# hyperplane field

# Hyperplane Fields

These are important because of their ties to [[foliations]].

*Example 1 of a Hyperplane Fields:*
Take $M=S^2$, so we have $\RR^2 \to TS^2 \to S^2$.
The tangent spaces are planes, and codimension 1 spaces are lines:


*Example 2 of a Hyperplane Fields:*
Let $M = \RR^3$ so $TM \cong \RR^3$. 
Write $T_pM = \spanof_\RR\theset{\partial x_1, \partial x_2, \partial x_3}$ and $T_p\dual M = \spanof_\RR\theset{dx_1, dx_2, dx_3}$ locally and define $\xi$ by the condition $$dx_3 = 0$$

> Idea: no movement in the $x_3$ direction, constrained to move only in $x_1, x_2$ directions. Assigns a "horizontal" hyperplane to each point in $\RR^3$.

> Remark: This is the kernel of a tangent covector at every point, i.e. a 1-form. This is a prototypical feature.

*Example 3 of a Hyperplane Fields:*
For $M=\RR^3$, write $T\dual \RR^3 = \spanof_\RR\theset{dx, dy, dz}$, and take $$\omega = dz + xdy \in \Omega^1(\RR^3)$$
This is the "standard contact structure" on $\RR^3$, and assigns hyperplanes that look like this:



> Remark:
> Note that this has a more twisted structure, which is what geometrically makes it contact -- no embedded (hyper) surface in $\RR^3$ can have an open subset $U$ such that $\xi$ is tangent to $p$ for every $p\in U$.