DOCSLIB.ORG

THE SPHERE THEOREM in RIEMANNIAN GEOMETRY Contents 1. Notation and Fundamental Notions 1 2. Some Necessary Lemmas 5 3. Proof Of

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
THE SPHERE THEOREM in RIEMANNIAN GEOMETRY Contents 1. Notation and Fundamental Notions 1 2. Some Necessary Lemmas 5 3. Proof Of THE SPHERE THEOREM IN RIEMANNIAN GEOMETRY THOMAS YU Abstract. We will give a proof of the Sphere Theorem in Riemannian Ge- ometry, which states that if M is a complete, simply-connected n-dimensional 1 manifold with sectional curvature KM such that 1 ≥ KM > 4 , then M is homeomorphic to the n-sphere Sn Contents 1. Notation and Fundamental Notions 1 2. Some Necessary Lemmas 5 3. Proof of the Sphere Theorem 6 4. Acknowledgements 7 References 7 1. Notation and Fundamental Notions For the sake of brevity, we will assume that the reader has completed a course in differential geometry. The proof of the Sphere Theorem given is taken largely from Comparison Theorems in Riemannian Geometry by Cheeger and Ebins; since this theorem requires 5 fairly dense chapters of background material, in this section we present without proof an overview of the most significant theorems and construc- tions used in the proof of the Sphere Theorem. Throughout the paper, assume M is a smooth, finite-dimensional manifold. For p 2 M, let Mp denote the tangent space to M at p, let ζ(M) denote the linear space of smooth vector fields on M, and let F(M) denote the ring of smooth, real-valued functions on M. Recall that tangent vectors are derivations, and we interpret them as giving directional derivatives. Definition 1.1. A Riemannian metric is an assigment to each p 2 M of a sym- metric, positive-definite bi-linear form h;ip on Mp such that for any smooth vector fields V; W , the function p ! hV; W ip is a smooth real-valued function on M. A manifold that possesses a Riemannian metric is called a Riemannian manifold. Note that this bi-linear form defines a norm on the tangent space; we shall denote 1 hV; V i 2 as kV k. Definition 1.2. An affine connecton is a connection on the tangent bundle T(M). We can view an affine connection as a bi-linear map r : ζ(M) × ζ(M) ! ζ(M)(1.3) Date: August 12, 2013. 1 2 THOMAS YU where ζ(M) is the linear space of smooth vector fields on M, which obeys the following properties: (1.4) rfV W = frV W (1.5) rV fW = (V f)W + frV W 8f 2 F(M); V; W 2 ζ(M). Affine connections provide a natural way of differentiat- ing arbitrary tensor fields on M since a connection on T (M) induces connections on arbitary tensor products of the tangent and co-tangent bundles. For this paper we will only require differentiating vector fields; rV W is called the covariant derivative of W in the direction V . We now state the fundamental theorem of Riemannian geometry. Theorem 1.6. For every Riemannian metric there is a unique affine connection called the Levi-Civita connection that satisfies the following properties (1.7) X hV; W i = hrX V; W i + hV; rX W i rV W − rW V − [V; W ] = 0(1.8) where [; ] denotes the Lie bracket [X; Y ]f = (XY − YX)f. The first condition in the theorem describes the compatibility of the connection and the metric; that is, norms of vectors transported along curves are preserved. The second condition is equivalent to saying that the connection is torsion-free. From here on, r will refer to the Levi-Civita connection. With the Riemannian metric we can define the length of a smooth curve c : [a; b] ! M; L[c] as Z b (1.9) L[c] = kc0(t)kdt: a Further, we can make M into a metric space by defining the distance between two points p; q 2 M as (1.10) ρ(p; q) = infc2C L[c]: where C is the set of all curves between p; q. We will now explore some conditions for the existence of a curve γ from p to q such that L[γ] = ρ(p; q). Readers familiar with the calculus of variations will recognize the following condition. For the following discussion assume that curves c are smooth, parametrized by arc length, and kc0(t)k = l, where l is some non-zero constant. Definition 1.11. Let α :[a; b] × [−, ] ! M be a smooth function such that αj[a;b]×0 = c :[a; b] ! M. Then α is a called a smooth variation of