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The above -sphere exists in (+) -dimensional Euclidean space and is an example of an -manifold. The volume form ω {\displaystyle \omega } of an n {\displaystyle n} -sphere of radius r {\displaystyle r} is given by
3-sphere S 3: Polar chart. Stereographic chart Mercator chart. Euclidean spaces: n-dimensional Euclidean space E n: Cartesian chart: Euclidean plane E 2: Bipolar coordinates. Biangular coordinates Two-center bipolar coordinates. Euclidean space E 3: Polar spherical chart. Cylindrical chart. Elliptical cylindrical, hyperbolic cylindrical ...
A manifold can be constructed by giving a collection of coordinate charts, that is, a covering by open sets with homeomorphisms to a Euclidean space, and patching functions [clarification needed]: homeomorphisms from one region of Euclidean space to another region if they correspond to the same part of the manifold in two different coordinate ...
By selecting this open set to be contained in a coordinate chart, one can reduce the claim to the well-known fact that, in Euclidean geometry, the shortest curve between two points is a line. In particular, as seen by the Euclidean geometry of a coordinate chart around p , any curve from p to q must first pass though a certain "inner radius."
In an isotropic chart (on a static spherically symmetric spacetime), the metric (aka line element) takes the form = + (+ (+ ())), < <, < <, < <, < < Depending on context, it may be appropriate to regard , as undetermined functions of the radial coordinate (for example, in deriving an exact static spherically symmetric solution of the Einstein field equation).
For a given sphere packing (arrangement of spheres) in a given space, a kissing number can also be defined for each individual sphere as the number of spheres it touches. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another.
An embedding of the Euclidean sphere into N +, as in the previous section, determines a conformal scale on S. Conversely, any conformal scale on S is given by such an embedding. Thus the line bundle N + → S is identified with the bundle of conformal scales on S : to give a section of this bundle is tantamount to specifying a metric in the ...
The same idea applies for any dimension n; the equation x 2 0 + x 2 1 + ⋯ + x 2 n = 1 produces the n-sphere as a geometric object in (n + 1)-dimensional space. For example, the 1-sphere S 1 is a circle. [2] Disk with collapsed rim: written in topology as D 2 /S 1; This construction moves from geometry to pure topology.