Curvatures on \(\mathrm{SU}(3)/T(k,l)\) (Q2846874)

The homogeneous, compact, simply connected manifolds with a Riemannian metric of positive sectional curvature are the compact symmetric spaces of rank \(1\), five exceptional manifolds and an infinite series \(W_{k,l}\), \((k,l) \in {\mathbb Z}^2 \setminus \{ (0,0) \}\), of \(7\)-dimensional manifolds, called Aloff-Wallach spaces. The manifolds \(W_{k,l} :=\mathrm{SU}(3) / T(k,l)\) were defined by Aloff and Wallach in 1975 as the quotients of \(\mathrm{SU}(3)\) by the images \(T(k,l)\) of the circle \(S^1\) under the diagonal embeddings \(i_{k,l} : S^1 \rightarrow\mathrm{SU}(3)\), \(i_{k,l} (e ^{2 \pi i \theta} ) = \text{diag} \left( e^{2 \pi i k \theta}, \, e^{2 \pi i l \theta}, \, e^{- 2 \pi i (k+l) \theta} \right)\) with \((k,l) \in {\mathbb Z}^2 \setminus \{ (0,0) \}\).NEWLINENEWLINE Let \(G\) be a compact connected semisimple Lie group, \(H\) a closed subgroup of \(G\) and \(H_o\) be the identity component of \(H\). The homogeneous space \(G/H\) is said to be reductive if there is a subspace \(\mathfrak{M}\) of the Lie algebra \(\mathrm{Lie} (G)\) of \(G\), such that \(\mathrm{Lie}(G) = \mathrm{Lie} (H_o) \oplus \mathfrak{M}\) is a direct sum of real vector spaces, which are invariant under the adjoint action \(\text{Ad} (H)\) of \(H\). The article under review shows that Aloff-Wallach spaces \(W_{k,l}\) are reductive homogeneous spaces with \(\text{Ad} (T(k,l))\)-invariant decomposition NEWLINE\[NEWLINE\mathrm{SU}(3) = \mathrm{Lie}(\mathrm{SU}(3)) =\mathrm{Lie} (T(k,l)) \oplus \mathfrak{M}_1 \oplus \mathfrak{M}_2 \oplus \mathfrak{M}_3 \oplus \mathfrak{M}_4NEWLINE\]NEWLINE into irreducible \(\text{Ad} (T(k,l))\)-modules of dimension NEWLINE\[NEWLINE\dim _{\mathbb R} \mathfrak{M}_1 = \dim _{\mathbb R} \mathfrak{M}_2 = \dim _{\mathbb R} \mathfrak{M} _3 =2,\quad \dim _{\mathbb R} \mathfrak{M}_4 =1.NEWLINE\]NEWLINE Let \(\mu _j : G \rightarrow\mathrm{GL}(V_j)\) be linear representations of a Lie group \(G\) in real vector spaces \(V_j\), \(1 \leq j \leq 2\). The representations \(\mu _1\) and \(\mu _2\) are equivalent if there is an \({\mathbb R}\)-linear isomorphism \(\rho : V_1 \rightarrow V_2\), which is equivariant with respect to \(\mu _1(G)\) and \(\mu _2 (G)\), i.e., \(\rho \mu _1 (x) = \mu _2 (x) \rho\) for all \(x \in G\). The author shows that the representations \(\mathfrak{M}_1\), \(\mathfrak{M}_2\), \(\mathfrak{M}_3\) of \(\text{Ad} (T(k,l))\) are mutually inequivalent if and only if \(k \neq 0\), \(l \neq 0\), \(k \neq \pm l\), \(k \neq - 2l\) and \(l \neq - 2k\). The \(G\)-invariant metrics on a reductive homogeneous space \(G/H\) are in a bijective correspondence with the \(\text{Ad} (H)\)-invariant inner products on \(\mathfrak{M}\).NEWLINENEWLINE Let \(B_o\) be the negative of the Killing form of the semisimple Lie algebra \(\mathrm{Lie}(G)\) and \(\mathfrak{M} = \mathfrak{M}'_1 \oplus \cdots \oplus \mathfrak{M}'_q\) be a \(B_o\)-orthogonal \(\text{Ad} (H)\)-invariant decomposition of \(\mathfrak{M}\) into irreducible \(\text{Ad} (H)\)-representations \(\mathfrak{M}'_i\). If \(\mathfrak{M}'_i\) are mutually inequivalent irreducible \(\text{Ad} (H)\)-representations then the \(G\)-invariant metrics on \(G/H\) are of the form \(\lambda _1 B_o | _{\mathfrak{M}'_1} + \cdots + \lambda _q B_o | _{\mathfrak{M}'_q}\) for some real positive numbers \(\lambda _1, \ldots , \lambda _q\). Let \(Y_{i, \alpha}\) with \(1 \leq \alpha \leq 2\) be an orthonormal basis of \(\mathfrak{M}_i\) for \(1 \leq i \leq 3\) and \(Y_4\) be a unit vector from \(\mathfrak{M}_4\) with respect to a metric \(g( \lambda _1, \lambda _2, \lambda _3)\) on \(W_{k,l}\), associated with \(\lambda _1 B_o | _{\mathfrak{M}_1} + \lambda _2 B_o | _{\mathfrak{M}_2} + \lambda _3 B_o | _{\mathfrak{M}_3} + B_o | _{\mathfrak{M}_4}\). Denote by \(\text{Ric}\) the Ricci tensor of the Levi-Civita connection \(\nabla\) of the metric \(g( \lambda _1, \lambda _2, \lambda _3)\) and put \(s( \lambda _1, \lambda _2, \lambda _3)\) for the scalar curvature of \(\nabla\). The article shows that \(\text{Ric} (Y_{i, \alpha}, Y_{j, \beta}) = 0\) for all \((i, \alpha) \neq (j, \beta)\) and \(\text{Ric} ( Y_{i, \alpha}, Y_4) = 0\) for all \((i, \alpha)\). For any (even) permutation \(i,j,k\) of \(1,2,3\) it computes that NEWLINE\[NEWLINE\text{Ric} (Y_{i, \alpha}, Y_{i, \alpha}) = \frac{\lambda _i ^2 - \lambda _j ^2 - \lambda _k ^2 + 6 \lambda _j \lambda _k}{12 \lambda _1 \lambda _2 \lambda _3} - \frac{\gamma _i}{8 \gamma \lambda _i ^2}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\text{Ric} (Y_4, Y_4) = \frac{1}{8 \gamma} \left( \frac{\gamma _1}{\lambda _1^2} + \frac{\gamma _2}{\lambda _2^2} + \frac{\gamma _3}{\lambda _3 ^2} \right)NEWLINE\]NEWLINE with NEWLINE\[NEWLINE\gamma _1 = (k+l)^2,\quad \gamma _2 = l^2,\quad \gamma _3 = k^2,\quad \gamma = k^2 + kl + l^2.NEWLINE\]NEWLINE A reductive homogeneous space \(G/H\) with a Riemannian metric \(g\) is called a normal homogeneous space if the metric \(g\) is induced by an \(\text{Ad} (G)\)-invariant inner product \(( \cdot, \cdot)\) on \(\mathrm{Lie}(G) =\mathrm{Lie}(H_o) \oplus \mathfrak{M}\) with \((\mathrm{Lie}(H_o), \mathfrak{M}) =0\). Let us suppose that \(W_{k,l}\) is a normal homogeneous space with mutually inequivalent irreducible \(\text{Ad} (T(k,l))\)-representations \(\mathfrak{M}_1, \ldots , \mathfrak{M}_4\). Then for any non-zero tangent vector \(v \in T_{\check{o}} W_{k,l}\) at the origin \(\check{o} \in W_{k,l}\) one has NEWLINE\[NEWLINE\frac{1}{4} \leq \frac{\text{Ric} (v,v)}{||v||^2} \leq \frac{10 k^2 + 10 kl + 7 l^2}{24( k^2 + kl + l^2)}NEWLINE\]NEWLINE and the metric \(g \) is not Einstein, i.e., the Ricci tensor \(\text{Ric} \neq c g \) is not proportional to the metric by a constant \(c\). The scalar curvature of such \(W_{k,l}\) is \(\frac{9}{4}\).