| Title: |
Tensor rectifiable G-flat chains |
| Authors: |
Goldman, Michael; Merlet, Benoît |
| Contributors: |
Centre de Mathématiques Appliquées de l'Ecole polytechnique (CMAP); Institut National de Recherche en Informatique et en Automatique (Inria)-École polytechnique (X); Institut Polytechnique de Paris (IP Paris)-Institut Polytechnique de Paris (IP Paris)-Centre National de la Recherche Scientifique (CNRS); Reliable numerical approximations of dissipative systems (RAPSODI); Laboratoire Paul Painlevé - UMR 8524 (LPP); Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centre Inria de l'Université de Lille; Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria); Université de Lille-Centre National de la Recherche Scientifique (CNRS); INRIA RAPSODI; ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018); ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011) |
| Source: |
ISSN: 0002-9947. |
| Publisher Information: |
CCSD; American Mathematical Society |
| Publication Year: |
2025 |
| Collection: |
LillOA (HAL Lille Open Archive, Université de Lille) |
| Subject Terms: |
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]; [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG] |
| Description: |
International audience ; A rigidity result for normal rectifiable $k$-chains in $\mathbb{R}^n$ with coefficients in an Abelian normed group is established. Given some decompositions $k=k_1+k_2$, $n=n_1+n_2$ and some rectifiable $k$-chain $A$ in $\mathbb{R}^n$, we consider the properties:(1) The tangent planes to $\mu_A$ split as $T_x\mu_A=L^1(x)\times L^2(x)$ 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) $A=A_{\vert\Sigma^1\times\Sigma^2}$ 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 $A$ is $(k_1,k_2)$-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 $(k_1,k_2)$-chains) in $\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$ 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 $(k_1,k_2)$-chain and that on the other hand any normal rectifiable $(k_1,k_2)$-chain is $(k_1,k_2)$-rectifiable. |
| Document Type: |
article in journal/newspaper |
| Language: |
English |
| Relation: |
info:eu-repo/semantics/altIdentifier/arxiv/2212.04753; ARXIV: 2212.04753 |
| DOI: |
10.1090/tran/9392 |
| Availability: |
https://hal.science/hal-03890966; https://hal.science/hal-03890966v2/document; https://hal.science/hal-03890966v2/file/TensorFlatChains_revised_dec_2024.pdf; https://doi.org/10.1090/tran/9392 |
| Rights: |
info:eu-repo/semantics/OpenAccess |
| Accession Number: |
edsbas.B53191BE |
| Database: |
BASE |