Isometry Groups of Homogeneous Spaces with Positive Sectional Curvature
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[Math.GT] 9 Oct 2020 Symmetries of Hyperbolic 4-Manifolds
Symmetries of hyperbolic 4-manifolds Alexander Kolpakov & Leone Slavich R´esum´e Pour chaque groupe G fini, nous construisons des premiers exemples explicites de 4- vari´et´es non-compactes compl`etes arithm´etiques hyperboliques M, `avolume fini, telles M G + M G que Isom ∼= , ou Isom ∼= . Pour y parvenir, nous utilisons essentiellement la g´eom´etrie de poly`edres de Coxeter dans l’espace hyperbolique en dimension quatre, et aussi la combinatoire de complexes simpliciaux. C¸a nous permet d’obtenir une borne sup´erieure universelle pour le volume minimal d’une 4-vari´et´ehyperbolique ayant le groupe G comme son groupe d’isom´etries, par rap- port de l’ordre du groupe. Nous obtenons aussi des bornes asymptotiques pour le taux de croissance, par rapport du volume, du nombre de 4-vari´et´es hyperboliques ayant G comme le groupe d’isom´etries. Abstract In this paper, for each finite group G, we construct the first explicit examples of non- M M compact complete finite-volume arithmetic hyperbolic 4-manifolds such that Isom ∼= G + M G , or Isom ∼= . In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic 4-space, on one hand, and the combinatorics of simplicial complexes, on the other. This allows us to obtain a universal upper bound on the minimal volume of a hyperbolic 4-manifold realising a given finite group G as its isometry group in terms of the order of the group. We also obtain asymptotic bounds for the growth rate, with respect to volume, of the number of hyperbolic 4-manifolds having a finite group G as their isometry group. -
Chapter 13 Curvature in Riemannian Manifolds
Chapter 13 Curvature in Riemannian Manifolds 13.1 The Curvature Tensor If (M, , )isaRiemannianmanifoldand is a connection on M (that is, a connection on TM−), we− saw in Section 11.2 (Proposition 11.8)∇ that the curvature induced by is given by ∇ R(X, Y )= , ∇X ◦∇Y −∇Y ◦∇X −∇[X,Y ] for all X, Y X(M), with R(X, Y ) Γ( om(TM,TM)) = Hom (Γ(TM), Γ(TM)). ∈ ∈ H ∼ C∞(M) Since sections of the tangent bundle are vector fields (Γ(TM)=X(M)), R defines a map R: X(M) X(M) X(M) X(M), × × −→ and, as we observed just after stating Proposition 11.8, R(X, Y )Z is C∞(M)-linear in X, Y, Z and skew-symmetric in X and Y .ItfollowsthatR defines a (1, 3)-tensor, also denoted R, with R : T M T M T M T M. p p × p × p −→ p Experience shows that it is useful to consider the (0, 4)-tensor, also denoted R,givenby R (x, y, z, w)= R (x, y)z,w p p p as well as the expression R(x, y, y, x), which, for an orthonormal pair, of vectors (x, y), is known as the sectional curvature, K(x, y). This last expression brings up a dilemma regarding the choice for the sign of R. With our present choice, the sectional curvature, K(x, y), is given by K(x, y)=R(x, y, y, x)but many authors define K as K(x, y)=R(x, y, x, y). Since R(x, y)isskew-symmetricinx, y, the latter choice corresponds to using R(x, y)insteadofR(x, y), that is, to define R(X, Y ) by − R(X, Y )= + . -
3. Sectional Curvature of Lorentzian Manifolds. 1
3. SECTIONAL CURVATURE OF LORENTZIAN MANIFOLDS. 1. Sectional curvature, the Jacobi equation and \tidal stresses". The (3,1) Riemann curvature tensor has the same definition in the rie- mannian and Lorentzian cases: R(X; Y )Z = rX rY Z − rY rX Z − r[X;Y ]Z: If f(t; s) is a parametrized 2-surface in M (immersion) and W (t; s) is a vector field on M along f, we have the Ricci formula: D DW D DW − = R(@ f; @ f)W: @t @s @s @t t s For a variation f(t; s) = γs(t) of a geodesic γ(t) (with variational vector field J(t) = @sfjs=0 along γ(t)) this leads to the Jacobi equation for J: D2J + R(J; γ_ )_γ = 0: dt2 ? The Jacobi operator is the self-adjoint operator on (γ _ ) : Rp[v] = Rp(v; γ_ )_γ. When γ is a timelike geodesic (the worldline of a free-falling massive particle) the physical interpretation of J is the relative displacement (space- like) vector of a neighboring free-falling particle, while the second covariant 00 derivative J represents its relative acceleration. The Jacobi operator Rp gives the \tidal stresses" in terms of the position vector J. In the Lorentzian case, the sectional curvature is defined only for non- degenerate two-planes Π ⊂ TpM. Definition. Let Π = spanfX; Y g be a non-degenerate two-dimensional subspace of TpM. The sectional curvature σXY = σΠ is the real number σ defined by: hR(X; Y )Y; Xi = σhX ^ Y; X ^ Y i: Remark: by a result of J. -
