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 Σ1⊂Rn1, Σ2⊂Rn2 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.
Fichier principal
TensorFlatChains_revised_dec_2024.pdf (646)
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TensorFlatChains_final.pdf (645)
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