partial...

decomposition Cartan-Iwasawa-Malcev theorem Cartan–Kähler theorem Cartan–Karlhede algorithm Cartan–Weyl theory Cartan–Weyl basis Cartan–Killing form Cartan–Kuranishi...

results such as Darboux's theorem and the Cartan-Kähler theorem. Despite being named for Ferdinand Georg Frobenius, the theorem was first proven by Alfred...

proposal for finding Kähler metrics of constant scalar curvature.[C82a] More broadly, Calabi introduced the notion of an extremal Kähler metric, and established...

these are the Cartan–Kähler theorem, which only works for real analytic differential systems, and the Cartan–Kuranishi prolongation theorem. See § Further...

In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can...

Ambrose–Singer theorem. The study of Riemannian holonomy has led to a number of important developments. Holonomy was introduced by Élie Cartan (1926) in order...

methods of differential geometry. For instance, we can apply the Cartan–Kähler_theorem to a system of partial differential equations by writing down the...

This relation is called the Cartan–Thullen theorem. See Oka's lemma Oka's proof uses Oka pseudoconvex instead of Cartan pseudoconvex. There are some...

Carpenter Cartan–Kähler theorem – Élie Cartan, Erich Kähler Casimir effect – Hendrik Casimir Catalan's conjecture (a.k.a. Mihăilescu's theorem), Catalan...

(J,g)} is called a Kähler structure, and a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex...

theorem for the cohomology of line bundles on compact Kähler manifolds, and Cartan's theorems A and B for the cohomology of coherent sheaves on affine...

On a Kähler manifold, the (p, q) components of a harmonic form are again harmonic. Therefore, for any compact Kähler manifold X, the Hodge theorem gives...

corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into...

Morse index, the Rauch comparison theorems, and the Cartan–Hadamard theorem. Then it ascends to complex manifolds, Kähler manifolds, homogeneous spaces,...

Geometry Worldsheet Kaluza–Klein theory Compactification Why 10 dimensions? Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3...

However, Kähler manifolds already possess holonomy in U ( n ) {\displaystyle U(n)} , and so the (restricted) holonomy of a Ricci-flat Kähler manifold...

following sequence of results. Complex analysis was revolutionized by Cartan's theorems A and B in 1953. These results say that if F {\displaystyle {\mathcal...

Riemannian metric, is called quaternion-Kähler symmetric space. An irreducible symmetric space G / K is quaternion-Kähler if and only if isotropy representation...

this work was later perfected by Élie Cartan. The final classification is often referred to as Killing-Cartan classification. Unfortunately, there is...

explicitly defined only in 1913 in a book by Hermann Weyl. Élie Cartan introduced the Cartan connection, one of the first concepts of a connection. Levi-Civita...

webs then on the Cartan-Kähler theory and invariant theory. He would often eat lunch and chat in German with fellow colleague Erich Kähler. He had a three-year...

which we now call g 2 {\displaystyle {\mathfrak {g}}_{2}} . In 1893, Élie Cartan published a note describing an open set in C 5 {\displaystyle \mathbb {C}...

Gauss–Bonnet theorem Hopf–Rinow theorem Cartan–Hadamard theorem Myers theorem Rauch comparison theorem Morse index theorem Synge theorem Weinstein theorem Toponogov...

{\displaystyle k=2n+2.} Calabi–Yau manifolds admit an Einstein metric that is also Kähler, with Einstein constant k = 0 {\displaystyle k=0} . Such metrics are not...