Tensor rectifiable $G$-flat chains

Abstract:

We prove a rigidity result for $k$-rectifiable sets $\Sigma$ in $\mathbb {R}^n$ (that is, up to an ${\mathcal {H}}^k$-negligible set, $\Sigma$ is covered by a countable union of $k$-manifolds of class $C^1$). Given some decompositions $k=k_1+k_2$, $n=n_1+n_2$, we consider the following properties.

1. For ${\mathcal {H}}^k$-almost every $x\in \Sigma$, the tangent plane to $\Sigma$ splits as $T_x\Sigma =L^{1}(x)\times L^{2}(x)$ for some $k_1$-plane $L^1(x)\subset \mathbb {R}^{n_1}$ and some $k_2$-plane $L^2(x)\subset \mathbb {R}^{n_2}$.
2. Up to an ${\mathcal {H}}^k$-negligible set, $\Sigma \subset \Sigma ^1\times \Sigma ^2$ for some sets $\Sigma ^1\subset \mathbb {R}^{n_1}$, $\Sigma ^2\subset \mathbb {R}^{n_2}$ such that $\Sigma ^1$ is $k_1$-rectifiable and $\Sigma ^2$ is $k_2$-rectifiable (we say that $\Sigma$ is $(k_1,k_2)$-rectifiable).

We always have $(2) \!\!\!\implies \!\!\! (1)$. We establish a partial converse: if $A=A{\text {\huge$\llcorner$}}\Sigma$ for some normal rectifiable $G$-flat $k$-chain $A$, then $(1) \!\!\implies \!\! (2)$ in the sense that $A=A{\text {\huge$\llcorner$}} \Sigma ^1\times \Sigma ^2$ with $\Sigma ^1$, $\Sigma ^2$ as in (2).

In the proof we introduce the new groups of tensor flat chains (or $(k_1,k_2)$-chains) in $\mathbb {R}^{n_1}\times \mathbb {R}^{n_2}$ generalizing Fleming’s $G$-flat chains. The other main tool is White’s rectifiable slices theorem.