论文标题
张量可调g-flat链
Tensor rectifiable G-flat chains
论文作者
论文摘要
建立了$ \ mathbb {r}^n $中正常可纠正的$ k $ chains的刚度结果，并在Abelian Normed组中具有系数。给定一些分解$ k = k_1+k_2 $，$ n = n_1+n_2 $和某些可重新配置的$ k $ - 链$ chain $ a $ in $ \ mathbb {r}^n $，我们考虑这些属性：（1）切好的平面至$μ_a$ split $μ_a$ split $μ_a$ y $t_xμ_________________1$ k_1（$ k_1） $ l^1（x）\ subset \ mathbb {r}^{n_1} $和一些$ k_2 $ -plane $ l^2（x）\ subset \ subset \ mathbb {r}^{n_2} $。 $Σ^1\subset\mathbb{R}^{n_1}$, $Σ^2\subset\mathbb{R}^{n_2}$ such that $Σ^1$ is $k_1$-rectifiable and $Σ^2$ is $k_2$-rectifiable (we say that $A$ is $（k_1，k_2）$ - 可总结）。主要结果是对于普通链，（1）表示（2），相反的是立即的。在证明中，我们介绍了$ \ Mathbb {r}^{n_1} \ times \ times \ athbb {r}^{n_2} $，以$ \ mathbb {r}^{r}^{n_2} $介绍新的张量平坦链（或$（k_1，k_2）$ - 链）。另一个主要工具是White的可纠正切片定理。我们表明，一方面，任何正常的可重新配置链令人满意〜（1）都以普通的$（k_1，k_2）$ - 链条识别，另一方面，又有任何普通的$（k_1，k_2）$ - 链为$（k_1，k_1，k_1，k_2）$ - 可总结。
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 $μ_A$ split as $T_xμ_A=L^1(x)\times L^2(x)$ 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) $A=A_{\vertΣ^1\timesΣ^2}$ for some sets $Σ^1\subset\mathbb{R}^{n_1}$, $Σ^2\subset\mathbb{R}^{n_2}$ such that $Σ^1$ is $k_1$-rectifiable and $Σ^2$ 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.
