DOCSLIB.ORG

612 CLASS LECTURE: HYPERBOLIC GEOMETRY 1. Conformal Metrics As a Vector Space, C Has a Canonical Norm, the Same As the Standard

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
612 CLASS LECTURE: HYPERBOLIC GEOMETRY 1. Conformal Metrics As a Vector Space, C Has a Canonical Norm, the Same As the Standard 612 CLASS LECTURE: HYPERBOLIC GEOMETRY JOSHUA P. BOWMAN 1. Conformal metrics As a vector space, C has a canonical norm, the same as the standard R2 norm. Denote this jdzj|one should think of dz as the identity map on C: ζ 7! ζ. This also means C has a standard norm as the tangent space to itself at any of its points. We will consistently use Roman letters (e.g., z) to represent a point in C when we're thinking of it as a topological ◦ space, and Greek letters (e.g., ζ) to represent a point in C thought of as a vector. Let U ⊂ C be a plane domain (non-empty, connected, but not necessarily simply connected). Definition 1.1. A conformal metric on U is a metric of the form ρ jdzj, where ρ is a smooth, positive function on U. We will also use ρ to denote the metric itself, not just the function. Almost always, \conformal" is used with respect to something else; here, we mean that angles in a conformal metric are the same as those in the background Euclidean metric. In other words, the unit ball of the norm in the tangent space at z is invariant under (Euclidean) rotations. Example 1.2. On U = C n f0g, ρ = jdzj=jzj is a conformal metric. U is preserved by multiplication by a non-zero complex number a; call such a map ma. The derivative of ma is again multiplication by a. Let z 2 U and let ζ 2 Tz(U) = C. The image of z by ma is az, and the image of ζ by the derivative is aζ 2 Taz(U). Hence jaζj jajjζj jζj ρ(az)jaζj = = = = ρ(z)jζj; jazj jajjzj jzj that is, the metric is preserved by ma. Definition 1.3. If ρU is a conformal metric on U and ρV is a conformal metric on V , then a conformal diffeomorphism f : U ! V is called an isometry if it sends tangent vectors to tangent vectors of the same length, i.e., 0 ρV (f(z))jf (z)ζj = ρU (z)jζj for all z 2 U, ζ 2 C. Definition 1.4. Let D be the open unit disk in C. The Poincar´emetric (or hyperbolic metric) on D is the conformal metric 2 jdzj ρ = : D 1 − jzj2 Theorem 1.5. Up to a global scaling factor, ρD is the unique conformal metric on D that is invariant under the action of Aut(D). Date: 13 March 2008. 1 Proof. Recall that Schwarz's Lemma implies that all automorphisms of D have the form z − a f (z) = eiθ ; θ 2 ; a 2 : θ;a 1 − az R D Suppose ρ is any conformal metric on D invariant under Aut(D). If z 2 D is any point and 0 ζ 2 Tz(D) is any tangent vector, then f0;z : w 7! (w − z)=(1 − zw) sends z to 0, and f0;z sends ζ ζ 7! f 0 (z)ζ = 2 T : 0;z 1 − jzj2 0D By the invariance of ρ, we must therefore have jζj ρ(z)jζj = ρ(0)jf 0 (z)ζj = ρ(0) : 0;z 1 − jzj2 Which shows that ρ is a multiple of ρD. Conversely, ρD has just been shown to be invariant iθ under f0;a for all a, and it is clearly invariant under multiplication by e , hence it is invariant under all of Aut(D). Corollary 1.6. The isometry group of ρD is precisely Aut(D). Proof. Immediate, as previously observed, from Schwarz's Lemma. We can determine a lot about the geometry of the metric space (D; ρD) just from the our 1 knowledge of ρD and Aut(D). Given any smooth (C ) curve γ :[a; b] ! D, its Poincar´e length is Z b 0 `D(γ) = ρD(γ(t)) jγ (t)j dt: a Using the definition of a geodesic as a \locally length-minimizing" curve, we can then com- pute the geodesics in D. Let z 2 D − f0g; we want to find a geodesic path from 0 to z. Because ρD is invariant under rotations, we can assume z lies in (0; 1). Let γ :[a; b] ! D be any path from 0 to z. The composition of γ with the rotational projection (r; θ) 7! (r; 0) is again a smooth path jγj whose radial component is the same as that of γ, but whose rotational component is zero; hence `D(jγj) ≤ `D(γ). Thus the geodesic from 0 to z is the segment [0; z]. The complete geodesics through 0 are therefore the diameters of D. To find the geodesic path between z1 and z2, we recall that we can send z1 to 0 by an isometry, find the geodesic from 0 to the image of z2, and reverse the isometry. Because all of the elements of Aut(D) are M¨obiustransformations, which are conformal and send circles to circles, we conclude that the geodesic from z1 to z2 is the arc of the circle passing through 1 z1 and z2 and orthogonal to @D = S . (If these properties of M¨obiustransformations are unfamiliar to you, then work through Exercise 30 in Chapter 14 of [4].) Definition 1.7. Let H be the open upper half-plane in C. The Poincar´emetric (or hyperbolic metric) on H is the conformal metric jdzj ρ = ; y = Im z: H y 2 