TENSOR RECTIFIABLE G-FLAT CHAINS

We prove a rigidity result for k-rectifiable sets Σ in ℝⁿ (that is, up to an H^k-negligible set, Σ is covered by a countable union of k-manifolds of class C¹). Given some decompositions k = k₁ + k₂, n = n₁ + n₂, we consider the following properties. (1) For H^k-almost every x ∊ Σ, the tangent plane to Σ splits as TxΣ = L¹(x)\timesL²(x) for some k₁-plane L¹(x) ⊂ ℝⁿ¹ and some k₂-plane L²(x) ⊂ ℝⁿ². (2) Up to an H^k-negligible set, Σ ⊂ Σ¹ \times Σ² for some sets Σ¹ ⊂ ℝⁿ¹, Σ² ⊂ ℝⁿ² such that Σ¹ is k₁-rectifiable and Σ² is k₂-rectifiable (we say that Σ is (k₁, k₂)-rectifiable). We always have (2)→(1). We establish a partial converse: if A = A⌊Σ for some normal rectifiable G-flat k-chain A, then (1) → (2) in the sense that A = A⌊Σ¹ × Σ² with Σ¹, Σ² as in (2). In the proof we introduce the new groups of tensor flat chains (or (k₁, k₂)chains) in ℝⁿ¹ × ℝⁿ² generalizing Fleming's G-flat chains. The other main tool is White's rectifiable slices theorem.