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Every Lie group $G$ is naturally contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$)

Is Hol$(G)$ always a Lie group?

If the answer is yes our main questions:

1.For a left invariant metric $g$ of Hol($G$), are there always two left invariant metrics $g_{1}$ and $g_{2}$ of $G$ and Aut($G$), respectively such that $g$ is conformal to the product of $g_{1}$ and $g_{2}$? (motivated by $G=\mathbb{R}$, as an example of this situation)

 

2. For what type of Lie groups, $G$ is a totally geodesic submanifold of its holomorph? As we see, $G=\mathbb{R}$ does not satisfies this property, but to what extent those Lie groups with this propery are studied?

 

3.Motivated by Poincare upper halph plane $\mathbb{H}^{2}\simeq \mathbb{R} \rtimes \mathbb{R}^{+}$, is it true to say that Aut($G$) is always a totally geodesic submanifold of Hol($G$)?

And finally: can one write the lie algebra of Hol($G$) in term of Lie algebras of $G$ and $Aut(G)$ and also the natural action of $Aut(G)$ on the Lie algebra of $G$?

Every Lie group $G$ is naturally contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$)

Is Hol$(G)$ always a Lie group?

If the answer is yes our main questions:

1.For a left invariant metric $g$ of Hol($G$), are there always two left invariant metrics $g_{1}$ and $g_{2}$ of $G$ and Aut($G$), respectively such that $g$ is conformal to the product of $g_{1}$ and $g_{2}$? (motivated by $G=\mathbb{R}$, as an example of this situation)

2. For what type of Lie groups, $G$ is a totally geodesic submanifold of its holomorph? As we see, $G=\mathbb{R}$ does not satisfies this property, but to what extent those Lie groups with this propery are studied?

3.Motivated by Poincare upper halph plane $\mathbb{H}^{2}\simeq \mathbb{R} \rtimes \mathbb{R}^{+}$, is it true to say that Aut($G$) is always a totally geodesic submanifold of Hol($G$)?

And finally: can one write the lie algebra of Hol($G$) in term of Lie algebras of $G$ and $Aut(G)$ and also the natural action of $Aut(G)$ on the Lie algebra of $G$?

In a natural way, every Lie group $G$ is contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$)

Is Hol$(G)$ always a Lie group?

If the answer is yes our main questions:

1.For a left invariant metric $g$ of Hol($G$), are there always two left invariant metrics $g_{1}$ and $g_{2}$ of $G$ and Aut($G$), respectively such that $g$ is conformal to the product of $g_{1}$ and $g_{2}$? (motivated by $G=\mathbb{R}$, as an example of this situation)

2. For what type of Lie groups, $G$ is a totally geodesic submanifold of its holomorph? As we see, $G=\mathbb{R}$ does not satisfies this property, but to what extent those Lie groups with this propery are studied?

3.Motivated by Poincare upper halph plane $\mathbb{H}^{2}\simeq \mathbb{R} \rtimes \mathbb{R}^{+}$, is it true to say that Aut($G$) is always a totally geodesic submanifold of Hol($G$)?

And finally: can one write the lie algebra of Hol($G$) in term of Lie algebras of $G$ and $Aut(G)$ and also the natural action of $Aut(G)$ on the Lie algebra of $G$?

Every Lie group $G$ is naturally contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$)

Is Hol$(G)$ always a Lie group?

If the answer is yes our main questions:

1.For a left invariant metric $g$ of Hol($G$), are there always two left invariant metrics $g_{1}$ and $g_{2}$ of $G$ and Aut($G$), respectively such that $g$ is conformal to the product of $g_{1}$ and $g_{2}$? (motivated by $G=\mathbb{R}$, as an example of this situation)

2. For what type of Lie groups, $G$ is a totally geodesic submanifold of its holomorph? As we see, $G=\mathbb{R}$ does not satisfies this property, but to what extent those Lie groups with this propery are studied?

3.Motivated by Poincare upper halph plane $\mathbb{H}^{2}\simeq \mathbb{R} \rtimes \mathbb{R}^{+}$, is it true to say that Aut($G$) is always a totally geodesic submanifold of Hol($G$)?

And finally: can one write the lie algebra of Hol($G$) in term of Lie algebras of $G$ and $Aut(G)$ and also the natural action of $Aut(G)$ on the Lie algebra of $G$?