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. -
Sufficient Conditions for Periodicity of a Killing Vector Field Walter C
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 38, Number 3, May 1973 SUFFICIENT CONDITIONS FOR PERIODICITY OF A KILLING VECTOR FIELD WALTER C. LYNGE Abstract. Let X be a complete Killing vector field on an n- dimensional connected Riemannian manifold. Our main purpose is to show that if X has as few as n closed orbits which are located properly with respect to each other, then X must have periodic flow. Together with a known result, this implies that periodicity of the flow characterizes those complete vector fields having all orbits closed which can be Killing with respect to some Riemannian metric on a connected manifold M. We give a generalization of this characterization which applies to arbitrary complete vector fields on M. Theorem. Let X be a complete Killing vector field on a connected, n- dimensional Riemannian manifold M. Assume there are n distinct points p, Pit' " >Pn-i m M such that the respective orbits y, ylt • • • , yn_x of X through them are closed and y is nontrivial. Suppose further that each pt is joined to p by a unique minimizing geodesic and d(p,p¡) = r¡i<.D¡2, where d denotes distance on M and D is the diameter of y as a subset of M. Let_wx, • ■ ■ , wn_j be the unit vectors in TVM such that exp r¡iwi=pi, i=l, • ■ • , n— 1. Assume that the vectors Xv, wx, ■• • , wn_x span T^M. Then the flow cptof X is periodic. Proof. Fix /' in {1, 2, •••,«— 1}. We first show that the orbit yt does not lie entirely in the sphere Sj,(r¡¿) of radius r¡i about p. -
Global Properties of Asymptotically De Sitter and Anti De Sitter Spacetimes
Global properties of asymptotically de Sitter and Anti de Sitter spacetimes Didier A. Solis May, 17, 2006 arXiv:1803.01171v1 [gr-qc] 3 Mar 2018 Ad Majorem Dei Gloriam. To all those who seek to fulfill God's will in their lives. iii ACKNOWLEDGEMENTS I would like to express my deepest gratitude to: • My advisor Dr. Gregory Galloway, for his constant support and help. This dissertation would not had seen the light of day without his continued encour- agement and guidance. • Dr. N. Saveliev, Dr. M. Cai and Dr. O. Alvarez for their valuable comments and thorough revision of this work. • The Faculty and staff at the Math Department for all the knowledge and affec- tion they shared with me. • CONACYT (Consejo Nacional de Ciencia y Tecnolog´ıa)for the financial support granted to the author. • My family: Douglas, Rosalinda, Douglas Jr, Rosil´uand Josu´efor all their prayers and unconditional love. They are indeed the greatest blessing in my life. • All my friends, especially Guille, for always being there for me to share the struggles and joys of life. • To the CSA gang, for being a living example of faith lived in love and hope. • Last but by no means least, to God, without whom there is no math, laughter or music. iv Contents 1 Introduction 1 1.1 Lorentz vector spaces . 1 1.2 Spacetimes . 6 1.2.1 The Levi-Civita connection . 7 1.2.2 Covariant derivative . 11 1.2.3 Geodesics . 12 1.3 Causal theory . 15 1.3.1 Definitions . 15 1.3.2 Global causality conditions . -
Lecture 8: the Sectional and Ricci Curvatures
LECTURE 8: THE SECTIONAL AND RICCI CURVATURES 1. The Sectional Curvature We start with some simple linear algebra. As usual we denote by ⊗2(^2V ∗) the set of 4-tensors that is anti-symmetric with respect to the first two entries and with respect to the last two entries. Lemma 1.1. Suppose T 2 ⊗2(^2V ∗), X; Y 2 V . Let X0 = aX +bY; Y 0 = cX +dY , then T (X0;Y 0;X0;Y 0) = (ad − bc)2T (X; Y; X; Y ): Proof. This follows from a very simple computation: T (X0;Y 0;X0;Y 0) = T (aX + bY; cX + dY; aX + bY; cX + dY ) = (ad − bc)T (X; Y; aX + bY; cX + dY ) = (ad − bc)2T (X; Y; X; Y ): 1 Now suppose (M; g) is a Riemannian manifold. Recall that 2 g ^ g is a curvature- like tensor, such that 1 g ^ g(X ;Y ;X ;Y ) = hX ;X ihY ;Y i − hX ;Y i2: 2 p p p p p p p p p p 1 Applying the previous lemma to Rm and 2 g ^ g, we immediately get Proposition 1.2. The quantity Rm(Xp;Yp;Xp;Yp) K(Xp;Yp) := 2 hXp;XpihYp;Ypi − hXp;Ypi depends only on the two dimensional plane Πp = span(Xp;Yp) ⊂ TpM, i.e. it is independent of the choices of basis fXp;Ypg of Πp. Definition 1.3. We will call K(Πp) = K(Xp;Yp) the sectional curvature of (M; g) at p with respect to the plane Πp. Remark. The sectional curvature K is NOT a function on M (for dim M > 2), but a function on the Grassmann bundle Gm;2(M) of M. -