c. Intuitively, α encodes perturbations of the curve c; that is, we can think of α as a family of curves parametrized by s 2 [−, ]. Now let T;V be the tangent vectors on [a; b] × [−, ] corresponding to the first and second variables respectively; note that T;V are just the derivative operators with respect to the associated variable. We now compute the change in arc length over the family of curves cs = αj[a;b]×s THE SPHERE THEOREM IN RIEMANNIAN GEOMETRY 3 around the base curve c0. This is given by b b d d Z 1 Z 1 0 0 2 2 (1.12) L[cs] = hcs(t); cs(t)i dt = V hT;T i dt ds ds a a b 1 Z −1 (1.13) = hT;T i 2 V hT;T i dt 2 a b Z −1 2 (1.14) = hT;T i hrV T;T i dt a Note that we used the properties of the Levi-Civita connection above. Since partial derivatives commute in the Euclidean rectangle [a; b]×[−, ], we can rewrite (1.14) as b Z −1 2 (1.15) hT;T i hrT V; T i dt a 0 Using the kc0k = l, we have that (1.16) Z b d −1 L[cs]js=0 = l hrT V; T i dt ds a Z b Z b ! −1 −1 b (1.17) = l (T hV; T i − hV; rT T i)dt = l hV; T i ja − hV; rT T i dt a a (1.17) is called the first variation formula. Note that if the family of curves share the same endpoints then the first term of (1.17) vanishes. Further, if c0 is the d shortest curve between the endpoints, we must have that ds L[cs]js=0 = 0 for any α. Thus, any vector field V which vanishes at the endpoints of c0 must satisfy the last term of (1.17) being 0. By choosing V = φ(t)rT T where φ(t) > 0 for a < t < b and φ(t) = 0 for t = a; b, we see that rT T = 0(1.18) This motivates the following definition of a geodesic curve. 0 0 Definition 1.19. A smooth curve is called a geodesic if rcc = 0. A geodesic is normal if kc0k = 1. Note that the defining property of a geodesic, written in a coordinate neighbor- hood is a second order differential equation in the parameter t of the geodesic; to see this, we simply need to translate the vector field c0 as well as the connection in terms of differential operators in the chosen coordinate basis. Hence, from existence and uniqueness theorems from the theory of ordinary differential equations, the speci- fication of a point p 2 M and v 2 Mp results in a unique geodesic γv through p whose tangent vector at p is v. We are now ready to define the exponential map. Definition 1.20. The exponential map exp(v), is defined by γv(1), for v 2 Mp such that 1 is in the domain of γv. expp is defined in a neighborhood of the origin of Mp, is smooth, and is a local diffeomorphism in a neighborhood of the origin by the Inverse Function Theorem since the differential of the exponential map is nonzero at the origin. An interesting theorem concerning the exponential map is the Hopf-Rinow Theorem 4 THOMAS YU Theorem 1.21. A connected, Riemannian manifold M is complete as a metric space with respect to the metric described above if and only if the exponential map is defined over all of Mp for every p 2 M. In particular, this implies that in a complete manifold, there is always a length minimizing geodesic between any two points in the manifold. We know that for each p 2 M, expp is injective on a sufficiently small ball; therefore, ρ(γ(t); γ(0)) = t for sufficiently small t(if we parametrize γ by arc length). A question that arises is how large t can be before this formula no longer holds. Definition 1.22. The injectivity radius i(M) of M is the supremum over all r such that 8p 2 M, expp is an embedding on the open ball of radius r in Mp Definition 1.23. Define d(M) to be supp;q2M ρ(p; q). We now define the curvature tensor which is a trilinear map R : Mp ×Mp ×Mp ! Mp. (1.24) R(X; Y )Z = rX rY Z − rY rX Z − r[X;Y ]Z We can think of the curvature as measuring the failure of commutativity of covariant differentiation. Definition 1.25. Given a plane σ in Mp and two spanning vectors v; w, we define the sectional curvature K(σ) as hR(v; w)w; vi (1.26) K(σ) = kv ^ wk2 where ^ is the wedge product and kv ^ wk2 denotes the square of the area of the parallelogram formed by v; w. It can be shown that K(σ) is independent of the choice of v; w. The notation KM > H denotes that for all plane sections at all points of M, the sectional curvature is bigger than the constant H.
