Riemannian vs. Lorentzian Geometry: A Multilevel Comparison

We present a comparative exposition of Riemannian and Lorentzian geometries, organized through successive levels of increasing structure. Beginning with scalar products on vector spaces, we proceed through smooth manifolds, left-invariant metrics on Lie groups, homogeneous spaces and contact metric geometry. At each level, we emphasize which notions and results survive the transition from positive-definite to Lorentzian signature, which ones fail, and the mechanisms responsible for these differences. Recurring themes include: the emergence of causal structures, the loss of diagonalizability of self-adjoint operators, the failure of the equivalence between metric and geodesic completeness, and the topological obstruction to the existence of Lorentzian metrics on compact manifolds. Particular attention is devoted to the classification of three-dimensional homogeneous spaces, including Bianchi–Cartan–Vranceanu spaces and their Lorentzian counterparts, and to the contact metric setting, where an additional geometric structure restores a correspondence between the Riemannian and Lorentzian theories.