UCLA Electronic Theses and Dissertations
UCLA UCLA Electronic Theses and Dissertations Title Shapes of Finite Groups through Covering Properties and Cayley Graphs Permalink https://escholarship.org/uc/item/09b4347b Author Yang, Yilong Publication Date 2017 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles Shapes of Finite Groups through Covering Properties and Cayley Graphs A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Yilong Yang 2017 c Copyright by Yilong Yang 2017 ABSTRACT OF THE DISSERTATION Shapes of Finite Groups through Covering Properties and Cayley Graphs by Yilong Yang Doctor of Philosophy in Mathematics University of California, Los Angeles, 2017 Professor Terence Chi-Shen Tao, Chair This thesis is concerned with some asymptotic and geometric properties of finite groups. We shall present two major works with some applications. We present the first major work in Chapter 3 and its application in Chapter 4. We shall explore the how the expansions of many conjugacy classes is related to the representations of a group, and then focus on using this to characterize quasirandom groups. Then in Chapter 4 we shall apply these results in ultraproducts of certain quasirandom groups and in the Bohr compactification of topological groups. This work is published in the Journal of Group Theory [Yan16]. We present the second major work in Chapter 5 and 6. We shall use tools from number theory, combinatorics and geometry over finite fields to obtain an improved diameter bounds of finite simple groups. We also record the implications on spectral gap and mixing time on the Cayley graphs of these groups. -
Curves of Constant Curvature and Torsion in the the 3-Sphere
CURVES OF CONSTANT CURVATURE AND TORSION IN THE 3-SPHERE DEBRAJ CHAKRABARTI, RAHUL SAHAY, AND JARED WILLIAMS Abstract. We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global behavior may be periodic or the curve may be dense in a Clifford torus embedded in the three- sphere. This behavior is very different from that of helices in three-dimensional Euclidean space, which also have constant curvature and torsion. 1. Introduction Let pM; x; yq be a three dimensional Riemannian manifold, let I Ď R be an open interval, and let γ : I Ñ M be a smooth curve in M, which we assume to be parametrized by the arc length t. It is well-known that the local geometry of γ is characterized by the curvature κ and the torsion τ. These are functions defined along γ and are the coefficients of the well-known Frenet-Serret formulas [2, Vol. IV, pp. 21-23]: D Tptq “ κptqNptq dt D Nptq “ ´κptqTptq `τptqBptq (1.1) dt D Bptq “ ´ τptqNptq; dt where the orthogonal unit vector fields T; N; B, with T “ γ1, along the unit-speed curve D γ, constitute its Frenet frame and dt denotes covariant differentiation along γ with respect to the arc length t. We will assume that each of the functions κ and τ is either nowhere zero or vanishes identically. Additionally, if κ is identically zero, then τ is also taken to be identically zero. -
A Review on Metric Symmetries Used in Geometry and Physics K
A Review on Metric Symmetries used in Geometry and Physics K. L. Duggala aUniversity of Windsor, Windsor, Ontario N9B3P4, Canada, E-mail address: [email protected] This is a review paper of the essential research on metric (Killing, homothetic and conformal) symmetries of Riemannian, semi-Riemannian and lightlike manifolds. We focus on the main characterization theorems and exhibit the state of art as it now stands. A sketch of the proofs of the most important results is presented together with sufficient references for related results. 