Timeline for Riemannian metrics preserved by diffeomorphisms
Current License: CC BY-SA 3.0
8 events
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| Aug 29, 2015 at 22:25 | comment | added | Vladimir S Matveev | 3 continued: I think you now see the phenomena: In order to make the set of metrics smaller, the orbit should be dense and should return close to a point with different differentials. I do not think that one can construct a two-dimensional example with one-dimensional set of metrics but I think I know an example, though complicated one, in dimension 4 | |
| Aug 29, 2015 at 22:22 | comment | added | Vladimir S Matveev | 3. Let me give an example, again on the torus, such that the set of the metrics is two-dimensional. It is similar to my example 2, but instead of shifting w.r.t vector $(\alpha_1, \alpha_2)$ take the sliding symmetry with respect to this vector. Since the square of the sliding symmetry is translation, the metric must have constant components in our coordinate system. The condition that it should be preserved by the sliding symmetry is an additional restriction on the metric that kills one from thee freedoms in changing the metric. | |
| Aug 29, 2015 at 22:12 | comment | added | Vladimir S Matveev | 2. Take any orbit and its closure. Denote by $d$ the distance to the orbit (in a background metric for which your mapping is an isometry). The function $d$ is preserved by isometries. Choose any positive function $f$ and conformally change the metric by multiplying it by $f(d)$. You will obtain a metric which is preserved by isometries, and you have a functional freedom, namely a choice of a function $f$. If the function $f$ is such that it is nonconstant only for small values, the resulting metric will be a smooth one. | |
| Aug 29, 2015 at 21:54 | comment | added | Vladimir S Matveev | 1. The ratio should be irrational because otherwise the orbit of any point will be discrete and therefore not dense. Like in my first example. | |
| Aug 28, 2015 at 23:33 | comment | added | Asaf Shachar | Very nice!A few points: 1) Why the ratio $\frac{α_1}{α_2}$ needs to be irrational? 2) In your example the differential of the the diffeomorphism was trivial (the identity, after canonically identifying all the tangent spaces). So preserving the metric amounted to the metric being invariant under translates. In general, even if we are guaranteed that there is no dense orbit, how can we deduce the preserved cone will always be infinitely-dimensional? 3) Ho can we make it one dimensional? Don't we always have freedom to choose the metric on one tangent space?(See my added answer below). | |
| Aug 28, 2015 at 22:06 | vote | accept | Asaf Shachar | ||
| Aug 24, 2015 at 7:47 | history | edited | Jochen Wengenroth | CC BY-SA 3.0 |
improved formatting
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| Aug 23, 2015 at 19:05 | history | answered | Vladimir S Matveev | CC BY-SA 3.0 |