Pseudosymmetric

Pseudosymmetric is a term used in differential geometry to describe a curvature condition for Riemannian manifolds that generalizes the notion of constant curvature. A Riemannian manifold (M,g) is called pseudosymmetric if its Riemann curvature tensor R satisfies a particular algebraic relation involving a smooth function L on M: R · R = L Q(g,R), where the operator · denotes the natural action of R on itself and Q(g,R) is a curvature-type tensor built from the metric g and R. The function L may vary from point to point; in homogeneous cases it is constant. When L is identically zero, the condition reduces to semisymmetry.

History and scope have shown pseudosymmetry to be a flexible generalization of uniform curvature conditions. The concept was developed in the study of curvature tensors by Deszcz and collaborators in the late 20th and early 21st centuries, as a way to classify and compare manifolds that are not of constant curvature but still exhibit a controlled, tensorial relationship between R and itself. Pseudosymmetric manifolds include spaces of constant curvature as well as a wider class that can display rich geometric behavior while maintaining a reference curvature constraint.

Examples and significance: many known examples arise in warped product constructions or homogeneous spaces, where the pseudosymmetric condition can be verified and used to derive geometric and topological consequences. The condition is typically weaker than local symmetry (where ∇R = 0) and stronger than generic curvature assumptions, making it a useful intermediate class for classification and study.

See also: curvature tensor, semisymmetric, Tachibana tensor.