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user11881
user11881
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Let $M$ be a differentiable manifold and suppose I have a group action $G \subseteq {\rm Diff}(M)$ where $G$ is a finite-dimensional Lie group (not necessarily compact). Does there exist a theory to find the Riemannian manifoldmetric $(M,g)$$g$ which has $G$ as isometries (assuming one exists)? For example if I set $M = \mathbb{S}^2 \subset \mathbb{R}^3$ and consider the matrix group $G = SO(3)$ acting in the usual way on $\mathbb{R}^3$ I would like to discover the round metric on $\mathbb{S}^2$ (or the Euclidean metric on $\mathbb{R}^3$).

Let $M$ be a differentiable manifold and suppose I have a group action $G \subseteq {\rm Diff}(M)$. Does there exist a theory to find the Riemannian manifold $(M,g)$ which has $G$ as isometries (assuming one exists)? For example if I set $M = \mathbb{S}^2 \subset \mathbb{R}^3$ and consider the matrix group $G = SO(3)$ acting in the usual way on $\mathbb{R}^3$ I would like to discover the round metric on $\mathbb{S}^2$ (or the Euclidean metric on $\mathbb{R}^3$).

Let $M$ be a differentiable manifold and suppose I have a group action $G \subseteq {\rm Diff}(M)$ where $G$ is a finite-dimensional Lie group (not necessarily compact). Does there exist a theory to find the metric $g$ which has $G$ as isometries (assuming one exists)? For example if I set $M = \mathbb{S}^2 \subset \mathbb{R}^3$ and consider the matrix group $G = SO(3)$ acting in the usual way on $\mathbb{R}^3$ I would like to discover the round metric on $\mathbb{S}^2$ (or the Euclidean metric on $\mathbb{R}^3$).

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user11881
user11881
  • 111
  • 2

Finding the metric associated with a group action

Let $M$ be a differentiable manifold and suppose I have a group action $G \subseteq {\rm Diff}(M)$. Does there exist a theory to find the Riemannian manifold $(M,g)$ which has $G$ as isometries (assuming one exists)? For example if I set $M = \mathbb{S}^2 \subset \mathbb{R}^3$ and consider the matrix group $G = SO(3)$ acting in the usual way on $\mathbb{R}^3$ I would like to discover the round metric on $\mathbb{S}^2$ (or the Euclidean metric on $\mathbb{R}^3$).