In mathematics, the Frobenius theorem states, in its smooth version of degree 1, the following:
Let U be an open set in Rn and F a submodule of Ω1(U) of constant rank r in U. Then F is integrable if and only if for every p ∈ U the stalk Fp is generated by rexact differential forms.
Geometrically, it states that an integrable module of 1-forms of rank r is the same thing as a codimension-r foliation. It is one of the basic tools for the study of vector fields and foliations.
The statement remains true for analytic 1-forms and in the holomorphic case, with C instead of R.
It can be generalized to differential forms of higher degree and, in some instances, to the singular case.
There is also a statement in terms of vector fields, which makes the sufficient condition for the existence of a submanifold of U of codimension r, tangent to vector fields
of any two of the given fields should lie in the space spanned by them. Since the Lie bracket can equally be taken on the submanifold, this condition is certainly necessary.
The relationship between the two aspects is because the Lie bracket and exterior derivative are connected.
Ferdinand Georg Frobenius has several theorems with his name in them, more than one of which have only his name.
Frobeniustheorem in differential geometry and topology for integrable subbundles;
Frobenius reciprocity theorem in group representation theory describing the reciprocity relation between restricted and induced representations on a subgroup;
In mathematics, the Perron–Frobenius theorem, named after Oskar Perron and Ferdinand Georg Frobenius, is a theorem in matrix theory about the eigenvalues and eigenvectors of a real positive n×n matrix:
This result has a natural interpretation in the theory of finite Markov chains (where it is the matrix-theoretic equivalent of the convergence of a finite Markov chain, formulated in terms of the transition matrix of the chain; see, for example, the article on the subshift of finite type).
The Perron–Frobenius theorem can be further generalized to the class of block-indecomposable non-negative matrices (called "irreducible" in reference [1] below, also called regular in the stochastic case).
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