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In dimension n > 3 a conformal metric is locally conformally flat if and only if its Weyl tensor vanishes; in dimension n = 3, if and only if the Cotton tensor vanishes. Conformal geometry has a number of features which distinguish it from (pseudo-)Riemannian geometry.
In mathematics, the conformal dimension of a metric space X is the infimum of the Hausdorff dimension over the conformal gauge of X, that is, the class of all metric spaces quasisymmetric to X. [ 1 ] Formal definition
In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space. Several specific conformal groups are particularly important: The conformal orthogonal group.
Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an n-dimensional base space R p,q to null vectors in R p+1,q+1.
This is a list of formulas encountered in Riemannian geometry. ... is the dimension of the manifold. ... is (pointwise) conformal to . Evidently, conformality of ...
A classical theorem of Joseph Liouville shows that there are far fewer conformal maps in higher dimensions than in two dimensions. Any conformal map from an open subset of Euclidean space into the same Euclidean space of dimension three or greater can be composed from three types of transformations: a homothety, an isometry, and a special ...
In mathematics, Liouville's theorem, proved by Joseph Liouville in 1850, [1] is a rigidity theorem about conformal mappings in Euclidean space.It states that every smooth conformal mapping on a domain of R n, where n > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions: they are Möbius transformations (in n dimensions).
The term conformal field theory has sometimes been used with the meaning of two-dimensional conformal field theory, as in the title of a 1997 textbook. [5] Higher-dimensional conformal field theories have become more popular with the AdS/CFT correspondence in the late 1990s, and the development of numerical conformal bootstrap techniques in the ...