2015/07/08 by Michel Bertrand Djiadeu Ngaha, Ngaha, M. B. Djiadeu, M. Boucetta +3
Mathematics · Physics and Astronomy · #17B30 #53C25 #53D05 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1507.02239
openalex publication_date 2015/07/08 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28
Let (G,h) be a nilpotent Lie group endowed with a left invariant Riemannian\nmetric, mathfrakg its Euclidean Lie algebra and Z( mathfrakg) the\ncenter of mathfrakg. By using an orthonormal basis adapted to the\nsplitting\n mathfrakg=(Z( mathfrakg)\∩[ mathfrakg, mathfrakg])\⊕\nO+\⊕ (Z( mathfrakg)\∩[ mathfrakg, mathfrakg]^\⊥)\⊕\n O-, where O+ (resp. O-) is the orthogonal of\nZ( mathfrakg)\∩[ mathfrakg, mathfrakg] in\n[ mathfrakg, mathfrakg] (resp. is the orthogonal of\nZ( mathfrakg)\∩[ mathfrakg, mathfrakg]^\⊥ in\n[ mathfrakg, mathfrakg]^\⊥), we show that the signature of the Ricci\noperator of (G,h) is determined by the dimensions of the vector spaces\nZ( mathfrakg)\∩[ mathfrakg, mathfrakg],\nZ( mathfrakg)\∩[ mathfrakg, mathfrakg]^\⊥ and the signature of a\nsymmetric matrix of order\n\dim[ mathfrakg, mathfrakg]-\dim(Z( mathfrakg)\∩[ mathfrakg, mathfrakg]).\nThis permits to associate to G a subset \Sign( mathfrakg) of\n\N3 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 \Sign( mathfrakg). We show also that\nfor any nilpotent Lie group of dimension less or equal to 6,\n\Sign( mathfrakg) 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