Hypersurface

Hypersurface of three-dimensional space

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. ${\displaystyle n}$ ${\displaystyle (n+1)}$

Differential geometry

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

Examples:

• The - sphere ${\displaystyle n}$

${\displaystyle S^{n}=\{x\in \mathbb {R} ^{n+1}\mid \|x\|=1\}\subset \mathbb {R} ^{n+1}.}$

• If a differentiable function is on a manifold and not a critical point of , then a hypersurface is in . ${\displaystyle f}$ ${\displaystyle M}$ ${\displaystyle c}$ ${\displaystyle f}$ ${\displaystyle f^{-1}(c)}$ ${\displaystyle M}$

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 .