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There is a theorem :

  1. 2-dim (pseudo-)Riemannian manifold must be local conformal flat;

  2. 3-dim (pseudo-)Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.

  3. n-dim (n>3) (pseudo-)Riemannian manifold is local conformal flat iff the Weyl tensor vanished.

Then I'm curious about the necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim (pseudo-)Riemannian manifold $(M,g)$, i.e. there exist a function $\Omega(x)$ defined in the whole manifold such that $g=\Omega^2 \eta$, where $\eta$ is the flat metric.

Are there some literature or textbooks covering this question? Thanks!

There is a theorem :

  1. 2-dim (pseudo-)Riemannian manifold must be local conformal flat;

  2. 3-dim (pseudo-)Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.

  3. n-dim (n>3) (pseudo-)Riemannian manifold is local conformal flat iff the Weyl tensor vanished.

Then I'm curious about the necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim (pseudo-)Riemannian manifold $(M,g)$, i.e. there exist a function $\Omega(x)$ defined in the whole manifold such that $g=\Omega^2 \eta$, where $\eta$ is the flat metric.

There is a theorem :

  1. 2-dim (pseudo-)Riemannian manifold must be local conformal flat;

  2. 3-dim (pseudo-)Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.

  3. n-dim (n>3) (pseudo-)Riemannian manifold is local conformal flat iff the Weyl tensor vanished.

Then I'm curious about the necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim (pseudo-)Riemannian manifold $(M,g)$, i.e. there exist a function $\Omega(x)$ defined in the whole manifold such that $g=\Omega^2 \eta$, where $\eta$ is the flat metric.

Are there some literature or textbooks covering this question? Thanks!

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346699
346699
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The necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim generalized Riemannian(pseudo-)Riemannian manifold

There is a theorem :

  1. 2-dim generalized Riemannian(pseudo-)Riemannian manifold must be local conformal flat;

  2. 3-dim generalized Riemannian(pseudo-)Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.

  3. n-dim (n>3) generalized Riemannian(pseudo-)Riemannian manifold is local conformal flat iff the Weyl tensor vanished.

Then I'm curious about the necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim generalized Riemannian(pseudo-)Riemannian manifold $(M,g)$, i.e. there exist a function $\Omega(x)$ defined in the whole manifold such that $g=\Omega^2 \eta$, where $\eta$ is the flat metric.

The necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim generalized Riemannian manifold

There is a theorem :

  1. 2-dim generalized Riemannian manifold must be local conformal flat;

  2. 3-dim generalized Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.

  3. n-dim (n>3) generalized Riemannian manifold is local conformal flat iff the Weyl tensor vanished.

Then I'm curious about the necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim generalized Riemannian manifold $(M,g)$, i.e. there exist a function $\Omega(x)$ defined in the whole manifold such that $g=\Omega^2 \eta$, where $\eta$ is the flat metric.

The necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim (pseudo-)Riemannian manifold

There is a theorem :

  1. 2-dim (pseudo-)Riemannian manifold must be local conformal flat;

  2. 3-dim (pseudo-)Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.

  3. n-dim (n>3) (pseudo-)Riemannian manifold is local conformal flat iff the Weyl tensor vanished.

Then I'm curious about the necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim (pseudo-)Riemannian manifold $(M,g)$, i.e. there exist a function $\Omega(x)$ defined in the whole manifold such that $g=\Omega^2 \eta$, where $\eta$ is the flat metric.

added 139 characters in body
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346699
346699
  • 977
  • 5
  • 13

There is a theorem :

  1. 2-dim generalized Riemannian manifold must be local conformal flat;

  2. 3-dim generalized Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.

  3. n-dim (n>3) generalized Riemannian manifold is local conformal flat iff the Weyl tensor vanished.

Then I'm curious about the necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim generalized Riemannian manifold $(M,g)$, i.e. there exist a function $\Omega(x)$ defined in the whole manifold such that $g=\Omega^2 \eta$, where $\eta$ is the flat metric.

There is a theorem :

  1. 2-dim generalized Riemannian manifold must be local conformal flat;

  2. 3-dim generalized Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.

  3. n-dim (n>3) generalized Riemannian manifold is local conformal flat iff the Weyl tensor vanished.

Then I'm curious about the necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim generalized Riemannian manifold.

There is a theorem :

  1. 2-dim generalized Riemannian manifold must be local conformal flat;

  2. 3-dim generalized Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.

  3. n-dim (n>3) generalized Riemannian manifold is local conformal flat iff the Weyl tensor vanished.

Then I'm curious about the necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim generalized Riemannian manifold $(M,g)$, i.e. there exist a function $\Omega(x)$ defined in the whole manifold such that $g=\Omega^2 \eta$, where $\eta$ is the flat metric.

Source Link
346699
346699
  • 977
  • 5
  • 13