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In space higher than two dimensions, conformal geometry may refer either to the study of conformal transformations of what are called "flat spaces" (such as Euclidean spaces or spheres), or to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics that are defined up to scale.
In conformal geometry, a conformal Killing vector field on a manifold of dimension n with (pseudo) Riemannian metric (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field whose (locally defined) flow defines conformal transformations, that is, preserve up to scale and preserve the conformal structure.
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.
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
We say that ~ is (pointwise) conformal to . Evidently, conformality of metrics is an equivalence relation. Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric.
In practice, the metric of the manifold has to be conformal to the flat metric , i.e., the geodesics maintain in all points of the angles by moving from one to the other, as well as keeping the null geodesics unchanged, [1] that means there exists a function () such that () = (), where () is known as the conformal factor and is a point on the ...
There is a close connection between 4 dimensional Conformal Geometry and 3 dimensional CR geometry associated with the study of CR manifolds.There is a naturally defined fourth order operator on CR manifolds introduced by C. Robin Graham and John Lee that has many properties similar to the classical Paneitz operator defined on 4 dimensional Riemannian manifolds. [1]
In higher dimensions, a manifold may have different curvatures in different directions, described by the Riemann curvature tensor. In mathematics , specifically differential geometry , the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point.