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In vector calculus, a complex lamellar vector field is a vector field which is orthogonal to a family of surfaces. In the broader context of differential geometry, complex lamellar vector fields are more often called hypersurface-orthogonal vector fields. They can be characterized in a number of different ways, many of which involve the curl.
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface.A hypersurface is a manifold or an algebraic variety of dimension n − 1, which is embedded in an ambient space of dimension n, generally a Euclidean space, an affine space or a projective space. [1]
Here, saying that = is irrotational means that the vorticity tensor of the corresponding timelike congruence vanishes; thus, this Killing vector field is hypersurface orthogonal. The fact that our spacetime admits an irrotational timelike Killing vector field is in fact the defining characteristic of a static spacetime .
A smooth quadric over a field k is a projective homogeneous variety for the orthogonal group (and for the special orthogonal group), viewed as linear algebraic groups over k. Like any projective homogeneous variety for a split reductive group, a split quadric X has an algebraic cell decomposition, known as the Bruhat decomposition. (In ...
Here, saying that = is irrotational means that the vorticity tensor of the corresponding timelike congruence vanishes; thus, this Killing vector field is hypersurface orthogonal. The fact that the spacetime admits an irrotational timelike Killing vector field is in fact the defining characteristic of a static spacetime .
In relativistic cosmology, Weyl's postulate stipulates that in the Friedmann model of the universe (a fluid cosmological model), the wordlines of fluid particles (modeling galaxies) should be hypersurface orthogonal. Meaning, they should form a 3-bundle of non-intersecting geodesics orthogonal to a series of spacelike hypersurfaces (hyperslices ...
An orthogonal array is simple if it does not contain any repeated rows. (Subarrays of t columns may have repeated rows, as in the OA(18, 7, 3, 2) example pictured in this section.) An orthogonal array is linear if X is a finite field F q of order q (q a prime power) and the rows of the array form a subspace of the vector space (F q) k. [2]
The Gauss map can be defined for hypersurfaces in R n as a map from a hypersurface to the unit sphere S n − 1 ⊆ R n. For a general oriented k-submanifold of R n the Gauss map can also be defined, and its target space is the oriented Grassmannian ~,, i.e. the set of all oriented k-planes in R n. In this case a point on the submanifold is ...