# Hypersurface

In mathematics , geometric objects of code dimension 1 are called **hypersurfaces** .

The eponymous special cases are all curved or flat surfaces in three-dimensional space and hyperplanes , i.e. -dimensional planes in a -dimensional affine space . Also curves in a plane are formally hypersurfaces.

## Differential geometry

In differential geometry , a hypersurface is a submanifold of codimension 1.

*Examples:*

- The - sphere

- If a differentiable function is on a manifold and not a critical point of , then a hypersurface is in .

## Algebraic Geometry

In algebraic geometry , a hypersurface is understood to be a sub-scheme of affine or projective space defined by a single (homogeneous) equation . Above a body, every closed sub- schema that has pure codimension 1 and has no embedded components - i.e. every effective divisor - has this form.

## literature

- John M. Lee:
*Riemannian Manifolds. An Introduction to Curvature*(=*Graduate Texts in Mathematics*176). Springer, New York NY et al. 1997, ISBN 0-387-98322-8 .