The Lie algebra of isometries of homogeneous spaces of nonpositive curvature
Let \(M\) denote a complete, simply connected homogeneous Riemannian manifold with sectional curvature \(K \leq 0\). Let \(G\) denote \(I_0(M)\), the identity component of the isometry group of \(M\), and let \(\mathfrak S\) denote the Lie algebra of \(G\). In this article the author finds sufficient conditions for \(M\) to be a symmetric space in terms of properties of \(G\) or \(\mathfrak G\). It is known that there exist simply transitive solvable subgroups of \(G\). Let \(S\) be such a group, and let \(\mathfrak G\) denote its Lie algebra. Then \(M\) is isometric to \(S\) with a left invariant metric \(\langle, \rangle\) and \(\{S, \langle, \rangle\}\) in turn is determined by the metric Lie algebra \(\{{\mathfrak S}, \langle, \rangle\}\) since \(S\) is simply connected and \({\mathfrak S} = T_eS\). Let \(K\) be a maximal compact subgroup of \(G\), and let \(\mathfrak K\) denote its Lie algebra. The Lie algebra \(\mathfrak S\) is said to be in standard position if \(B({\mathfrak K}, {\mathfrak A}) = 0\), where \(B\) denotes the Killing form of \(\mathfrak G\). It is always possible to find a simply transitive solvable group \(S\) whose Lie algebra is in standard position. Let \({\mathfrak A} = [{\mathfrak S}, {\mathfrak S}]^\perp\), the orthogonal complement in \(\mathfrak S\) of the commutator ideal \({\mathfrak S}' = [{\mathfrak S}, {\mathfrak S}]\). By work of Azencott and Wilson it is known that \(\mathfrak A\) is an abelian subalgebra of \(\mathfrak S\). Typically one now studies the geometry of \(M\) through the algebraic properties of the action of \(\text{ad}({\mathfrak A})\) on the complexifications \({\mathfrak G}^\mathbb{C}\) and \({\mathfrak S}^\mathbb{C}\) of \(\mathfrak G\) and \(\mathfrak S\), respectively. For each element \(\lambda \in\Hom({\mathfrak A},\mathbb{C})\) let \({\mathfrak G}^\mathbb{C}_n\) (respectively \({\mathfrak S}^\mathbb{C}_n)\) denote the largest subspace of \({\mathfrak G}^\mathbb{C}\) (respectively \({\mathfrak S}^\mathbb{C}_\lambda)\) on which \(\text{ad }A - \lambda(A) \text{Id}\) is nilpotent for all \(A \in {\mathfrak A}\). Let \(\Lambda_{\mathfrak G} = \{\lambda \in \text{Hom}({\mathfrak A}, \mathbb{C}) : {\mathfrak G}^\mathbb{C}_\lambda \neq \{0\}\}\), and let \(\Lambda_{\mathfrak G}\) be defined in a similar way. Clearly \(\Lambda_{\mathfrak S} \subseteq \Lambda_{\mathfrak G}\) but equality does not hold in general. Given \(\lambda = \alpha + i\beta \in \Lambda_{\mathfrak G}\) let \({\mathfrak G}_{\alpha, \beta} = {\mathfrak G} \cap \{{\mathfrak G}^\mathbb{C}_\lambda \oplus {\mathfrak G}^\mathbb{C}_{\overline{\lambda}}\}\) and let \({\mathfrak G}_- = \oplus {\mathfrak G}_{\alpha, \beta} : \lambda = \alpha + i\beta \in \Lambda_{\mathfrak G} - \Lambda_{\mathfrak S}\). The first main result of the article is the following algebraic characterization of symmetric spaces. Theorem: Let \(M\) be an irreducible nonflat, simply connected homogeneous space of sectional curvature \(K \leq 0\), and let \(G = I_0(M)\). Fix a basepoint \(p \in M\) and let \(K = G_p\) be the corresponding maximal compact subgroup with Lie algebra \(\mathfrak K\). Let \(S\) be a simply transitive solvable subgroup of \(G\) such that its Lie algebra \(\mathfrak S\) is in standard position relative to \(\mathfrak K\). Then \(\dim({\mathfrak S}_-) \leq \dim([{\mathfrak G}, {\mathfrak S}])\) with equality if and only if \(m\) is a symmetric space of noncompact type. Using the same notation as above we say that an element \(H\) of \({\mathfrak A} = [{\mathfrak S}, {\mathfrak S}]^\perp\) is \({\mathfrak S}\)-regular if \((\text{Re }\lambda) (H) > 0\) for all \(\lambda \in \Lambda_{\mathfrak S}\) with \(\text{Re }\lambda \neq 0\). Elements \(H\) that are \(\mathfrak S\)-regular exist, and one can show that the point \(x = \gamma_H(\infty)\) in the boundary sphere \(M(\infty)\) represented by the geodesic \(\gamma_H(t) = \text{exp}(tH)(p)\) is fixed by \(S\). If \(y = \gamma_H(-\infty) \in M(\infty)\), then the proof of the above result follows from a previous result of the author, which states that \(M\) is symmetric if and only if \(G_y = \{g \in G : g(y) = y\}\) acts transitively on \(M\). The orbit structure of \(G\) on \(M(\infty)\) characterizes those spaces \(M\) which satisfy the visibility axiom; that is, given a point \(p\) of \(M\) and a positive number \(\varepsilon\) there exists a positive number \(R\) such that any geodesic segment of distance at least \(R\) from \(p\) subtends an angle at most \(\varepsilon\) at \(p\). The second main result of the article is the following. Theorem: Let \(M\) be a simply connected homogeneous space of sectional curvature \(K \leq 0\) that has no Euclidean de Rham factor. Let \(G\), \(p\), \(K\), \(S\), \(\mathfrak K\) and \(\mathfrak S\) be as in the result stated above. Let \(H\) be an \(\mathfrak S\)-regular element of \({\mathfrak A} = [{\mathfrak S}, {\mathfrak S}]^\perp\) and let \(x = \gamma_H(\infty)\). Then the following statements are equivalent: 1) \(M\) satisfies the visibility axiom. 2) \(G(y)\) is dense in \(M(\infty)\) for all \(y \neq x\) in \(M(\infty)\). 3) \(G(y)\) is dense in \(M(\infty)\) for some \(y \in M(\infty)\) that can be joined to \(x\) by a geodesic of \(M\) (i.e., there exists a geodesic \(\sigma\) of \(M\) such that \(x = \sigma(\infty)\) and \(y = \sigma(-\infty))\). If \(M\) has negative sectional curvature, which implies the visibility axiom, then one can show that \(S(y) = M(\infty) - \{x\}\) for all \(y \neq x\) in \(M(\infty)\) and \(M\) is symmetric if and only if \(G\) acts transitively on \(M(\infty)\).
scientific article; zbMATH DE number 806202
scientific article