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Ali Taghavi
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Edit: According to comment of Andre HerniquesHenriques we revise the question: In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that this possible counter example comes from a Riemannian structure not merely from a metric space view point.

Is there an example of an analytic manifold $M$ with two real analytic Riemanian metrics $g_1,g_2$ such that $M$ has a compact subset which is a fractal set with respect to $g_1$ but is not a fractal set with respect to $g_2$?

Edit: According to comment of Andre Herniques we revise the question: In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that this possible counter example comes from a Riemannian structure not merely from a metric space view point.

Is there an example of an analytic manifold $M$ with two real analytic Riemanian metrics $g_1,g_2$ such that $M$ has a compact subset which is a fractal set with respect to $g_1$ but is not a fractal set with respect to $g_2$?

Edit: According to comment of Andre Henriques we revise the question: In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that this possible counter example comes from a Riemannian structure not merely from a metric space view point.

Is there an example of an analytic manifold $M$ with two real analytic Riemanian metrics $g_1,g_2$ such that $M$ has a compact subset which is a fractal set with respect to $g_1$ but is not a fractal set with respect to $g_2$?

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Source Link
Ali Taghavi
  • 356
  • 8
  • 31
  • 123

Edit: According to comment of Andre Herniques we revise the question: In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that this possible counter example comes from a Riemannian structure not merely from a metric space view point.

Is there an example of an analytic manifold $M$ with two real analytic Riemanian metrics $g_1,g_2$ such that $M$ has a compact subset which is a fractal set with respect to $g_1$ but is not a fractal set with respect to $g_2$?

Is there an example of an analytic manifold $M$ with two real analytic Riemanian metrics $g_1,g_2$ such that $M$ has a compact subset which is a fractal set with respect to $g_1$ but is not a fractal set with respect to $g_2$?

Edit: According to comment of Andre Herniques we revise the question: In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that this possible counter example comes from a Riemannian structure not merely from a metric space view point.

Is there an example of an analytic manifold $M$ with two real analytic Riemanian metrics $g_1,g_2$ such that $M$ has a compact subset which is a fractal set with respect to $g_1$ but is not a fractal set with respect to $g_2$?

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Ali Taghavi
  • 356
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  • 31
  • 123

Does fractallity depend on the Riemannian metric?

Is there an example of an analytic manifold $M$ with two real analytic Riemanian metrics $g_1,g_2$ such that $M$ has a compact subset which is a fractal set with respect to $g_1$ but is not a fractal set with respect to $g_2$?