Load more

Recommended publications
  • Examples of Manifolds
    Examples of Manifolds Example 1 (Open Subset of IRn) Any open subset, O, of IRn is a manifold of dimension n. One possible atlas is A = (O, ϕid) , where ϕid is the identity map. That is, ϕid(x) = x. n Of course one possible choice of O is IR itself. Example 2 (The Circle) The circle S1 = (x,y) ∈ IR2 x2 + y2 = 1 is a manifold of dimension one. One possible atlas is A = {(U , ϕ ), (U , ϕ )} where 1 1 1 2 1 y U1 = S \{(−1, 0)} ϕ1(x,y) = arctan x with − π < ϕ1(x,y) <π ϕ1 1 y U2 = S \{(1, 0)} ϕ2(x,y) = arctan x with 0 < ϕ2(x,y) < 2π U1 n n n+1 2 2 Example 3 (S ) The n–sphere S = x =(x1, ··· ,xn+1) ∈ IR x1 +···+xn+1 =1 n A U , ϕ , V ,ψ i n is a manifold of dimension . One possible atlas is 1 = ( i i) ( i i) 1 ≤ ≤ +1 where, for each 1 ≤ i ≤ n + 1, n Ui = (x1, ··· ,xn+1) ∈ S xi > 0 ϕi(x1, ··· ,xn+1)=(x1, ··· ,xi−1,xi+1, ··· ,xn+1) n Vi = (x1, ··· ,xn+1) ∈ S xi < 0 ψi(x1, ··· ,xn+1)=(x1, ··· ,xi−1,xi+1, ··· ,xn+1) n So both ϕi and ψi project onto IR , viewed as the hyperplane xi = 0. Another possible atlas is n n A2 = S \{(0, ··· , 0, 1)}, ϕ , S \{(0, ··· , 0, −1)},ψ where 2x1 2xn ϕ(x , ··· ,xn ) = , ··· , 1 +1 1−xn+1 1−xn+1 2x1 2xn ψ(x , ··· ,xn ) = , ··· , 1 +1 1+xn+1 1+xn+1 are the stereographic projections from the north and south poles, respectively.
    [Show full text]
  • An Introduction to Topology the Classification Theorem for Surfaces by E
    An Introduction to Topology An Introduction to Topology The Classification theorem for Surfaces By E. C. Zeeman Introduction. The classification theorem is a beautiful example of geometric topology. Although it was discovered in the last century*, yet it manages to convey the spirit of present day research. The proof that we give here is elementary, and its is hoped more intuitive than that found in most textbooks, but in none the less rigorous. It is designed for readers who have never done any topology before. It is the sort of mathematics that could be taught in schools both to foster geometric intuition, and to counteract the present day alarming tendency to drop geometry. It is profound, and yet preserves a sense of fun. In Appendix 1 we explain how a deeper result can be proved if one has available the more sophisticated tools of analytic topology and algebraic topology. Examples. Before starting the theorem let us look at a few examples of surfaces. In any branch of mathematics it is always a good thing to start with examples, because they are the source of our intuition. All the following pictures are of surfaces in 3-dimensions. In example 1 by the word “sphere” we mean just the surface of the sphere, and not the inside. In fact in all the examples we mean just the surface and not the solid inside. 1. Sphere. 2. Torus (or inner tube). 3. Knotted torus. 4. Sphere with knotted torus bored through it. * Zeeman wrote this article in the mid-twentieth century. 1 An Introduction to Topology 5.
