Conformale

Conformale is a mathematical and physical framework that studies conformal structures and angle-preserving transformations on manifolds. It emphasizes analysis and geometry that are invariant under local rescaling of the metric, rather than under rigid changes of length.

In this view, a conformal class is a set of metrics that are equivalent up to multiplication by a positive function. A space equipped with such a class is called a Conformale space, and conformal maps between these spaces preserve angles but not necessarily distances. The framework links ideas from differential geometry, complex analysis, and theoretical physics.

Origins and scope: The concept builds on classical conformal geometry and has grown to encompass techniques in representation theory, global analysis, and quantum field theory. It provides a unifying language for describing geometric structures modulo scaling and has influenced both abstract theory and applied methods.

Key concepts: Conformal metric, Weyl transformation, and conformal density describe how quantities transform under scaling. The Weyl tensor encodes the part of curvature that remains after factoring out conformal changes. In two dimensions, local angle preservation is particularly constrained, while higher dimensions reveal richer conformal structures through tools such as Cartan connections and ambient metric methods.

Applications: In physics, conformal invariance underpins conformal field theory and aspects of the AdS/CFT correspondence. In geometry, conformal techniques address problems related to scalar curvature and conformal compactification of spacetime. Computationally, conformal mappings are used in texture mapping, meshing, and image processing to preserve angular relationships.

See also: Conformal geometry, Conformal field theory, Weyl tensor, Möbius transformation, Riemann surface.