In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system...

mathematical theorems named after Ferdinand Georg Frobenius. They include: Frobenius theorem (differential topology) in differential geometry and topology for...

eversion Frobenius theorem (differential topology) Distribution (differential geometry) integral curve foliation integrability conditions for differential systems...

endomorphism Frobenius inner product Frobenius norm Frobenius method Frobenius group Frobenius theorem (differential topology) Georg Ludwig Frobenius (1566–1645)...

classes of differential 1-forms, partially generalizing the Frobenius integration theorem. It is named after Jean Gaston Darboux who established it as...

Cauchy–Kovalevskaya theorem Complete vector fields Frobenius theorem (differential topology) Integrability conditions for differential systems Newton's method...

reciprocity Frobenius solution to the hypergeometric equation Frobenius splitting Frobenius theorem (differential topology) Frobenius theorem (real division...

for complete integrability of a Pfaffian system are given by the Frobenius theorem. One version states that if the ideal I {\displaystyle {\mathcal {I}}}...

theorem (symplectic topology) Euler's theorem (differential geometry) Four-vertex theorem (differential geometry) Frobenius theorem (foliations) Gauss's...

foliation on the manifold, whose equivalence is the content of the Frobenius theorem. Contact geometry is in many ways an odd-dimensional counterpart of...

equal to the whole manifold   M {\displaystyle \ M} . Frobenius theorem (differential topology) Jurdjevic, Velimir (1997). Geometric control theory. Cambridge...

In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with...

that L {\displaystyle L} is an involutive distribution. Frobenius theorem (differential topology) – On finding a maximal set of solutions of a system of...

where immersed submanifolds provide the right context to prove the Frobenius theorem. An embedded submanifold (also called a regular submanifold), is an...

Since the Frobenius is not a cdh-covering, the cdh-topology is also a useful replacement for the h-topology in the study of differentials in positive...

Poisson geometry, non-commutative geometry, sub-Riemannian geometry, differential topology. Even though they share the same name, distributions presented in...

_{q}g_{ij}-S_{ij}} Let Fω,g denote the tensor on the right-hand side. The Frobenius theorem states that the above equation is locally solvable if and only if...

the Frobenius norm, Schatten 2-norm, or Hilbert–Schmidt norm of ⁠ M . {\displaystyle \mathbf {M} .} ⁠ Direct calculation shows that the Frobenius norm...

example, the Riemann–Roch theorem (Roch was a student of Riemann) says something about the number of linearly independent differentials (with known conditions...

other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones...

plays an important role in differential geometry and differential topology, for instance in the Frobenius integrability theorem, and is also fundamental...

ϕ ∗ {\displaystyle \phi _{*}} is the differential of ϕ {\displaystyle \phi } at the identity. Lie's third theorem says that every finite-dimensional real...

important. For real-analytic J, the Newlander–Nirenberg theorem follows from the Frobenius theorem; for C∞ (and less smooth) J, analysis is required (with...

homology, and partial differential equations. 3-manifold theory is considered a part of low-dimensional topology or geometric topology. A key idea in the...

injective anchor (hence foliation algebroids) are always integrable (by Frobenius theorem) Lie algebra bundle are always integrable Action Lie algebroids are...

value of any eigenvalue α of Frobenius on a fiber of E as follows. For any integer k, αk is an eigenvalue of Frobenius on a stalk of Ek, which for k...

stretching and bending, like dimension. Glossary of differential geometry and topology Glossary of general topology Glossary of Riemannian and metric geometry...

one associates its Frobenius tensor S 2 P ↦ T M / P {\displaystyle S^{2}P\mapsto TM/P} (since P is odd, the skew-symmetric Frobenius tensor is a symmetric...

Manin, Yu.I. (2002). "Three constructions of Frobenius manifolds: a comparative study". Surveys in Differential Geometry. 7: 497–554. arXiv:math/9801006....

ingredient in characterizing integrable systems is the Frobenius theorem, which states that a system is Frobenius integrable (i.e., is generated by an integrable...