The signature of the Ricci curvature of left-invariant Riemannian metrics on nilpotent Lie groups

Abstract

Let $(G,h)$ be a nilpotent Lie group endowed with a left invariant Riemannian metric, $\mathfrak{g}$ its Euclidean Lie algebra and $Z(\mathfrak{g})$ the center of $\mathfrak{g}$. By using an orthonormal basis adapted to the splitting $\mathfrak{g}=(Z(\mathfrak{g})\cap [\mathfrak{g},\mathfrak{g}])\oplus O^{+}\oplus (Z(\mathfrak{g})\cap [\mathfrak{g},\mathfrak{g}{]}^{\perp })\oplus O^{-}$, where $O^{+}$ (resp. $O^{-}$) is the orthogonal of $Z(\mathfrak{g})\cap [\mathfrak{g},\mathfrak{g}]$ in $[\mathfrak{g},\mathfrak{g}]$ (resp. is the orthogonal of $Z(\mathfrak{g})\cap [\mathfrak{g},\mathfrak{g}{]}^{\perp }$ in $[\mathfrak{g},\mathfrak{g}{]}^{\perp }$), we show that the signature of the Ricci operator of $(G,h)$ is determined by the dimensions of the vector spaces $Z(\mathfrak{g})\cap [\mathfrak{g},\mathfrak{g}],$ $Z(\mathfrak{g})\cap [\mathfrak{g},\mathfrak{g}{]}^{\perp }$ and the signature of a symmetric matrix of order $\dim [\mathfrak{g},\mathfrak{g}]-\dim (Z(\mathfrak{g})\cap [\mathfrak{g},\mathfrak{g}])$. This permits to associate to $G$ a subset $\mathbf{Sign}(\mathfrak{g})$ of ${\mathbf{N}}^3$ depending only on the Lie algebra structure, easy to compute and such that, for any left invariant Riemannian metric on $G$, the signature of its Ricci operator belongs to $\mathbf{Sign}(\mathfrak{g})$. We show also that for any nilpotent Lie group of dimension less or equal to 6, $\mathbf{Sign}(\mathfrak{g})$ is actually the set of signatures of the Ricci operators of all left invariant Riemannian metrics on $G$. We give also some general results which support the conjecture that the last result is true in any dimension.