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Zbl 1150.53015
Iannuzzi, Andrea; Halverscheid, Stefan
On naturally reductive left-invariant metrics of $SL(2,\Bbb R)$. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 5, No. 2, 171-187 (2006). ISSN 0391-173X; ISSN 2036-2145/e

Let $G$ be a noncompact connected semisimple Lie group and ${\frak k} \oplus {\frak p}$ be a Cartan decomposition of the Lie algebra ${\frak g}$ of $G$. Denote by $B$ the Killing form of ${\frak g}$ and define a left-invariant metric on $G$ by $\nu_m(X,Y) = -mB(X_{\frak k},Y_{\frak k}) + B(X_{\frak p},Y_{\frak p})$, $m \in {\Bbb R}$, $X,Y \in {\frak g}$. If $G$ is a compact connected semisimple Lie group, choose a noncompact real form $G^\prime$ of the complexification $G^{\Bbb C}$ of $G$, define $K = G \cap G^\prime$, and consider the induced Cartan decomposition ${\frak g}^\prime = {\frak k} \oplus {\frak p}^\prime$ and the dual decomposition ${\frak g} = {\frak k} \oplus {\frak p}$ with ${\frak p} = i{\frak p}^\prime$. Analogously to the noncompact case one gets a left-invariant metric on $G$.\par The authors investigate curvature properties and isometry groups of these left-invariant metrics. A thorough discussion is carried out in case of $SL_2({\Bbb R})$, its universal covering group, and $PSL_2({\Bbb R})$. In particular, it is shown that a left-invariant Riemannian metric on any of these three Lie groups is naturally reductive if and only if the dimension of the isometry group is $4$.
[Jürgen Berndt (Cork)]

MSC 2000:: *53C30 Homogeneous manifolds
53C50 Lorentz manifolds, manifolds with indefinite metrics
53C55 Complex differential geometry (global)

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Zentralblatt MATH Berlin [Germany]