    [Show full text]
  • The Sphere Project
    ;OL :WOLYL 7YVQLJ[ +XPDQLWDULDQ&KDUWHUDQG0LQLPXP 6WDQGDUGVLQ+XPDQLWDULDQ5HVSRQVH ;OL:WOLYL7YVQLJ[ 7KHULJKWWROLIHZLWKGLJQLW\ ;OL:WOLYL7YVQLJ[PZHUPUP[PH[P]L[VKL[LYTPULHUK WYVTV[LZ[HUKHYKZI`^OPJO[OLNSVIHSJVTT\UP[` /\THUP[HYPHU*OHY[LY YLZWVUKZ[V[OLWSPNO[VMWLVWSLHMMLJ[LKI`KPZHZ[LYZ /\THUP[HYPHU >P[O[OPZ/HUKIVVR:WOLYLPZ^VYRPUNMVYH^VYSKPU^OPJO [OLYPNO[VMHSSWLVWSLHMMLJ[LKI`KPZHZ[LYZ[VYLLZ[HISPZO[OLPY *OHY[LYHUK SP]LZHUKSP]LSPOVVKZPZYLJVNUPZLKHUKHJ[LK\WVUPU^H`Z[OH[ YLZWLJ[[OLPY]VPJLHUKWYVTV[L[OLPYKPNUP[`HUKZLJ\YP[` 4PUPT\T:[HUKHYKZ This Handbook contains: PU/\THUP[HYPHU (/\THUP[HYPHU*OHY[LY!SLNHSHUKTVYHSWYPUJPWSLZ^OPJO HUK YLÅLJ[[OLYPNO[ZVMKPZHZ[LYHMMLJ[LKWVW\SH[PVUZ 4PUPT\T:[HUKHYKZPU/\THUP[HYPHU9LZWVUZL 9LZWVUZL 7YV[LJ[PVU7YPUJPWSLZ *VYL:[HUKHYKZHUKTPUPT\TZ[HUKHYKZPUMV\YRL`SPMLZH]PUN O\THUP[HYPHUZLJ[VYZ!>H[LYZ\WWS`ZHUP[H[PVUHUKO`NPLUL WYVTV[PVU"-VVKZLJ\YP[`HUKU\[YP[PVU":OLS[LYZL[[SLTLU[HUK UVUMVVKP[LTZ"/LHS[OHJ[PVU;OL`KLZJYPILwhat needs to be achieved in a humanitarian response in order for disaster- affected populations to survive and recover in stable conditions and with dignity. ;OL:WOLYL/HUKIVVRLUQV`ZIYVHKV^ULYZOPWI`HNLUJPLZHUK PUKP]PK\HSZVMMLYPUN[OLO\THUP[HYPHUZLJ[VYH common language for working together towards quality and accountability in disaster and conflict situations ;OL:WOLYL/HUKIVVROHZHU\TILYVMºJVTWHUPVU Z[HUKHYKZ»L_[LUKPUNP[ZZJVWLPUYLZWVUZL[VULLKZ[OH[ OH]LLTLYNLK^P[OPU[OLO\THUP[HYPHUZLJ[VY ;OL:WOLYL7YVQLJ[^HZPUP[PH[LKPU I`HU\TILY VMO\THUP[HYPHU5.6ZHUK[OL9LK*YVZZHUK 9LK*YLZJLU[4V]LTLU[ ,+0;065 7KHB6SKHUHB3URMHFWBFRYHUBHQBLQGG The Sphere Project Humanitarian Charter and Minimum Standards in Humanitarian Response Published by: The Sphere Project Copyright@The Sphere Project 2011 Email: [email protected] Website : www.sphereproject.org The Sphere Project was initiated in 1997 by a group of NGOs and the Red Cross and Red Crescent Movement to develop a set of universal minimum standards in core areas of humanitarian response: the Sphere Handbook.
