View source for Sphere

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The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s0866401.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s0866402.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s0866403.png" /> that are situated at a constant distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s0866404.png" /> (the radius of the sphere) from a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s0866405.png" /> (the centre of the sphere), i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s0866406.png" /></td> </tr></table>

The sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s0866407.png" /> is a pair of points, the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s0866408.png" /> is the circle, the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s0866409.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664010.png" />, is sometimes called a hypersphere. The volume of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664011.png" /> (the length when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664012.png" />, the surface when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664013.png" />) is given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664014.png" /></td> </tr></table>

in particular,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664015.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664016.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664017.png" /> is the [[Gamma-function|gamma-function]].

The equation of a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664018.png" /> in the Cartesian coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664019.png" /> takes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664020.png" /></td> </tr></table>

(here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664023.png" />, are the coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664024.png" /> respectively), i.e. the sphere is a (hyper-)quadric or a surface of the second order of special form.

The position of any point in space relative to a sphere is characterized by the power of the point (cf. [[Degree of a point|Degree of a point]]). The totality of all spheres (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664025.png" />-space) relative to which a given point has a fixed power forms a [[Web of spheres|web of spheres]]. The totality of all spheres relative to which the points of a straight line (the radical axis) have an identical power (different for different points), forms a [[Net|net]] of spheres. The totality of all spheres relative to which the points of a plane (the radical plane) have an identical degree (different for different points), forms a pencil of spheres.

From the point of view of differential geometry, the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664026.png" /> is a [[Riemannian space|Riemannian space]] of constant curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664027.png" />. (This curvature is Gaussian when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664028.png" /> and Riemannian when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664029.png" />.) All geodesics of a sphere are closed and have constant length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664030.png" /> — these are known as great circles, i.e. the intersections with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664031.png" /> of two-dimensional planes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664032.png" /> that pass through its centre. The exterior-geometric properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664033.png" /> are: all normals intersect at one point; the curvature of any normal section is one and the same and does not depend on the point at which it is examined, in particular, it has constant mean curvature, whereby the complete mean curvature of the sphere is the least among the convex surfaces of identical area; and all points of the sphere are umbilical (cf. [[Umbilical point|Umbilical point]]).

Certain of these properties, which are taken to be fundamental, have been used as the starting point for a generalization of the concept of a sphere. For example, an [[Affine sphere|affine sphere]] is defined by the fact that all its (affine) normals intersect at one point; a [[Pseudo-sphere|pseudo-sphere]] is a surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664034.png" /> of constant Gaussian curvature (although negative); one of the interpretations of a [[Horosphere|horosphere]] (limit sphere) is as the set of points within <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664035.png" /> defined by an equation that is also of the second degree:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664036.png" /></td> </tr></table>

The orthogonal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664037.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664038.png" /> operates doubly-transitive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664039.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664041.png" />-transitivity means that for any two pairs of points with equal distances between them there is a rotation — an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664042.png" /> — that maps one pair onto the other); this group is the complete group of isometries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664043.png" />; finally, a sphere is a [[Homogeneous space|homogeneous space]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664044.png" />.

From the point of view of (differential) topology, the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664045.png" /> is a closed differentiable manifold that divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664046.png" /> into two domains and that is their common boundary; the bounded domain homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664047.png" /> is then an (open) [[Ball|ball]]; thus, the sphere can be defined as its boundary.

The homology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664049.png" />, are:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664050.png" /></td> </tr></table>

in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664051.png" /> does not contract into a point, i.e. the identity mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664052.png" /> onto itself is essential (cf. [[Essential mapping|Essential mapping]]).

The homotopy groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664054.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664055.png" /> are:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664056.png" /></td> </tr></table>

In addition one has, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664057.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664058.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664059.png" />. Generally, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664062.png" />, the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664063.png" /> have not been calculated (see [[Spheres, homotopy groups of the|Spheres, homotopy groups of the]]).

The concept of a sphere also has a generalization here. For example, a [[Wild sphere|wild sphere]] is a topological sphere (see below) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664064.png" /> that does not bound a domain homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664065.png" />; a [[Milnor sphere|Milnor sphere]] (an exotic sphere) is a manifold that is homeomorphic, but not diffeomorphic, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664066.png" />.

A topological space homeomorphic to a sphere is called a topological sphere. One of the basic problems here is the question of the conditions under which a space is a topological sphere.

Examples. a) No topologically invariant characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664067.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664068.png" /> is known (1990). For the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664069.png" />, see [[One-dimensional manifold|One-dimensional manifold]]. In order that a [[Continuum|continuum]] be homeomorphic to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664070.png" />, it is necessary and sufficient that it be locally connected, that it contain at least one simple closed curve and that every such curve that lies in it divides it into two domains having this curve as their common boundary (Wilder's theorem).

b) A complete simply-connected Riemannian space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664071.png" /> whose curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664072.png" /> for all tangent two-dimensional planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664073.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664074.png" />-bounded with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664075.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664076.png" />, is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664077.png" /> (the sphere theorem, see [[Riemannian geometry|Riemannian geometry]]).

c) A simply-connected closed smooth manifold whose (integral) homology groups coincide with the homology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664078.png" /> is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664079.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664080.png" /> (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664081.png" />, it is unknown (1990)). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664082.png" />, it is also diffeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664083.png" /> (the generalized Poincaré conjecture), when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664084.png" />, the diffeomorphism result does not hold.

A sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664085.png" /> in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664086.png" /> is defined in exactly the same way: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664087.png" />. However, this set, generally speaking, may have a fairly-complicated structure (it may even be empty).

In a normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664088.png" /> with norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664089.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664090.png" /> is called a sphere; this is, generally speaking, essentially an arbitrary, infinite-dimensional, convex (hyper)surface, and does not always possess the properties of, for example, smoothness, roundedness and other useful properties of ordinary spheres. One of the variants used in topology — the so-called infinite-dimensional sphere — is the strict inductive limit, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664091.png" />, of a sequence of nested spheres:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664092.png" /></td> </tr></table>

another definition: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664093.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664094.png" /> is an infinite-dimensional [[Stiefel manifold|Stiefel manifold]]. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664095.png" />, it turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664096.png" />.

The applications of the concept of a sphere are remarkably varied. For example, a sphere is used in constructing new spaces or supplementary structures on them. For example, the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664097.png" /> can be interpreted as a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664098.png" /> with diametrically-opposite points identified; a sphere with handles and holes is used in [[Handle theory|handle theory]]; see also [[Cohomotopy group|Cohomotopy group]]; [[Spherical map|Spherical map]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.A. Rozenfel'd,   "Multi-dimensional spaces" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Rozenfel'd,   "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Lévy,   "Problèmes concrets d'analyse fonctionelle" , Gauthier-Villars  (1951)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> , ''Introduction to topology'' , Moscow  (1980)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H. Busemann,   "The geometry of geodesics" , Acad. Press  (1955)</TD></TR></table>



====Comments====
A simply-connected topological manifold whose homology is like that of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s08664099.png" />-sphere is sometimes called a Poincaré manifold. It was recently shown that a smooth Poincaré <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s086640100.png" />-manifold is not necessarily diffeomorphic to the standard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s086640101.png" />.

For a survey of recent results on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s086640102.png" />-manifolds, including the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086640/s086640103.png" />-sphere, see [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Milnor,   "On manifolds homeomorphic to the 7-sphere"  ''Ann. of Math.'' , '''64'''  (1956)  pp. 399–405</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''I''' , Springer  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.H. Freedman,   F. Luo,   "Selected applications of geometry to low-dimensional topology" , Amer. Math. Soc.  (1987)</TD></TR></table>