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R.S.
R.S.
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I would like to see an example of a non-abelian compact lie group admitting a bi/left/right-invariant flat metric.

Is there any non-abelian compact lie group admitting a flat metric that is bi or one-sided invariant?

The motivation is the following: as a manifold a torus does admit a flat metric, but it is also a abelian group. The first impression would be that the triviality of the lie algebra could imply the existence of a flat metric, or even be a necessary condition. Of course it is reasonable to ask for some relationship between the metric and the algebraic structure; thus some invariance would be expected (even though I would be happy to see a less restrictive assumption and a more powerful restriction dictated by the algebraic structure in the compact case).

Thus one could ask for an example of a non-abelian lie group admitting a flat metric.

A easy counter example (flat $\Rightarrow$ abelian) is the group of affine transformations of the line. The connected component of the identity is diffeomorphic to the real plane, so as a manifold (forgetting about the group structure) it does admit a flat metric. But if one considers a one-sided invariant metric one gets the Lobachevsky plane.

Since I could not think of a counter example of a non-abelian compact lie group admitting a flat metric (not necessary related to the group structure), I will also be happy to see a counter example of it. I will keep the invariance of the metric on the question though.

P.S.: Since for bi-invariant metrics $R(X,Y)Z=\frac{1}{4}[[X,Y],Z]$ (see e.g.: do Carmo, Riemannian Geometry), it is true in this case that abelian $\Rightarrow$ flat.

EDITED

Due to Igor's comment I will also include another question:

Is there any non-abelian lie group admitting a flat metric that is bi or one-sided invariant?

I would like to see an example of a non-abelian compact lie group admitting a bi/left/right-invariant flat metric.

Is there any non-abelian compact lie group admitting a flat metric that is bi or one-sided invariant?

The motivation is the following: as a manifold a torus does admit a flat metric, but it is also a abelian group. The first impression would be that the triviality of the lie algebra could imply the existence of a flat metric, or even be a necessary condition. Of course it is reasonable to ask for some relationship between the metric and the algebraic structure; thus some invariance would be expected (even though I would be happy to see a less restrictive assumption and a more powerful restriction dictated by the algebraic structure in the compact case).

Thus one could ask for an example of a non-abelian lie group admitting a flat metric.

A easy counter example (flat $\Rightarrow$ abelian) is the group of affine transformations of the line. The connected component of the identity is diffeomorphic to the real plane, so as a manifold (forgetting about the group structure) it does admit a flat metric. But if one considers a one-sided invariant metric one gets the Lobachevsky plane.

Since I could not think of a counter example of a non-abelian compact lie group admitting a flat metric (not necessary related to the group structure), I will also be happy to see a counter example of it. I will keep the invariance of the metric on the question though.

P.S.: Since for bi-invariant metrics $R(X,Y)Z=\frac{1}{4}[[X,Y],Z]$ (see e.g.: do Carmo, Riemannian Geometry), it is true in this case that abelian $\Rightarrow$ flat.

I would like to see an example of a non-abelian compact lie group admitting a bi/left/right-invariant flat metric.

Is there any non-abelian compact lie group admitting a flat metric that is bi or one-sided invariant?

The motivation is the following: as a manifold a torus does admit a flat metric, but it is also a abelian group. The first impression would be that the triviality of the lie algebra could imply the existence of a flat metric, or even be a necessary condition. Of course it is reasonable to ask for some relationship between the metric and the algebraic structure; thus some invariance would be expected (even though I would be happy to see a less restrictive assumption and a more powerful restriction dictated by the algebraic structure in the compact case).

Thus one could ask for an example of a non-abelian lie group admitting a flat metric.

A easy counter example (flat $\Rightarrow$ abelian) is the group of affine transformations of the line. The connected component of the identity is diffeomorphic to the real plane, so as a manifold (forgetting about the group structure) it does admit a flat metric. But if one considers a one-sided invariant metric one gets the Lobachevsky plane.

Since I could not think of a counter example of a non-abelian compact lie group admitting a flat metric (not necessary related to the group structure), I will also be happy to see a counter example of it. I will keep the invariance of the metric on the question though.

P.S.: Since for bi-invariant metrics $R(X,Y)Z=\frac{1}{4}[[X,Y],Z]$ (see e.g.: do Carmo, Riemannian Geometry), it is true in this case that abelian $\Rightarrow$ flat.

EDITED

Due to Igor's comment I will also include another question:

Is there any non-abelian lie group admitting a flat metric that is bi or one-sided invariant?

Source Link
R.S.
R.S.
  • 641
  • 7
  • 13

Lie groups admitting flat (bi)invariant metrics.

I would like to see an example of a non-abelian compact lie group admitting a bi/left/right-invariant flat metric.

Is there any non-abelian compact lie group admitting a flat metric that is bi or one-sided invariant?

The motivation is the following: as a manifold a torus does admit a flat metric, but it is also a abelian group. The first impression would be that the triviality of the lie algebra could imply the existence of a flat metric, or even be a necessary condition. Of course it is reasonable to ask for some relationship between the metric and the algebraic structure; thus some invariance would be expected (even though I would be happy to see a less restrictive assumption and a more powerful restriction dictated by the algebraic structure in the compact case).

Thus one could ask for an example of a non-abelian lie group admitting a flat metric.

A easy counter example (flat $\Rightarrow$ abelian) is the group of affine transformations of the line. The connected component of the identity is diffeomorphic to the real plane, so as a manifold (forgetting about the group structure) it does admit a flat metric. But if one considers a one-sided invariant metric one gets the Lobachevsky plane.

Since I could not think of a counter example of a non-abelian compact lie group admitting a flat metric (not necessary related to the group structure), I will also be happy to see a counter example of it. I will keep the invariance of the metric on the question though.

P.S.: Since for bi-invariant metrics $R(X,Y)Z=\frac{1}{4}[[X,Y],Z]$ (see e.g.: do Carmo, Riemannian Geometry), it is true in this case that abelian $\Rightarrow$ flat.