    [Show full text]
  • Quick Reference Guide - Ansi Z80.1-2015
    QUICK REFERENCE GUIDE - ANSI Z80.1-2015 1. Tolerance on Distance Refractive Power (Single Vision & Multifocal Lenses) Cylinder Cylinder Sphere Meridian Power Tolerance on Sphere Cylinder Meridian Power ≥ 0.00 D > - 2.00 D (minus cylinder convention) > -4.50 D (minus cylinder convention) ≤ -2.00 D ≤ -4.50 D From - 6.50 D to + 6.50 D ± 0.13 D ± 0.13 D ± 0.15 D ± 4% Stronger than ± 6.50 D ± 2% ± 0.13 D ± 0.15 D ± 4% 2. Tolerance on Distance Refractive Power (Progressive Addition Lenses) Cylinder Cylinder Sphere Meridian Power Tolerance on Sphere Cylinder Meridian Power ≥ 0.00 D > - 2.00 D (minus cylinder convention) > -3.50 D (minus cylinder convention) ≤ -2.00 D ≤ -3.50 D From -8.00 D to +8.00 D ± 0.16 D ± 0.16 D ± 0.18 D ± 5% Stronger than ±8.00 D ± 2% ± 0.16 D ± 0.18 D ± 5% 3. Tolerance on the direction of cylinder axis Nominal value of the ≥ 0.12 D > 0.25 D > 0.50 D > 0.75 D < 0.12 D > 1.50 D cylinder power (D) ≤ 0.25 D ≤ 0.50 D ≤ 0.75 D ≤ 1.50 D Tolerance of the axis Not Defined ° ° ° ° ° (degrees) ± 14 ± 7 ± 5 ± 3 ± 2 4. Tolerance on addition power for multifocal and progressive addition lenses Nominal value of addition power (D) ≤ 4.00 D > 4.00 D Nominal value of the tolerance on the addition power (D) ± 0.12 D ± 0.18 D 5. Tolerance on Prism Reference Point Location and Prismatic Power • The prismatic power measured at the prism reference point shall not exceed 0.33Δ or the prism reference point shall not be more than 1.0 mm away from its specified position in any direction.
    [Show full text]
  • Sphere Vs. 0°/45° a Discussion of Instrument Geometries And
    Sphere vs. 0°/45° ® Author - Timothy Mouw Sphere vs. 0°/45° A discussion of instrument geometries and their areas of application Introduction When purchasing a spectrophotometer for color measurement, one of the choices that must be made is the geometry of the instrument; that is, whether to buy a spherical or 0°/45° instrument. In this article, we will attempt to provide some information to assist you in determining which type of instrument is best suited to your needs. In order to do this, we must first understand the differences between these instruments. Simplified schematics of instrument geometries We will begin by looking at some simple schematic drawings of the two different geometries, spherical and 0°/45°. Figure 1 shows a drawing of the geometry used in a 0°/45° instrument. The illumination of the sample is from 0° (90° from the sample surface). This means that the specular or gloss angle (the angle at which the light is directly reflected) is also 0°. The fiber optic pick-ups are located at 45° from the specular angle. Figure 1 X-Rite World Headquarters © 1995 X-Rite, Incorporated Doc# CA00015a.doc (616) 534-7663 Fax (616) 534-8960 September 15, 1995 Figure 2 shows a drawing of the geometry used in an 8° diffuse sphere instrument. The sphere wall is lined with a highly reflective white substance and the light source is located on the rear of the sphere wall. A baffle prevents the light source from directly illuminating the sample, thus providing diffuse illumination. The sample is viewed at 8° from perpendicular which means that the specular or gloss angle is also 8° from perpendicular.
    [Show full text]
  • Recognizing Surfaces
    RECOGNIZING SURFACES Ivo Nikolov and Alexandru I. Suciu Mathematics Department College of Arts and Sciences Northeastern University Abstract The subject of this poster is the interplay between the topology and the combinatorics of surfaces. The main problem of Topology is to classify spaces up to continuous deformations, known as homeomorphisms. Under certain conditions, topological invariants that capture qualitative and quantitative properties of spaces lead to the enumeration of homeomorphism types. Surfaces are some of the simplest, yet most interesting topological objects. The poster focuses on the main topological invariants of two-dimensional manifolds—orientability, number of boundary components, genus, and Euler characteristic—and how these invariants solve the classification problem for compact surfaces. The poster introduces a Java applet that was written in Fall, 1998 as a class project for a Topology I course. It implements an algorithm that determines the homeomorphism type of a closed surface from a combinatorial description as a polygon with edges identified in pairs. The input for the applet is a string of integers, encoding the edge identifications. The output of the applet consists of three topological invariants that completely classify the resulting surface. Topology of Surfaces Topology is the abstraction of certain geometrical ideas, such as continuity and closeness. Roughly speaking, topol- ogy is the exploration of manifolds, and of the properties that remain invariant under continuous, invertible transforma- tions, known as homeomorphisms. The basic problem is to classify manifolds according to homeomorphism type. In higher dimensions, this is an impossible task, but, in low di- mensions, it can be done. Surfaces are some of the simplest, yet most interesting topological objects.
