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.
* Hopf's pinching problem: The question whether a compact, simply connected manifold with suitably pinched curvature is topologically a sphere.
* 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]; Szczęsny et al IJGMP(09)-a0810 [classification of mappings in same dimension]; Barmak & Minian math/06 [finite spaces with the same homotopy groups as the spheres]; Brendle & Schoen BAMS(11) [Hopf's pinching problem and the Differentiable Sphere Theorem].
> 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. Hopf Sphere Theorem; integration; laplacian; spherical harmonics; spherical symmetry; trigonometry [spherical].
* Circle, S1: A possible parametrization is x = ±(t2–1)/(t2+1), y = 2t/(t2+1); covers half the circle for t ∈ [–∞,∞].
* Sets of circles: Some interesting arrangements of circles in the Euclidean plane are the Apollonian circles (> see Wikipedia page); > s.a. fractals.
* Sphere, S2: The scalar curvature of a unit 2-sphere is R = 2; 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) = 2π a sin(r/a) ,   A(r) = 2π r2 [1–cos(r/a)] .

(For an approximation, cut 1 triangle out of hexagons and paste together to get an 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 (θa and φa are unit vectors)

ma = 2–1/2 exp{iφ cosθ}(θa + i φa) ,   m*a = 2–1/2 exp{–iψ cosθ}(θa – i φa) .

* S3: The Ricci tensor of a unit 3-sphere is Rψψ = 2, Rθθ = 2 sin2ψ, Rφφ = 2 sin2ψ sin2θ, and the scalar curvature R = 6; In a 3-sphere of radius of curvature a, the volume of a ball of radius r is

V(B3) = 4π a3 {$$1\over2$$arcsin(r/a) – (r/2a) [1 – (r/a)2]1/2} ≈ (4π r3/3) [1 + O(r/a)2] .

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

S(Sn) = 2π(n+1)/2 / Γ[(n+1)/2] ;   R(Sn) = n(n–1) ;
V(Bn) = S(Sn–1)/n = 2πn/2/[nΓ(n/2)] = (2π)n/2/n!!  for n even ,  2(2π)(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]; Brauchart & Grabner JCompl(15)-a1407 [spherical designs and minimal-energy point configurations].
> Online resources: J C Polking's spherical site [geometry]; John Baez's pages on rolling balls and circles (2012).

Shere Packings
* Kepler's conjecture: In $$\mathbb R$$3, 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 Annals of Mathematics, 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]; Hales AM(05) [proof of Kepler's conjecture + resources]; Aste & Weaire 08 [I].
@ Random packings: Shlosman & Tsfasman mp/00; Radin JSP(08) [phase transition]; Kallus PRE(13)-a1305 [Monte Carlo approach to the d-dimensional lattice sphere packing problem].

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