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Concept
Frobenius theorem (differential topology)
Summary
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields. The theorem generalizes the existence theorem for ordinary differential equations, which guarantees that a single vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of r vector fields mesh into coordinate grids on r-dimensional integral manifolds. The theorem is foundational in differential topology and calculus on manifolds.
Contact geometry studies 1-forms that maximally violates the assumptions of Frobenius' theorem. An example is shown on the right.
Suppose we are to find the trajectory of a particle in a subset of 3D space, but we do not know its trajectory formula. Instead, we know only that its trajectory satisfies , where are smooth functions of . Then, we can know only for sure that, if at some moment in time, the particle is at location , then its velocity at that moment is restricted within the plane with equation
In other words, we can draw a "local plane" at each point in 3D space, and we know that the particle's trajectory must be tangent to the local plane at all times.
If have two equationsthen we can draw two local planes at each point, and their intersection is generically a line, allowing us to uniquely solve for the curve starting at any point. In other words, with two 1-forms, we can foliate the domain into curves.
If we have only one equation , then we might be able to foliate into surfaces, in which case, we can be sure that a curve starting at a certain surface must be restricted to wander within that surface. If not, then a curve starting at any point might end up at any other point in .
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