Spheres |

**Topological**

* __Annulus conjecture__:
If *S* and *S'* are
two disjoint locally flat (*n*–1)-spheres in S^{n},
the closure of the region between them is homeomorphic to S^{n–1} × [0,1].

* __Smith conjecture (theorem)__:
The background is that the set of fixed points of a periodic homeomorphism
S^{3} → S^{3} is ≅ S^{1};
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 S^{n}/*H*, *H* a
finite group acting freely on S^{n}.

@ __References__: Peterson AJP(79)dec [visualizing];
Morgan & Bass ed-84 [Smith
conjecture]; Chen & Lin JDG(01) [scalar curvature];
Schueth JDG(01) [S^{3}, 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, S__^{1}: A possible parametrization is *x* = ±(*t*^{2}–1)/(*t*^{2}+1), *y* =
2*t*/(*t*^{2}+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, S__^{2}:
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 *r*^{2} [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 ____S__^{2}:
There can be no non-vanishing vector field on S^{2},
let alone an orthonormal dyad in the ordinary sense, but a complex dyad (*m*^{a},
*m**^{a}) satisfying *m*^{a} · *m*_{a} =
0, *m**^{a} · *m**_{a} =
0, *m*^{a} · *m**_{a} =
1, can be defined by (*θ*^{a} and* φ^{a}* are
unit vectors)

*m*^{a} = 2^{–1/2} exp{i*φ* cos*θ*}(*θ*^{a} +
i* φ^{a}*) ,

* __S__^{3}:
The Ricci tensor of a unit 3-sphere is *R*_{ψψ}
= 2, *R*_{θθ}
= 2 sin^{2}*ψ*, *R*_{φφ}
= 2 sin^{2}*ψ* sin^{2}*θ*,
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*(B^{3}) = 4π *a*^{3} {\(1\over2\)arcsin(*r*/*a*)
– (*r*/2*a*) [1 – (*r*/*a*)^{2}]^{1/2}}
≈ (4π* r*^{3}/3)
[1 + *O*(*r*/*a*)^{2}] .

* __S__^{n}:
Area and scalar curvature of (*n*–1)-surface, and volume of interior *n*-ball:

*S*(S^{n}) =
2π^{(n+1)/2} / Γ[(*n*+1)/2]
; *R*(S^{n})
= *n*(*n*–1) ;

*V*(B^{n}) = *S*(S^{n–1})/*n* =
2π^{n/2}/[*n*Γ(*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 S^{3}];
Abdel-Khalek mp/00 [S^{7}];
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
S^{n}]; lie
groups; quaternions; Trinion.

@ __Monge metric__: Życzkowski & Słomczyński JPA(01)qp/00 [and quantum states]; > s.a.
types of distances.

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