Spheres| 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; 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 Sn/H, with H
a finite group acting freely on Sn.
@ 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(S2)].
> 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: Chen & Lin JDG(01) [scalar curvature];
Schueth JDG(01)
[S5, 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 Sn]; lie
groups; quaternions; Trinion.
@ Monge metric:
Życzkowski & Słomczyński JPA(01)qp/00 [and quantum states];
> s.a. types of distances.
main page
– abbreviations
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– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 31 may 2031