2015·Prépublication·Accès ouvert
Michel Bertrand Djiadeu Ngaha, M. Boucetta, J. Wouafo Kamga
arXiv (Cornell University)
Let $(G,h)$ be a nilpotent Lie group endowed with a left invariant Riemannian\nmetric, $\\mathfrak{g}$ its Euclidean Lie algebra and $Z(\\mathfrak{g})$ the\ncenter of $\\mathfrak{g}$. By using an orthonormal basis adapted to the\nsplitting\n $\\mathfrak{g}=(Z(\\mathfrak{g})\\cap[\\mathfrak{g},\\mathfrak{g}])\\oplus\nO^+\\oplus (Z(\\mathfrak{g})\\cap[\\mathfrak{g},\\mathfrak{g}]^\\perp)\\oplus\n O^-$, where $O^+$ (resp. $O^-$) is the orthogonal of\n$Z(\\mathfrak{g})\\cap[\\mathfrak{g},\\mathfrak{g}]$ in\n$[\\mathfrak{g},\\mathfrak{g}]$ (resp. is the orthogonal of\n$Z(\\mathfrak{g})\\cap[\\mathfrak{g},\\mathfrak{g}]^\\perp$ in\n$[\\mathfrak{g},\\mathfrak{g}]^\\perp$), we show that the signature of the Ricci\noperator of $(G,h)$ is determined by the dimensions of the vector spaces\n$Z(\\mathfrak{g})\\cap[\\mathfrak{g},\\mathfrak{g}],$\n$Z(\\mathfrak{g})\\cap[\\mathfrak{g},\\mathfrak{g}]^\\perp$ and the signature of a\nsymmetric matrix of order\n$\\dim[\\mathfrak{g},\\mathfrak{g}]-\\dim(Z(\\mathfrak{g})\\cap[\\mathfrak{g},\\mathfrak{g}])$.\nThis permits to associate to $G$ a subset $\\mathbf{Sign}(\\mathfrak{g})$ of\n$\\mathbf{N}^3$ depending only on the Lie algebra structure, easy to compute and\nsuch that, for any left invariant Riemannian metric on $G$, the signature of\nits Ricci operator belongs to $\\mathbf{Sign}(\\mathfrak{g})$. We show also that\nfor any nilpotent Lie group of dimension less or equal to 6,\n$\\mathbf{Sign}(\\mathfrak{g})$ is actually the set of signatures of the Ricci\noperators of all left invariant Riemannian metrics on $G$. We give also some\ngeneral results which support the conjecture that the last result is true in\nany dimension.\n
