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I'm posting this not really for information, but as I'm trying to get my own head around it.  It's a theorem in differential geometry, giving conditions for the existence of integral submanifolds of a given space, given a 'distribution' of that space.

So what the hell is all that about?  Start with a 2-dimensional space (eg a sheet of paper) which has a vector field defined on it, such that each point (on the paper) has a little arrow defined at it.  A bit like a weather map with wind speed drawn on it.  It's pretty intuitive that you can join up those arrows with smooth curves, such that at each point there is a single curve that goes through it, and the curves fill the whole space (page).  This is demonstrably true as long as the vector is never zero.

Frobenius' theorem takes that up to an arbitrary dimension.  Given any space, a 'distribution' defines a 'plane' at each point - a plane here can be of arbitrary dimension, not just 2.  Frobenius' theorem gives the conditions that mean that the distribution define a 'foliation' of the space, such that the defined planes join to form a surface (or hyper-surface) through each point.  In 3D this is understandable - at each point in a 3D space a 2D plane can be defined, and these planes may be tangential to a 2D curved surface in the space, such that the whole space is filled by these curved 'sheets' - this is what is meant by a foliation.

How do you define the plane at each point?  There are 3-ways.  You can define a set of basis vectors at each point (in a 3D space defining a 2D plane would mean giving 2 vectors).  You can define a 'characterising (n-m) form' or you can define a set of basis constraint 1-forms.  What the hell are 'forms'?  basically they are formulae that map vectors to the real numbers.  A 1-form takes a single vector as an argument, a 2-form takes 2 vectors etc.  'Forms' form a vector space themselves, and in the normal 3D Euclidean space we're all familiar with a 1-form is just a vector, and it defines a plane via the set of vectors it has a zero dot product with.

So, given a space of arbitrary dimension, and a rule for defining a characterising form at each point, Frobenius' theorem tells us whether or not the distribution gives a foliation.  If the form at each point is then Frobenius theorem states that .  This gives necessary and sufficient conditions for the foliation to exist.

Why is this important?  Don't know yet, but it is important I think with regard to the solution of sets of differential equations.  I think I'm getting it though!