The usual story goes like this:

Smooth picture (?):For a smooth bijection $\phi: M \to N$ between $n$-manifolds the following is true:

$\phi^{-1}$ is a local diffeomorphism a.e.

Given an open set $U \subset N$ and a form $\omega \in \Omega^k(U)$ we have the equality: $\int_U \omega= \int_{\phi^{-1}(U)} \phi^*\omega$.

Satisfied with this simple and very intuitive picture I slowly came to believe that this is the most general change of variables theorem there could be. That is, until I met the following theorem in a measure theoretic context.

Theorem 1:Let $U\subset\mathbb{R}^n$ be a measurable subset and $\phi :U\to \mathbb{R}^n$ be injective and differentiable.

$\implies$ $\int_{\phi(U)} f = \int_U f\circ\phi |\det D\phi|$ for all real valued functions $f$.

Then upon searching the internet I came by a weaker version of the inverse function theorem for everywhere differentiable functions:

Theorem 2:Let $U \subset \mathbb{R}^n$ and $f:U \to \mathbb{R}^n$ be differentiable s.t. $Df_{x_0}$ has full rank for all $x_0 \in U$

$\implies$ $f$ is a local differentiable homeomorphism.

Which leads to the following generalization:

Differentiable picture (conjecture):For a differentiable bijection $\phi: M \to N$ between $n$-manifolds the following is true:

$\phi^{-1}$ is a local differentiable homeomorphism a.e.

Given an open set $U \subset N$ and a form $\omega \in \Omega^k(U)$ we have the equality: $\int_U \omega= \int_{\phi^{-1}(U)} \phi^*\omega$.

In search of a unifying picture i listed the properties a function $\phi$ must have so that the pullback wont change the value of the integral.

**I.** $\phi$ must be locally absolutely continuous. (otherwise it could send a null set to a positive measure set). This also establishes the almost everywhere differentiability of $\phi$.

**II.** $D \phi$ must have full rank almost everywhere.

If we can preform change of variables with $\phi$. What more properties must it have?

Following the connection with lipschitz functions i arrived at the following unifying conjecture:

The great conjecture:For a locally lipschitz bijection $\phi: M \to N$ between $n$-manifolds the following is true:

$\phi^{-1} $ is locally bi-lipschitz (a.e.?)

Given an open set $U \subset N$ and a form $\omega \in \Omega^k(U)$ we have the equality: $\int_U \omega= \int_{\phi^{-1}(U)} \phi^*\omega$.

Is this true?