    [Show full text]
  • The Real Projective Spaces in Homotopy Type Theory
    The real projective spaces in homotopy type theory Ulrik Buchholtz Egbert Rijke Technische Universität Darmstadt Carnegie Mellon University Email: [email protected] Email: [email protected] Abstract—Homotopy type theory is a version of Martin- topology and homotopy theory developed in homotopy Löf type theory taking advantage of its homotopical models. type theory (homotopy groups, including the fundamen- In particular, we can use and construct objects of homotopy tal group of the circle, the Hopf fibration, the Freuden- theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key thal suspension theorem and the van Kampen theorem, players in homotopy theory, as certain higher inductive types for example). Here we give an elementary construction in homotopy type theory. The classical definition of RPn, in homotopy type theory of the real projective spaces as the quotient space identifying antipodal points of the RPn and we develop some of their basic properties. n-sphere, does not translate directly to homotopy type theory. R n In classical homotopy theory the real projective space Instead, we define P by induction on n simultaneously n with its tautological bundle of 2-element sets. As the base RP is either defined as the space of lines through the + case, we take RP−1 to be the empty type. In the inductive origin in Rn 1 or as the quotient by the antipodal action step, we take RPn+1 to be the mapping cone of the projection of the 2-element group on the sphere Sn [4].
    [Show full text]
  • The Sphere Packing Problem in Dimension 8 Arxiv:1603.04246V2
    The sphere packing problem in dimension 8 Maryna S. Viazovska April 5, 2017 8 In this paper we prove that no packing of unit balls in Euclidean space R has density greater than that of the E8-lattice packing. Keywords: Sphere packing, Modular forms, Fourier analysis AMS subject classification: 52C17, 11F03, 11F30 1 Introduction The sphere packing constant measures which portion of d-dimensional Euclidean space d can be covered by non-overlapping unit balls. More precisely, let R be the Euclidean d vector space equipped with distance k · k and Lebesgue measure Vol(·). For x 2 R and d r 2 R>0 we denote by Bd(x; r) the open ball in R with center x and radius r. Let d X ⊂ R be a discrete set of points such that kx − yk ≥ 2 for any distinct x; y 2 X. Then the union [ P = Bd(x; 1) x2X d is a sphere packing. If X is a lattice in R then we say that P is a lattice sphere packing. The finite density of a packing P is defined as Vol(P\ Bd(0; r)) ∆P (r) := ; r > 0: Vol(Bd(0; r)) We define the density of a packing P as the limit superior ∆P := lim sup ∆P (r): r!1 arXiv:1603.04246v2 [math.NT] 4 Apr 2017 The number be want to know is the supremum over all possible packing densities ∆d := sup ∆P ; d P⊂R sphere packing 1 called the sphere packing constant. For which dimensions do we know the exact value of ∆d? Trivially, in dimension 1 we have ∆1 = 1.
    [Show full text]
  • Hyperbolic Geometry
    Flavors of Geometry MSRI Publications Volume 31,1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a This work was supported in part by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry.
