Frobenius theorem (differential topology)
theorem that a distribution is integrable iff it arises from a regular foliation
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 violate the assumptions of Frobenius' theorem. An example is shown on the right.
Introduction
One-form version
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
a
d
x
+
b
d
y
+
c
d
z
=
0
{\displaystyle adx+bdy+cdz=0}
, where
a
,
b
,
c
{\displaystyle a,b,c}
are smooth functions of
(
x
,
y
,
z
)
{\displaystyle (x,y,z)}
. Thus, our only certainty is that if at some moment in time the particle is at location
(
x
0
,
y
0
,
z
0
)
{\displaystyle (x_{0},y_{0},z_{0})}
, then its velocity at that moment is restricted within the plane with equation
a
(
x
0
,
y
0
,
z
0
)
[
x
−
x
0
]
+
b
(
x
0
,
y
0
,
z
0
)
[
y
−
y
0
]
+
c
(
x
0
,
y
0
,
z
0
)
[
z
−
z
0
]
=
0
{\displaystyle a(x_{0},y_{0},z_{0})[x-x_{0}]+b(x_{0},y_{0},z_{0})[y-y_{0}]+c(x_{0},y_{0},z_{0})[z-z_{0}]=0}
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 we have two equations
{
a
d
x
+
b
d
y
+
c
d
z
=
0
a
′
d
x
+
b
′
d
y
+
c
′
d
z
=
0
{\displaystyle {\begin{cases}adx+bdy+cdz=0\\a'dx+b'dy+c'dz=0\end{cases}}}
then 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.
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