Hypersurface

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] Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally.

A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface.

For example, the equation

x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}-1=0

defines an algebraic hypersurface of dimension $n - 1$ in the Euclidean space of dimension $n$ . This hypersurface is also a smooth manifold, and is called a hypersphere or an $(n - 1)$ -sphere.

^ Lee, Jeffrey (2009). "Curves and Hypersurfaces in Euclidean Space". Manifolds and Differential Geometry. Providence: American Mathematical Society. pp. 143–188. ISBN 978-0-8218-4815-9.