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Conformal geometry has a number of features which distinguish it from (pseudo-)Riemannian geometry. The first is that although in (pseudo-)Riemannian geometry one has a well-defined metric at each point, in conformal geometry one only has a class of metrics. Thus the length of a tangent vector cannot be defined, but the angle between two ...
The point x = 0 in R p,q maps to n o in R p+1,q+1, so n o is identified as the (representation) vector of the point at the origin. A vector in R p+1,q+1 with a nonzero n ∞ coefficient, but a zero n o coefficient, must (considering the inverse map) be the image of an infinite vector in R p,q. The direction n ∞ therefore represents the ...
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.
Working within GA, Euclidean space (along with a conformal point at infinity) is embedded projectively in the CGA (,) via the identification of Euclidean points with 1D subspaces in the 4D null cone of the 5D CGA vector subspace. This allows all conformal transformations to be performed as rotations and reflections and is covariant, extending ...
Several specific conformal groups are particularly important: The conformal orthogonal group. If V is a vector space with a quadratic form Q, then the conformal orthogonal group CO(V, Q) is the group of linear transformations T of V for which there exists a scalar λ such that for all x in V = ()
If the metric in a conformal class is replaced by the conformally rescaled metric of the same class ^ =, then the Levi-Civita connection transforms according to the rule [12] ^ = + + (,) (). where () is the gradient vector field of i.e. the vector field -dual to , in local coordinates given by ().
In conformal geometry, the tractor bundle is a particular vector bundle constructed on a conformal manifold whose fibres form an effective representation of the conformal group (see associated bundle).
This is a list of formulas encountered in Riemannian geometry. ... (pointwise) conformal ... Note that this transformation formula is for the mean curvature vector, ...