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...

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

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

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...

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

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...

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

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...

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

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...

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...

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

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...

(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...

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...

manifold. Kähler metrics remain Kähler under Ricci flow, and so Ricci flow has also been studied in this setting, where it is called Kähler–Ricci flow...

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

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

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...

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

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

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...

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}...

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

seminars of major importance: the Séminaire Cartan–Chevalley of the academic year 1955-6, with Henri Cartan and the Séminaire Chevalley of 1956-7 and 1957-8...

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...

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...