Spheres  

Topological
* Annulus conjecture: If S and S' are two disjoint locally flat (n–1)-spheres in Sn, the closure of the region between them is homeomorphic to Sn–1 × [0,1].
* Smith conjecture (theorem): The background is that the set of fixed points of a periodic homeomorphism S3 → S3 is S1; Then this circle cannot be knotted; Has been shown to be true if the homeomorphism is smooth enough.
* Spherical space: A manifold of the type Sn/H, H a finite group acting freely on Sn.
@ References: Peterson AJP(79)dec [visualizing]; Morgan & Bass ed-84 [Smith conjecture]; Chen & Lin JDG(01) [scalar curvature]; Schueth JDG(01) [S3, isospectral]; Szczesny et al a0810 [classification of mappings in same dimension].
> Special cases: see 2-manifolds; 3-manifolds; Hopf Fibration; types of manifolds [parallelizable].
> Related topics and results: see Brouwer Theorem; euler number; Smale Conjecture.

Metric > s.a. integration; laplacian; spherical harmonics; spherical symmetry; trigonometry [spherical].
* S1: A possible parametrization is x = (t2–1)/(t2+1), y = 2t/(t2+1); covers half the circle for t [–,].
* S2: If a is the radius of the sphere (as embedded in flat space), the circumference of a circle of radius r on the sphere, and the surface area of the spherical cap enclosed by it are, respectively,

C(r) = 2a sin(r/a) ,   A(r) = 2r2 [1–cos(r/a)] .

(For an approximation, cut 1 triangle out of hexagons and paste together to get icosahedron; Add 1 triangle to get a pseudosphere).
* Complex dyad on S2: There can be no non-vanishing vector field on S2, let alone an orthonormal dyad in the ordinary sense, but a complex dyad (ma, m*a) satisfying ma · ma = 0, m*a · m*a = 0, ma · m*a = 1, can be defined by (a and a are unit vectors)

ma = 2–1/2 exp{i cos}(a + i a) ,   m*a = 2–1/2 exp{–i cos}(a – i a) .

* S3: In a 3-sphere of radius of curvature a, the volume of a ball of radius r is

V(B3) = 4a3 { arcsin(r/a) – (r/2a) [1 – (r/a)2]1/2} (4r3/3) [1 + O(r/a)2] .

* Sn: Area and scalar curvature of (n–1)-surface, and volume of interior n-ball:

S(Sn) = 2(n+1)/2 / [(n+1)/2] ;   R(Sn) = n(n–1) ;
V(Bn) = S(Sn–1)/n = 2n/2/[n(n/2)] = (2)n/2/n!! for n even, 2(2)(n–1)/2/n!! for n odd.

@ General references: Dowker CQG(90) [volume-preserving diffeomorphisms on S3]; Abdel-Khalek mp/00 [S7]; Boya et al RPMP(03)mp/02 [volumes].
@ Related topics: Joachim & Wraith BAMS(08) [curvature of exotic spheres].
> Online resources: J C Polking's spherical site [geometry].

Shere Packings
* Kepler's conjecture: In R3, the usual packing (which fills about 74% of the total available space) is the tightest one; The proof given in 1998 by Thomas Hales (after a proof "outline" published in 1993 by W-Y Hsiang) relied on computer use; In 2004 AM after an exhausting reviewing process (editor MacPherson likened it to proofreading a phone book) decided to publish the analytical parts only, while the rest will be published in DCG.
* Random or amorphous packing: Packing fractions are only about 64% of the total available space (this is for spheres, while spheroids – like M&Ms – can randomly pack more densely to fill between 68 to 71% of the total available space, and cigar-shaped ellipsoids could be randomly packed with a density of almost 74%).
* Apollonian or osculatory arrangement: An arrangement of d-dimensional spheres, each one of which touches d + 1 others.
@ General references: Moraal JPA(94); Weaire 99 [I]; Aste & Weaire 00 [I]; Hales AM(05) [proof of Kepler's conjecture + resources].
@ Random packings: Shlosman & Tsfasman mp/00; Radin JSP(08) [phase transition].

Other Structures and Related Spaces > s.a. complex structures; differentiable manifolds [exotic Sn]; lie groups; Quaternions; Trinion.
@ Monge metric: Zyczkowski & Slomczynski JPA(01)qp/00 [and quantum states]; > s.a. types of distances.

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