Timeline for Sheaf-theoretically characterize a Riemannian structure?
Current License: CC BY-SA 4.0
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Jan 28, 2019 at 10:05 | comment | added | Liviu Nicolaescu | Given an elliptic partial differential operator $L:C^\infty(E_0)\to C^\infty(E_1)$, $E_0,E_1$ smooth vector bundles over the compact manifold $M$, one can form the sheaf $\ker L$, $$\ker L(U)=\{ s\in C^\infty(E_0|_U):\;\;Ls=0\}$$ one can show that the Euler characteristic of the cohomology of $\ker L$ is equal to the index of $L$. In fact $H^0(M,\ker L)= \ker L(M)$ | |
| Jan 28, 2019 at 5:06 | comment | added | Zhiqiang Sun | Your Paper is very interesting, do you have further works about how we can get more information about the geometry, the topology of a manifold from the Sheaf structure(for example, Cohomology) ? | |
| Jun 5, 2018 at 13:29 | history | edited | Liviu Nicolaescu | CC BY-SA 4.0 |
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| Jun 5, 2018 at 12:18 | comment | added | Liviu Nicolaescu | If $M$ is compact and connected, then the only harmonic functions are constant. The same happens for other noncompact Riemann manifolds. | |
| Jun 5, 2018 at 12:10 | comment | added | Elle Najt | Does it ever suffice to consider only the sections over the open set $M$ (just the global harmonic functions, and not the entire sheaf)? Or maybe there are problems with extension of harmonic functions? (I know that some harmonic functions cannot be extended, but that wouldn't rule out a stronger result as far as I can tell...) | |
| Jun 5, 2018 at 11:51 | history | edited | Liviu Nicolaescu | CC BY-SA 4.0 |
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| Jun 3, 2018 at 12:31 | comment | added | Liviu Nicolaescu | I tried to be cautious in my statement. | |
| Jun 3, 2018 at 11:37 | comment | added | Denis Nardin | Don't harmonic functions form a sheaf? I thought that the vanishing of the Laplacian was a local condition | |
| Jun 3, 2018 at 11:15 | history | answered | Liviu Nicolaescu | CC BY-SA 4.0 |