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Stefan Kohl ♦
Stefan Kohl ♦
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I'm trying to find an example when G isof a lieLie group that$G$ which admits a bi invariant riemannian-invariant Riemannian metric, and Hwhich has a is closed subgroup wich$H$ such that the manifold G/H$G/H$ does not admit a G$G$-invariant riemannianRiemannian metric.

I know so far that if G/H$G/H$ admits aan invariant Riemannian metric, then H$H$ has to be compact. So I'm searching for a lieLie group G that$G$ which admits a bi invariant riemannian-invariant Riemannian metric and H a closed subgroup that$H$ which is not compact, since this will solve the problem. Does anyone know such group?

Thank you for any help!

I'm trying to find an example when G is a lie group that admits a bi invariant riemannian metric and H a is closed subgroup wich the manifold G/H does not admit a G-invariant riemannian metric.

I know so far that if G/H admits a invariant Riemannian metric, then H has to be compact. So I'm searching for a lie group G that admits a bi invariant riemannian metric and H a closed subgroup that is not compact, this will solve the problem. Does anyone know such group?

Thank you for any help!

I'm trying to find an example of a Lie group $G$ which admits a bi-invariant Riemannian metric, and which has a closed subgroup $H$ such that the manifold $G/H$ does not admit a $G$-invariant Riemannian metric.

I know so far that if $G/H$ admits an invariant Riemannian metric, then $H$ has to be compact. So I'm searching for a Lie group $G$ which admits a bi-invariant Riemannian metric and a closed subgroup $H$ which is not compact, since this will solve the problem. Does anyone know such group?

Thank you for any help!

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user42999
user42999
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Bi invariant Riemannian metric on a Lie Group

I'm trying to find an example when G is a lie group that admits a bi invariant riemannian metric and H a is closed subgroup wich the manifold G/H does not admit a G-invariant riemannian metric.

I know so far that if G/H admits a invariant Riemannian metric, then H has to be compact. So I'm searching for a lie group G that admits a bi invariant riemannian metric and H a closed subgroup that is not compact, this will solve the problem. Does anyone know such group?

Thank you for any help!