Timeline for Relation between harmonic vector field and harmonic 1-form
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22 events
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| Mar 31, 2020 at 15:26 | history | edited | YCor | CC BY-SA 4.0 |
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| Mar 31, 2020 at 14:53 | history | edited | C.F.G | CC BY-SA 4.0 |
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| Feb 25, 2020 at 21:50 | history | edited | C.F.G | CC BY-SA 4.0 |
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| May 9, 2016 at 18:15 | vote | accept | C.F.G | ||
| May 9, 2016 at 15:18 | answer | added | Robert Bryant | timeline score: 14 | |
| May 7, 2016 at 20:31 | comment | added | Robert Bryant | @C.F.G: Oh. The answer to that is 'no'. For example, if one takes a left-invariant unit vector field $X$ on $M^3=\mathrm{SU}(2)\simeq S^3$ endowed with its bi-invariant metric as a Lie group, then $X:M\to S(M)$ is harmonic as a map between Riemannian manifolds (and as a section of $S(M)$ as well), but $X^\flat$ is not a harmonic $1$-form on $M$, since the only harmonic $1$-form on $M$ is the one that vanishes identically. | |
| May 6, 2016 at 18:45 | comment | added | C.F.G | Many thanks R. Bryant. I still did not get the answer my question. is this true: if $X:M\to S(M)$ is Harmonic in the sense of a mapping between two Riemannian manifolds if and only if $\omega= X^\flat$ is Harmonic as 1-form.? | |
| May 6, 2016 at 9:11 | comment | added | Robert Bryant | I think that the OP is mixing up a couple of different things: The definition of harmonic for a 1-form is standard, but for unit vector fields, there is another notion of 'harmonic': Regard $S(M)$, the unit sphere bundle of $(M,g)$, as a Riemannian manifold in the natural way and then ask whether a unit vector field $X:M\to S(M)$ is harmonic as a mapping between two Riemannian manifolds. There is actually an additional subtlety, in that one can ask that $X$ be a critical point of the energy functional when one only varies $X$ through sections of $S(M)\to M$ (instead of through all maps). | |
| May 6, 2016 at 7:51 | history | edited | C.F.G | CC BY-SA 3.0 |
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| May 3, 2016 at 21:33 | comment | added | Willie Wong | I meant Harmonic in the sense of in the kernel of the Laplace-Beltrami operator, which is the sense given by both of your definitions. | |
| May 3, 2016 at 20:59 | comment | added | Deane Yang | Perhaps you don't mean to have the word "unit" in your definition of a harmonic vector field? | |
| May 3, 2016 at 20:20 | comment | added | Ryan Budney | Harmonic forms are almost never unit length, as far as I am aware. It would appear to me that Definition 1 is a very special definition, largely unrelated to Definition 2. You can of course talk about harmonic vector fields (i.e. dual to harmonic forms) but this results in a different object than your "harmonic unit vector fields". | |
| May 3, 2016 at 20:12 | history | edited | C.F.G | CC BY-SA 3.0 |
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| May 3, 2016 at 19:54 | comment | added | C.F.G | @Willie Wong thanks for your Hint. what is your means of harmonic? Def. 1 or Def. 2? | |
| May 3, 2016 at 19:46 | comment | added | Willie Wong | Note that $0 = \Delta_g(1) = \Delta_g (g(X,X)) = 2 g(X,\Delta_g X) + 2 g(\nabla X, \nabla X)$ you actually have that a unit harmonic vector field on a Riemannian manifold must be parallel. It seems very strange to require that $X$ is unit in the definition. | |
| May 3, 2016 at 19:44 | comment | added | C.F.G | Thanks Ben McKay. But in every reference i can't find this relation. | |
| May 3, 2016 at 19:44 | comment | added | Willie Wong | @BenMcKay: maybe not exactly a reference on Hodge theory; the OP may be slightly confused by Weitzenbock if we bring the form Laplacian into the picture. | |
| May 3, 2016 at 19:35 | comment | added | Ben McKay | The duality from the metric between 1-forms and vector fields shows that your energy is the usual one on 1-forms for which the critical 1-forms are the harmonic ones (see any reference on Hodge theory). But the restriction to unit vector fields is a bit unusual, and doesn't give the same Euler-Lagrange equations, I imagine. | |
| May 3, 2016 at 19:07 | history | edited | C.F.G | CC BY-SA 3.0 |
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| May 3, 2016 at 18:54 | comment | added | Saal Hardali | It would help if you could provide more context. What is "the energy function" here? | |
| May 3, 2016 at 18:53 | review | First posts | |||
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| May 3, 2016 at 18:51 | history | asked | C.F.G | CC BY-SA 3.0 |