    [Show full text]
  • Sphere Packing, Lattice Packing, and Related Problems
    Sphere packing, lattice packing, and related problems Abhinav Kumar Stony Brook April 25, 2018 Sphere packings Definition n A sphere packing in R is a collection of spheres/balls of equal size which do not overlap (except for touching). The density of a sphere packing is the volume fraction of space occupied by the balls. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ In dimension 1, we can achieve density 1 by laying intervals end to end. In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´oth1940] Sphere packing problem n Problem: Find a/the densest sphere packing(s) in R . In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´oth1940] Sphere packing problem n Problem: Find a/the densest sphere packing(s) in R . In dimension 1, we can achieve density 1 by laying intervals end to end. Sphere packing problem n Problem: Find a/the densest sphere packing(s) in R . In dimension 1, we can achieve density 1 by laying intervals end to end. In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´oth1940] Sphere packing problem II In dimension 3, the best possible way is to stack layers of the solution in 2 dimensions. This is Kepler's conjecture, now a theorem of Hales and collaborators. mmm m mmm m There are infinitely (in fact, uncountably) many ways of doing this! These are the Barlow packings. Face centered cubic packing Image: Greg A L (Wikipedia), CC BY-SA 3.0 license But (until very recently!) no proofs. In very high dimensions (say ≥ 1000) densest packings are likely to be close to disordered.
    [Show full text]
  • 17 Measure Concentration for the Sphere
    17 Measure Concentration for the Sphere In today’s lecture, we will prove the measure concentration theorem for the sphere. Recall that this was one of the vital steps in the analysis of the Arora-Rao-Vazirani approximation algorithm for sparsest cut. Most of the material in today’s lecture is adapted from Matousek’s book [Mat02, chapter 14] and Keith Ball’s lecture notes on convex geometry [Bal97]. n n−1 Notation: We will use the notation Bn to denote the ball of unit radius in R and S n to denote the sphere of unit radius in R . Let µ denote the normalized measure on the unit sphere (i.e., for any measurable set S ⊆ Sn−1, µ(A) denotes the ratio of the surface area of µ to the entire surface area of the sphere Sn−1). Recall that the n-dimensional volume of a ball n n n of radius r in R is given by the formula Vol(Bn) · r = vn · r where πn/2 vn = n Γ 2 + 1 Z ∞ where Γ(x) = tx−1e−tdt 0 n−1 The surface area of the unit sphere S is nvn. Theorem 17.1 (Measure Concentration for the Sphere Sn−1) Let A ⊆ Sn−1 be a mea- n−1 surable subset of the unit sphere S such that µ(A) = 1/2. Let Aδ denote the δ-neighborhood n−1 n−1 of A in S . i.e., Aδ = {x ∈ S |∃z ∈ A, ||x − z||2 ≤ δ}. Then, −nδ2/2 µ(Aδ) ≥ 1 − 2e . Thus, the above theorem states that if A is any set of measure 0.5, taking a step of even √ O (1/ n) around A covers almost 99% of the entire sphere.
    [Show full text]
  • CIRCLE and SPHERE GEOMETRY. [July
    464 CIRCLE AND SPHERE GEOMETRY. [July, CIRCLE AND SPHERE GEOMETRY. A Treatise on the Circle and the Sphere. By JULIAN LOWELL COOLIDGE. Oxford, the Clarendon Press, 1916. 8vo. 603 pp. AN era of systematizing and indexing produces naturally a desire for brevity. The excess of this desire is a disease, and its remedy must be treatises written, without diffuseness indeed, but with the generous temper which will find room and give proper exposition to true works of art. The circle and the sphere have furnished material for hundreds of students and writers, but their work, even the most valuable, has been hitherto too scattered for appraisement by the average student. Cyclopedias, while indispensable, are tan­ talizing. Circles and spheres have now been rescued from a state of dispersion by Dr. Coolidge's comprehensive lectures. Not a list of results, but a well digested account of theories and methods, sufficiently condensed by judicious choice of position and sequence, is what he has given us for leisurely study and enjoyment. The reader whom the author has in mind might be, well enough, at the outset a college sophomore, or a teacher who has read somewhat beyond the geometry expected for en­ trance. Beyond the sixth chapter, however, or roughly the middle of the book, he must be ready for persevering study in specialized fields. The part most read is therefore likely to be the first four chapters, and a short survey of these is what we shall venture here. After three pages of definitions and conventions the author calls "A truce to these preliminaries!" and plunges into inversion.
    [Show full text]