Hypersurface

For differential geometry usage, see glossary of differential geometry and topology.

In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface. Equivalently, the codimension of a hypersurface is one.

In algebraic geometry, a hypersurface in projective space of dimension n is an algebraic set that is purely of dimension n − 1. It is then defined by a single equation F = 0, a homogeneous polynomial in the homogeneous coordinates. It may have singularities, so not in fact be a submanifold in the strict sense. "Primal" is an old term for an irreducible hypersurface.

See also

• hypersphere
• hyperspace

References

• Hazewinkel, Michiel, ed. (2001), "Hypersurface", Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/H/h048520.htm 

• Kobayashi and Nomizu (1969), Foundations of Differential Geometry Vol II ,John Wiley & Sons