produktgeometrier

Produktgeometrier describe geometric structures obtained by forming products of simpler geometric objects. In its broad sense, the term covers the geometry of product spaces in differential geometry, metric geometry and related fields, including the Cartesian product of manifolds, graphs and metric spaces. The central idea is that the geometry of the composite object reflects the geometries of its factors through a product operation.

In differential geometry, the standard construction is the Riemannian product. If (M,g) and (N,h) are Riemannian manifolds, the product M×N carries the metric g⊕h, defined by g⊕h((v,w),(v,w)) = g(v,v) + h(w,w). Distances satisfy d((p,q),(p',q'))^2 = d_M(p,p')^2 + d_N(q,q')^2. Geodesics in a product are pairs of geodesics, and the sectional curvature of a product equals the curvature of the corresponding factor for planes contained in a single factor; mixed planes have zero curvature.

In discrete settings, product geometries arise from the Cartesian product of graphs or metric spaces, where distance is typically defined by the shortest-path metric. Product grids and lattice-like structures are common examples.

Applications span physics, where spacetime models can be written as products of manifolds; robotics and motion planning, computer graphics, and data analysis, where product spaces model multi-parameter or multi-modal data.

Etymology and terminology: in many languages the term exists as Produktgeometrien or produktgeometri, often used interchangeably with product manifolds or product spaces. See also product manifold, product metric, Riemannian geometry, and graph products.