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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]
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 .
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,
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 ...
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.
For hypersurfaces in higher-dimensional Euclidean spaces, the principal curvatures may be defined in a directly analogous fashion. The principal curvatures are the eigenvalues of the matrix of the second fundamental form (,) in an orthonormal basis of the tangent space. The principal directions are the corresponding eigenvectors.
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 ...