In mathematics, and especially differential geometry, an affine sphere is a hypersurface

Hypersurface

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...

 for which the affine normals all intersect in a single point. The term affine sphere is used because they play an analogous role in affine differential geometry

Affine differential geometry

Affine differential geometry, is a type of differential geometry in which the differential invariants are invariant under volume-preserving affine transformations. The name affine differential geometry follows from Klein's Erlangen program...

 to that of ordinary spheres in Euclidean differential geometry.

An affine sphere is called improper if all of the affine normals are constant. In that case, the intersection point mentioned above lies on the hyperplane at infinity.

Affine spheres have been the subject of much investigation, with many hundreds of research articles devoted to their study.

Examples

• All quadric
  Quadric
  In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
  
  s are affine spheres; the quadrics that are also improper affine spheres are the paraboloid
  Paraboloid
  In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....
  
  s.

• If ƒ is a smooth function
  Smooth function
  In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
  
   on the plane and the determinant
  Determinant
  In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
  
   of the Hessian matrix
  Hessian matrix
  In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...
  
  is ±1 then the graph of ƒ in three-space is an improper affine sphere.