Hypersurface

In geometry, a hypersurface is a generalization of the concepts of hyperplane 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.

Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation. When this equation is a multivariate polynomial the hypersurface is an algebraic hypersurface.

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 a submanifold of the Euclidean space, which is also called an hypersphere or a (n – 1)-sphere.

Hypersurfaces occur frequently in multivariable calculus as level sets.

In Rⁿ, every closed hypersurface is orientable. Every connected compact hypersurface is a level set, and separates Rⁿ in two connected components, which is related to the Jordan–Brouwer separation theorem.

In algebraic geometry, a hypersurface in projective space of dimension n is an algebraic set (algebraic variety) that is purely of dimension n − 1. It is then defined by a single equation f(x₁, x₂, ..., x_n) = 0, a homogeneous polynomial in the homogeneous coordinates.

Thus, it generalizes those algebraic curves f(x₁, x₂) = 0 (dimension one), and those algebraic surfaces f(x₁, x₂, x₃) = 0 (dimension two), when they are defined by homogeneous polynomials.

A hypersurface may have singularities, and hence is not necessarily a submanifold in the strict sense. "Primal" is an old term for an irreducible hypersurface.

See also

• Affine sphere
• Coble hypersurface
• Dwork family
• Null hypersurface
• Polar hypersurface

References

• Hazewinkel, Michiel, ed. (2001), "Hypersurface", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
• Shoshichi Kobayashi and Katsumi Nomizu (1969), Foundations of Differential Geometry Vol II, Wiley Interscience
• P.A. Simionescu & D. Beal (2004) Visualization of hypersurfaces and multivariable (objective ) functions by partial globalization, The Visual Computer 20(10):665–81.