Article Dans Une Revue Transactions of the American Mathematical Society Année : 2025

Tensor rectifiable G-flat chains

Résumé

A rigidity result for normal rectifiable k-chains in Rn with coefficients in an Abelian normed group is established. Given some decompositions k=k1+k2, n=n1+n2 and some rectifiable k-chain A in Rn, we consider the properties: (1) The tangent planes to μA split as TxμA=L1(x)×L2(x) for some k1-plane L1(x)Rn1 and some k2-plane L2(x)Rn2. (2) A=A|Σ1×Σ2 for some sets Σ1Rn1, Σ2Rn2 such that Σ1 is k1-rectifiable and Σ2 is k2-rectifiable (we say that A is (k1,k2)-rectifiable). The main result is that for normal chains, (1) implies (2), the converse is immediate. In the proof we introduce the new groups of tensor flat chains (or (k1,k2)-chains) in Rn1×Rn2 which generalize Fleming's G-flat chains. The other main tool is White's rectifiable slices theorem. We show that on the one hand any normal rectifiable chain satisfying~(1) identifies with a normal rectifiable (k1,k2)-chain and that on the other hand any normal rectifiable (k1,k2)-chain is (k1,k2)-rectifiable.
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Dates et versions

hal-03890966 , version 1 (08-12-2022)
hal-03890966 , version 2 (20-01-2025)

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Michael Goldman, Benoît Merlet. Tensor rectifiable G-flat chains. Transactions of the American Mathematical Society, In press, ⟨10.1090/tran/9392⟩. ⟨hal-03890966v2⟩
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