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  2. 3-sphere - Wikipedia

    en.wikipedia.org/wiki/3-sphere

    The 3-sphere is homeomorphic to the one-point compactification of R 3. In general, any topological space that is homeomorphic to the 3-sphere is called a topological 3-sphere. The homology groups of the 3-sphere are as follows: H 0 (S 3, Z) and H 3 (S 3, Z) are both infinite cyclic, while H i (S 3, Z) = {} for all other indices i.

  3. Shape of the universe - Wikipedia

    en.wikipedia.org/wiki/Shape_of_the_universe

    Zero curvature (flat) – a drawn triangle's angles add up to 180° and the Pythagorean theorem holds; such 3-dimensional space is locally modeled by Euclidean space E 3. Positive curvature – a drawn triangle's angles add up to more than 180°; such 3-dimensional space is locally modeled by a region of a 3-sphere S 3.

  4. Spherical harmonics - Wikipedia

    en.wikipedia.org/wiki/Spherical_harmonics

    An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). In turn, SU(2) is identified with the group of unit quaternions, and so coincides with the 3-sphere. The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the ...

  5. Homotopy group - Wikipedia

    en.wikipedia.org/wiki/Homotopy_group

    For example, if two topological objects have different homotopy groups, they cannot have the same topological structure—a fact that may be difficult to prove using only topological means. For example, the torus is different from the sphere: the torus has a "hole"; the sphere doesn't. However, since continuity (the basic notion of topology ...

  6. Homogeneous space - Wikipedia

    en.wikipedia.org/wiki/Homogeneous_space

    A homogeneous space of N dimensions admits a set of ⁠ 1 / 2 ⁠ N(N + 1) Killing vectors. [5] For three dimensions, this gives a total of six linearly independent Killing vector fields; homogeneous 3-spaces have the property that one may use linear combinations of these to find three everywhere non-vanishing Killing vector fields ξ (a) i,

  7. Homotopy groups of spheres - Wikipedia

    en.wikipedia.org/wiki/Homotopy_groups_of_spheres

    This is the set of points in 3-dimensional Euclidean space found exactly one unit away from the origin. It is called the 2-sphere, S 2, for reasons given below. 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.

  8. Friedmann equations - Wikipedia

    en.wikipedia.org/wiki/Friedmann_equations

    k = +1, 0 or −1 depending on whether the shape of the universe is a closed 3-sphere, flat (Euclidean space) or an open 3-hyperboloid, respectively. [10] If k = +1, then a is the radius of curvature of the universe. If k = 0, then a may be fixed to any arbitrary positive number at one particular time.

  9. Lorentz group - Wikipedia

    en.wikipedia.org/wiki/Lorentz_group

    The group Sim(2) is the stabilizer of a null line; i.e., of a point on the Riemann sphere—so the homogeneous space SO + (1, 3) / Sim(2) is the Kleinian geometry that represents conformal geometry on the sphere S 2.