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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]
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 ...
where is the Gauss map, and the differential of regarded as a vector-valued differential form, and the brackets denote the metric tensor of Euclidean space. More generally, on a Riemannian manifold, the second fundamental form is an equivalent way to describe the shape operator (denoted by S ) of a hypersurface,
Considered extrinsically, as a hypersurface embedded in (+) -dimensional Euclidean space, an -sphere is the locus of points at equal distance (the radius) from a given center point. Its interior , consisting of all points closer to the center than the radius, is an ( n + 1 ) {\displaystyle (n+1)} -dimensional ball .
Furthermore, the theorem can be extended from Bonnet's local formulation to a global formulation, allowing D to be any connected and simply-connected smooth manifold, with the result asserting the existence and uniqueness (up to a rigid motion) of a smooth immersion of D as a hypersurface of Euclidean space with first fundamental form g and ...
More generally, one can formulate a similar trick using the normal bundle to define the Laplace–Beltrami operator of any Riemannian manifold isometrically embedded as a hypersurface of Euclidean space. One can also give an intrinsic description of the Laplace–Beltrami operator on the sphere in a normal coordinate system.
A notable exception is when the manifold is given a priori as a hypersurface of Euclidean space. The second fundamental form, which determines the full curvature via the Gauss–Codazzi equation, is itself determined by the Ricci tensor and the principal directions of the hypersurface are also the eigendirections of the Ricci tensor. The tensor ...
The special case for a hypersurface (an (n-1)-dimensional submanifolds in an n-dimensional Euclidean space) was proved by H. Hopf in which the integrand is the Gauss–Kronecker curvature (the product of all principal curvatures at a point of the hypersurface). This was generalized independently by Allendoerfer in 1939 and Fenchel in 1940 to a ...