## It’s better with ‘theory’

It seems to me that adding the word theory tends to increase my interest in a subject.  Consider puddles, they’re fun…but ‘Puddle Theory’ what’s that?  Sounds intriguing.

Definition: An n-puddle is a n-dimensional manifold with boundary such that the boundary is the union of a sky and ground.  The ground is a n-1 manifold with boundary equal to the n-2 boundary of the sky.  The sky boundary resides within a hyperplane of dimension n-1.

We can use standard geometric structures on $\mathbb{R}^n$ because in general an n-puddle will not be smooth.

Definition: Define a splash to be a family of functions $F_{\alpha}:P\to P$ indexed by $[0,1]$.  $F_{\alpha}$ need not be continuous, in fact will seldom be so.

It’s possible for $P$ to become disconnected during the splash.  Splashes can be divided into volume-preserving and non-volume-preserving splashes.  These are called dry and wet respectably.

Definition: A splash is dry if $\lim_{\alpha\to 1} F_{\alpha}\to id_P$.

It’s possible to compose a series of splashes, $F^1_{\alpha_1}, F^2_{\alpha_2},...F^k_{\alpha_k},...$ $k\in\mathbb{N}$   such that $F_1^k = F_0^{k+1}$.  It was discovered by A. Child that for any such sequence there exists an integer N such that for all $n>N$ $F^n_{\alpha}$ is the empty splash. This is known as Child’s Theorem.  We leave this as an exercise.