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; It 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*, with *H*
a finite group acting freely on S^{n}.

@ __References__:
Peterson AJP(79)dec [visualizing];
Morgan & Bass ed-84 [Smith conjecture];
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];
Enríquez-Rojo et al a2105
[algebra of vector fields, deformations and extensions of Diff(S^{2})].

> __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}) ,
*m**^{a}
= 2^{−1/2 }exp{−i*ψ*
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__: Chen & Lin JDG(01) [scalar curvature];
Schueth JDG(01)
[S^{5}, isospectral]; 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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