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In mathematics, an n-sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional, and the sphere 2-dimensional, because the surfaces themselves are 1- and 2-dimensional respectively, not because they ...
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.
Simple examples of Einstein manifolds include: All 2D manifolds admit Einstein metrics. In fact, in this dimension, a metric is Einstein if and only if it has constant Gauss curvature. The classical uniformization theorem for Riemann surfaces guarantees that there is such a metric in every conformal class on any 2-manifold.
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system—shown here in the mathematics convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored ...
For example, the class of two-dimensional Euclidean space forms includes Riemannian metrics on the Klein bottle, the Möbius strip, the torus, the cylinder S 1 × ℝ, along with the Euclidean plane. Unlike the case of two-dimensional spherical space forms, in some cases two space form structures on the same manifold are not homothetic.
The n-dimensional model is the celestial sphere of the (n + 2)-dimensional Lorentzian space R n+1,1. Here the model is a Klein geometry: a homogeneous space G/H where G = SO(n + 1, 1) acting on the (n + 2)-dimensional Lorentzian space R n+1,1 and H is the isotropy group of a fixed null ray in the light cone.
Embed M in some high-dimensional Euclidean space. (Use the Whitney embedding theorem.) Take a small neighborhood of M in that Euclidean space, N ε. Extend the vector field to this neighborhood so that it still has the same zeroes and the zeroes have the same indices.
The Borsuk–Ulam theorem is equivalent to the following statement: A continuous odd function from an n-sphere into Euclidean n-space has a zero. PROOF: PROOF: If the theorem is correct, then it is specifically correct for odd functions, and for an odd function, g ( − x ) = g ( x ) {\displaystyle g(-x)=g(x)} iff g ( x ) = 0 {\displaystyle g(x ...