1. Introduction The measurement of distances in a Euclidean space R3 is represented by the distance element ds2 = dx2 + dy2 + dz2 with respect to a rectangular coordinate system (x, y, z). Back in 1854, Riemann generalized this idea for n-dimensional spaces and he defined element of length by means of a quadratic 2 i j differential form ds = gijdx dx on a differentiable manifold M, where the coefficients gij are functions of the coordinates system (x1, . , xn), which represent a symmetric tensor field g of type (0, 2). Since then much of the subsequent differential geometry was developed on a real smooth manifold (M, g), called a Riemannian manifold, where g is a positive definite metric tensor field. Marcel Berger’s book [1] includes the major developments of Riemannian geome- try since 1950, citing the works of differential geometers of that time. On the other hand, we refer standard book of O’Neill [2] on the study of semi-Riemannian geometry where the metric g is indefinite and, in particular, Beem-Ehrlich [3] on the global Lorentzian geometry used in relativity. -
Differential Geometry Lecture 18: Curvature
Differential geometry Lecture 18: Curvature David Lindemann University of Hamburg Department of Mathematics Analysis and Differential Geometry & RTG 1670 14. July 2020 David Lindemann DG lecture 18 14. July 2020 1 / 31 1 Riemann curvature tensor 2 Sectional curvature 3 Ricci curvature 4 Scalar curvature David Lindemann DG lecture 18 14. July 2020 2 / 31 Recap of lecture 17: defined geodesics in pseudo-Riemannian manifolds as curves with parallel velocity viewed geodesics as projections of integral curves of a vector field G 2 X(TM) with local flow called geodesic flow obtained uniqueness and existence properties of geodesics constructed the exponential map exp : V ! M, V neighbourhood of the zero-section in TM ! M showed that geodesics with compact domain are precisely the critical points of the energy functional used the exponential map to construct Riemannian normal coordinates, studied local forms of the metric and the Christoffel symbols in such coordinates discussed the Hopf-Rinow Theorem erratum: codomain of (x; v) as local integral curve of G is d'(TU), not TM David Lindemann DG lecture 18 14. July 2020 3 / 31 Riemann curvature tensor Intuitively, a meaningful definition of the term \curvature" for 3 a smooth surface in R , written locally as a graph of a smooth 2 function f : U ⊂ R ! R, should involve the second partial derivatives of f at each point. How can we find a coordinate- 3 free definition of curvature not just for surfaces in R , which are automatically Riemannian manifolds by restricting h·; ·i, but for all pseudo-Riemannian manifolds? Definition Let (M; g) be a pseudo-Riemannian manifold with Levi-Civita connection r. -
Structure of Isometry Group of Bilinear Spaces ୋ
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 416 (2006) 414–436 www.elsevier.com/locate/laa Structure of isometry group of bilinear spaces ୋ Dragomir Ž. Ðokovic´ Department of Pure Mathematics, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 Received 5 November 2004; accepted 27 November 2005 Available online 18 January 2006 Submitted by H. Schneider Abstract We describe the structure of the isometry group G of a finite-dimensional bilinear space over an algebra- ically closed field of characteristic not two. If the space has no indecomposable degenerate orthogonal summands of odd dimension, it admits a canonical orthogonal decomposition into primary components and G is isomorphic to the direct product of the isometry groups of the primary components. Each of the latter groups is shown to be isomorphic to the centralizer in some classical group of a nilpotent element in the Lie algebra of that group. In the general case, the description of G is more complicated. We show that G is a semidirect product of a normal unipotent subgroup K with another subgroup which, in its turn, is a direct product of a group of the type described in the previous paragraph and another group H which we can describe explicitly. The group H has a Levi decomposition whose Levi factor is a direct product of several general linear groups of various degrees. We obtain simple formulae for the dimensions of H and K. © 2005 Elsevier Inc. All rights reserved. Keywords: Bilinear space; Isometry group; Asymmetry; Gabriel block; Toeplitz matrix 1.