Manifolds in GMT settings

In this post, I will demonstrate how we translate the concept of Differential Manifolds in the measure-theoretical setting and have partially some similar features $$\\$$

Differentiable Manifolds

$$\\$$ A set $M$ is called a Differentiable Manifold if there exist injective maps $\pi_{i} : U_{i} \subseteq R^{n} \to M$ satisfying the following $$\\$$ $$\bigcup_{i} \pi_{i} (U_{i}) = M$$ $\\$ $\\$ If $\pi_{i} (U_{i}) \cap \pi_{j} (U_{j}) =W \ne \phi $ then $\pi^{-1}_{i}(W)$ and $\pi^{-1}_{j}(W)$ are both open and $\pi^{-1}_{j} \pi_{i}: \pi^{-1}_{i}(W) \to \pi^{-1}_{j}(W)$ and $\pi^{-1}_{i} \pi_{j}: \pi^{-1}_{j}(W) \to \pi^{-1}_{i}(W)$ are both differentiable $$\\$$

Tangent Spaces

Defining differentiability on a Manifold is simply by using the parametrization defined on it to get the regular definition of differentiability on the Euclidean space $$\\$$ Defining the Tangent space at some point $p$ to be the set of all Tangent vectors at that point, and by simple Calculus knowledge you can expect to use the concept of Derivates. $$\\$$ A Tangent Vector at a point $p \in M$ is the Tangent Vector at $0$ of some curve $\alpha:(-\epsilon,\epsilon) \to M$ such that $\alpha(0)=p$. We write the Tangent Vector of $\alpha$ at $p$ as $\alpha'(0)$ such that the differentiability here is defined With the idea we introduced above. $$\\$$ We write the Tangent Vector $\alpha'(0) = \sum_{i} x_{i} {(\frac{\partial}{\partial x_{i}})}_{0}$

Rectifiable Sets

$$\\$$ The function $f:E \subseteq R^{n} \to R^{m}$ is said to be $\textbf{Lipschitz}$ on $E$ if and only if it satisfies the following inequality: $$ |f(y)-f(x)| \leq C|y-x| $$ For some constant $C \in R^{+}$ for all $x,y \in E$ $\\$ The infimum of the set of constants $C$ is the $\textbf{Lipschitz constant}$ and denoted by $\mathbf{lip(f;E)}$ $$\\$$ If $E=R^{n}$ we simply write $\mathbf{lip(f)}$ for the $\textbf{Lipschitz constant}$ $$\\$$

Rademacher's Theorem

$$\\$$ If $f:R^{n} \to R^{m}$ is a Lipschitz function, it is differentiable $\mathfrak{L}^{n}$-almost everywhere. $$\\$$ With this theorem, we can see that Lipschitz functions can replace differentiable functions in measure-theoretical view. $$\\$$

Definition of Rectifiable Sets

$$\\$$ $M \subset R^{n}$ is called $H^{m}-\textbf{ Rectifiable Set}$ if $H^{m}(M) < \infty$ and there exist Countably $\textbf{Lipschitz maps} \pi_{i}:R^{m} \to R^{n}$ such that $H^{m}(M-\bigcup_{i \in N} \pi_{i}(R^{k}) )=0$ $$\\$$ We can see that that the manifold we defined above is a $H^{n}$-rectifiable set if we only add that $H^{n}(M) < \infty$

(Approximate) Tangent Spaces to Rectifiable sets

let $C_{c}^{0}(R^{m})$ be the set of compactly supported Continoues functions on $R^{n}$ and $M$ be our rectifiable set $$\\$$ The m-dimensional subspace $T_{x}M$ that satisfies the following $\forall \alpha \in C_{c}^{0}(R^{m})$ $$\lim_{r \to 0^{+}} \frac{1}{r^{m}} \int_{M} \alpha (\frac{y-x}{r})dH^{m}(y) = \int_{T_{x}M} \alpha dH^{m}$$ $$\\$$ is the approximate tangent space. $$\\$$ The approximate tangent space for a manifold is the same as the tangent space defined above.