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proved by Samuil Shefel (1979) and rediscovered by Dennis DeTurck and Jerry Kazdan (1981)
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Anton Petrunin
Anton Petrunin
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  1. NO. Given a Riemannian manifold, it might be possible to improve smoothness by changing atlas. It is proved by Shefel, that theThe atlas with harmonic functions as coordinates is the best [proved by Samuil Shefel (1979) and rediscovered by Dennis DeTurck and Jerry Kazdan (1981)]. But, the obtained metric might be worse than $C^\infty$.

  2. There is no local-global issue here, harmonic atlas is defined locally and it is the best one globally. So you get problems starting with dimension 2.

  1. NO. Given a Riemannian manifold, it might be possible to improve smoothness by changing atlas. It is proved by Shefel, that the atlas with harmonic functions as coordinates is the best. But, the obtained metric might be worse than $C^\infty$.

  2. There is no local-global issue here, harmonic atlas is defined locally and it is the best one globally. So you get problems starting with dimension 2.

  1. NO. Given a Riemannian manifold, it might be possible to improve smoothness by changing atlas. The atlas with harmonic functions as coordinates is the best [proved by Samuil Shefel (1979) and rediscovered by Dennis DeTurck and Jerry Kazdan (1981)]. But, the obtained metric might be worse than $C^\infty$.

  2. There is no local-global issue here, harmonic atlas is defined locally and it is the best one globally. So you get problems starting with dimension 2.

deleted 24 characters in body
Source Link
Anton Petrunin
Anton Petrunin
  • 45k
  • 14
  • 135
  • 299

  1. NO. Given a Riemannian manifold, it might be possible to improve smoothness by changing atlas. It is proved by Shefel, that the atlas with harmonic functions as coordinates is the best. But, the obtained metric might be worse than $C^\infty$.

  2. It not true even in dimension 2. In this caseThere is no local-global issue here, harmonic atlas is defined locally and it is the best atlas use isotropic coordinatesone globally. So the metric is described by a function; if this function is not smooth then no hopeyou get problems starting with dimension 2.

  1. NO. Given a Riemannian manifold, it might be possible to improve smoothness by changing atlas. It is proved by Shefel, that the atlas with harmonic functions as coordinates is the best. But, the obtained metric might be worse than $C^\infty$

  2. It not true even in dimension 2. In this case, the best atlas use isotropic coordinates. So the metric is described by a function; if this function is not smooth then no hope.

  1. NO. Given a Riemannian manifold, it might be possible to improve smoothness by changing atlas. It is proved by Shefel, that the atlas with harmonic functions as coordinates is the best. But, the obtained metric might be worse than $C^\infty$.

  2. There is no local-global issue here, harmonic atlas is defined locally and it is the best one globally. So you get problems starting with dimension 2.

Source Link
Anton Petrunin
Anton Petrunin
  • 45k
  • 14
  • 135
  • 299

  1. NO. Given a Riemannian manifold, it might be possible to improve smoothness by changing atlas. It is proved by Shefel, that the atlas with harmonic functions as coordinates is the best. But, the obtained metric might be worse than $C^\infty$

  2. It not true even in dimension 2. In this case, the best atlas use isotropic coordinates. So the metric is described by a function; if this function is not smooth then no hope.