Timeline for Subsupermanifolds defined using ideal, transversal example
Current License: CC BY-SA 4.0
22 events
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| Apr 28, 2020 at 13:02 | comment | added | Jan Vysoky | such that $\lambda_{\alpha} \cdot g_{i} \neq 0$ only for $i \in I_{\alpha}$. Also note that $\lambda_{\alpha} \cdot g_{i} \in \mathcal{J}_{N}$. Then $\sum_{i \in I} (g_{i} \circ \psi) h_{i} = \sum_{i \in I} ((\sum_{\alpha=0}^{m}\lambda_{\alpha}) \cdot g_{i} \circ \psi) h_{i} = (\sum_{i \in I} (g_{i} \circ \psi) h_{i}) \cdot (\lambda_{0} \circ \psi)$ $+ \sum_{\alpha=1}^{m} \sum_{i \in I_{\alpha}} (\lambda_{\alpha} \cdot g_{i} \circ \psi) h_{i}$. The first summand is in $\mathcal{I}$ as $\lambda_{0} \in \mathcal{J}_{N}$ and the rest is a finite sum of elements in $\mathcal{I}$. | |
| Apr 28, 2020 at 12:52 | comment | added | Jan Vysoky | Version 2 can be significantly simplified. By the modification of Lemma 1 above, you can assume that $f_{i} = (g_{i} \circ \psi) h_{i}$, where $\{ supp(g_{i}) \}_{i \in I}$ is locally finite. It then suffices to cover $N$ by finitely many precompact sets $\{U_{\alpha} \}_{\alpha=1}^{m}$ (e.g. coordinate balls). Let $U_{0} := M - N$ and let $\{ \lambda_{\alpha} \}_{\alpha = 0}^{m}$ be the corresponding partition of unity. Note that $\lambda_{0} \in \mathcal{J}_{N}$. Now, as $U_{\alpha}$ are compact for $\alpha > 0$, there is a finite subset $I_{\alpha} \subseteq I$ such that | |
| Apr 27, 2020 at 14:46 | history | edited | Pavel | CC BY-SA 4.0 |
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| Apr 27, 2020 at 10:27 | history | edited | Pavel | CC BY-SA 4.0 |
A previous version added
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| Apr 21, 2020 at 13:53 | comment | added | Jan Vysoky | Your answer related to the original formulation of this question (I have edited it significantly). In a summary - you give an answer for the case where all the manifolds are ordinary manifolds. However, in supermanifolds, one does not have a full apparautus of tubular neighborhoods, or the fact that k-dimensional manifold can be covered by k+1 charts. | |
| Apr 15, 2020 at 5:32 | history | edited | Pavel | CC BY-SA 4.0 |
Details
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| Apr 14, 2020 at 21:21 | history | edited | Pavel | CC BY-SA 4.0 |
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| Apr 14, 2020 at 21:14 | history | edited | Pavel | CC BY-SA 4.0 |
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| Apr 14, 2020 at 21:06 | history | edited | Pavel | CC BY-SA 4.0 |
WLOG explained
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| Apr 14, 2020 at 20:59 | history | edited | Pavel | CC BY-SA 4.0 |
WLOG explained
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| Apr 12, 2020 at 9:15 | history | edited | Pavel | CC BY-SA 4.0 |
Clarity improved.
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| Apr 12, 2020 at 9:02 | history | edited | Pavel | CC BY-SA 4.0 |
Typo corrected.
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| Apr 11, 2020 at 18:36 | history | edited | Pavel | CC BY-SA 4.0 |
Minor corrections
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| Apr 11, 2020 at 18:26 | history | edited | Pavel | CC BY-SA 4.0 |
General answer
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| Apr 11, 2020 at 16:16 | history | edited | Pavel | CC BY-SA 4.0 |
General answer
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| Apr 11, 2020 at 15:37 | history | edited | Pavel | CC BY-SA 4.0 |
Proof extended to bigger generality.
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| Apr 7, 2020 at 18:43 | history | edited | Pavel | CC BY-SA 4.0 |
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| Apr 7, 2020 at 18:20 | history | edited | Pavel | CC BY-SA 4.0 |
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| Apr 7, 2020 at 17:58 | history | edited | Pavel | CC BY-SA 4.0 |
Old version replaced by a general proof.
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| Apr 7, 2020 at 17:29 | history | edited | Pavel | CC BY-SA 4.0 |
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| Apr 7, 2020 at 16:18 | history | edited | Pavel | CC BY-SA 4.0 |
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| Apr 7, 2020 at 15:03 | history | answered | Pavel | CC BY-SA 4.0 |