N-sphere

Under a sphere is understood in mathematics, the surface of a sphere and the generalization of it to arbitrarily high dimensions. Of considerable importance for many studies here is the unit sphere, that is the surface of the unit sphere in the n-dimensional Euclidean space. More generally, particularly in topology and differential geometry, also called each homeomorphic topological space as a sphere to the spherical surface, see topological sphere.

• 5.1 spheres in normed spaces
• 5.2 spheres in metric spaces

Definition

Unit sphere

The unit sphere is the set of points in - dimensional Euclidean space with distance one from the origin. It is defined as

The Euclidean norm. The unit sphere may be regarded as a boundary of the unit ball, and is therefore also referred to as.

General spheres

Is an arbitrary point in the now -dimensional space, the sphere having a radius of about that point is defined by

Each sphere is formed from the corresponding unit sphere by scaling by the factor and translation by the vector.

Examples

The closed n-dimensional unit sphere can be assigned to each of a (n-1 )-dimensional sphere as a manifold edge:

• The 1- ball is the interval [-1,1]. 0 Accordingly, the sphere consists of only the two points 1 and -1. It is not contiguous as a single sphere.

• The 2- sphere is the circular disk of radius 1 in the plane. 1, the sphere is the unit circle, in other words, the edge of the unit circle. The unit circle is connected but not simply connected. They can be described by complex numbers of magnitude 1 and obtained by multiplying them a group structure, the circle group.

• The 3- sphere is the solid sphere in three-dimensional space. The 2-sphere, the surface of the unit sphere. It is simply connected - like all higher-dimensional spheres. She is described by spherical coordinates.

• The 3- sphere is no more vividly imagine. It is a 3- dimensional submanifold in a 4 -dimensional space. The 3- sphere can be conceived as a set of quaternions from the amount obtained by multiplying them 1 and a group structure corresponding straight.

Content and volume

The surface area or the volume of an arbitrary ( n-1) - sphere with radius in Euclidean space can be represented by the formula

Calculate the volume of the dimensional unit sphere, and the gamma function, respectively.

The sphere in the topology and geometry

In mathematics, particularly in differential geometry and topology, the term sphere is usually used with another ( more general ) Meaning: the n- dimensional sphere is the n-dimensional topological manifold that is homeomorphic to the unit sphere in.

A as defined above sphere with the Euclidean metric induced by the Riemannian metric is referred to in the differential geometry of a round sphere.

Generalizations

Spheres in normed spaces

More generally, can the concept of sphere in normed spaces grasp. Is a vector space over the real or complex numbers with the associated norm, then the norm sphere by the vector is defined with radius than the amount

The resulting spheres are point-symmetrical with respect to true, but not necessarily circular ( as in the case of the Euclidean norm ), but may also have, for example, edges and corners ( as in the case of the maximum norm of the sum of standard). If the zero vector and the radius, then one speaks again of a unit sphere. All standard spheres arise from the associated unit sphere by scaling by the factor and translation by the vector. The unit sphere is again the edge of the associated unit sphere.

Spheres in metric spaces

Still more can be put into spheres metric spaces. Is any amount of a metric, the metric sphere to the point of radius is defined as the amount of

In contrast to spheres in normed spaces metric spheres are not translationally invariant in general and, accordingly, the metric unit sphere has no special significance. In certain metric spaces can be even empty the unit sphere. Furthermore, a metric sphere in general can no longer be regarded as the edge of the associated metric ball.