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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]
for the projection tensor which projects tensors into their transverse parts; for example, the transverse part of a vector is the part orthogonal to . This tensor can be seen as the metric tensor of the hypersurface whose tangent vectors are orthogonal to X. Thus, we have shown that:
In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to a unit vector that is orthogonal to the surface at that point. Namely, given a surface X in Euclidean space R 3 , the Gauss map is a map N : X → S 2 (where S 2 is the unit sphere ) such that for each p in X , the function value N ( p ) is ...
Similarly, if M is a hypersurface in a Riemannian manifold N, then the principal curvatures are the eigenvalues of its second-fundamental form. If k 1 , ..., k n are the n principal curvatures at a point p ∈ M and X 1 , ..., X n are corresponding orthonormal eigenvectors (principal directions), then the sectional curvature of M at p is given by
In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace.Formally, starting with a linearly independent set of vectors {v 1, ... , v k} in an inner product space (most commonly the Euclidean space R n), orthogonalization results in a set of orthogonal vectors {u 1, ... , u k} that generate the same subspace as the vectors v 1 ...
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).
In geometry, a hyperplane of an n-dimensional space V is a subspace of dimension n − 1, or equivalently, of codimension 1 in V.The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can ...