Recall that, according to Hodge, de-Rham cohomology classes of "nice enough" manifolds have "special" representatives - namely, harmonic forms.

Now, how does one choose a "special" one among embeddings that represent the same homology class? For example, there are numerous embeddings of $S^1$ in $S^1\times S^1$ representing the same homology class, but if the torus is equipped with the usual metric and you require the representative to be a geodesic and to pass through a particular fixed point on the torus then you get uniqueness.

Thus the questions: What would be a good definition of a "special" representative of a homology class that would narrow down the choice among its representatives to just one (for a choice of fixed point perhaps)? What would be the requirements on the manifolds to make that possible?


As requested by Henry, a few more words on the above example. Imagine a 2D torus embedded in 3D as usual. Which oriented curves on the torus would represent the "meridian" homology class? There are too many of them; however, if you impose the "special" requirement that they are geodesics you get exactly 1 passing through each point on the torus. The questions were about the possibility of generalizing that to any homology class, not necessarily $H_1$.

  • $\begingroup$ Could you just raise an example, since I don't quite get it... $\endgroup$ – Henry.L Aug 24 '15 at 20:51
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    $\begingroup$ You could take a mass minimizer in the homology class (at least for closed manifolds) -- this need not be unique but is certainly natural. It would be the unique minimizer (after fixing a point) if it was calibrated. $\endgroup$ – foliations Aug 24 '15 at 21:16

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