1 As subsets of CP = C [ f1g, we know that D and H are conformally equivalent, via the M¨obiustransformations 1 + z ' : z 7! i ! ; 1 − z D H w − i : w 7! ! : w + i H D We want to show that, in fact, these maps are isometries. We'll do this by \pulling back" ρH via ' and showing that the result coincides with ρD. (We won't formally define \pull-back", but it basically means cooking up a metric so that a diffeomorphism becomes an isometry. The definition should be clear after an example.) Given z 2 D, ζ 2 C, we have 2 0 1 2i j1 − zj 2 jζj ρ ('(z))j' (z)ζj = ζ = = ρ (z)jζj: H Im '(z) (1 − z)2 1 − jzj2 j1 − zj2 D Definition 1.8. We call any smooth Riemannian manifold that is isometric to (D; ρD) a model of the hyperbolic plane. Definition 1.9. If U is any simply connected domain in C, not equal to all of C, then its Poincar´emetric ρU is the conformal metric obtained by pulling back the Poincar´emetric on D via the Riemann map. (Note: this is well-defined because the Riemann map is defined up to an element of Aut(D).) Often, pictures are easier to draw in H than in D, because the geodesics are simply the circles and lines in C orthogonal to the real axis (again, this follows from properties of M¨obiustransformations). Also, the isometry group in this model simply becomes PSL2(R) ⊂ PSL2(C), because these are the M¨obiustransformations that preserve the real axis. This model distinguishes a single point 1 on the boundary, as we occasionally wish to do. Other models are also useful. For example, if we wanted to distinguish two points on the boundary, we could use the band model. Let B be the infinite strip B := fz 2 C j jIm zj < πg ; and define its Poincar´emetric by jdzj ρ = ; y = Im z: B cos y Then all translations in the x-direction are isometries: moreover, they preserve the two boundary points ±∞ and the geodesic between them. The remaining geodesics in this model are much more difficult to describe, however, except to say that they meet the boundary orthogonally (or have one end limiting to ±∞). Exercise 1. (1) Show that the map z 7! ez is an isometry from B with the conformal metric jdzj to C n (−∞; 0] with the conformal metric jdzj=jzj. [Note: the same calculation shows that the covering map z 7! ez is a local isometry from (C; jdzj) to (C n f0g; jdzj=jzj).] (2) Show that (B; ρB) is isometric to (D; ρD). (3) Find the Poincar´emetric on C n [0; 1). (4) Using the idea of \local isometry" from (1), find the Poincar´emetric on D n f0g. 3 2. Curvature The most geometric way to describe the qualitative difference between conformal metrics is via their curvature. Roughly speaking, we want to compare how the area of a circle grows in terms of its radius, and compare this to the Euclidean growth rate πr2. More precisely, we define the following. Definition 2.1. Let ρ jdzj be a conformal metric on U ⊂ C. Then the Gaussian curvature, or simply curvature, of ρ jdzj at z 2 U is ∆ log ρ(z) K (z) = − ; ρ ρ2(z) where ∆ is the standard Laplacian @2=@x2 + @2=@y2. Here's an attempt at a geometric explanation, somewhat hand-wavey for now: ∆/ρ2 = divρ2 gradρ is the Laplace{Beltrami operator for (U; ρ), hence is defined intrinsically. Note that the volume form on each tangent space to points in U is given by ρ2 dx ^ dy. ρx • gradρ log ρ = ρ measures the change in ρ with respect to itself ; the ρ-length of ρy 2p 2 2 this vector is ρ ρx + ρy , which takes into account both the current volume form and how fast it's changing. ρx • divρ2 ρ measures how the volume form changes when moving along flow lines ρy of the gradient of log ρ, i.e., in the direction that the metric is increasing most quickly.
Load more

Recommended publications
  • MAT 362 at Stony Brook, Spring 2011
    MAT 362 Differential Geometry, Spring 2011 Instructors' contact information Course information Take-home exam Take-home final exam, due Thursday, May 19, at 2:15 PM. Please read the directions carefully. Handouts Overview of final projects pdf Notes on differentials of C1 maps pdf tex Notes on dual spaces and the spectral theorem pdf tex Notes on solutions to initial value problems pdf tex Topics and homework assignments Assigned homework problems may change up until a week before their due date. Assignments are taken from texts by Banchoff and Lovett (B&L) and Shifrin (S), unless otherwise noted. Topics and assignments through spring break (April 24) Solutions to first exam Solutions to second exam Solutions to third exam April 26-28: Parallel transport, geodesics. Read B&L 8.1-8.2; S2.4. Homework due Tuesday May 3: B&L 8.1.4, 8.2.10 S2.4: 1, 2, 4, 6, 11, 15, 20 Bonus: Figure out what map projection is used in the graphic here. (A Facebook account is not needed.) May 3-5: Local and global Gauss-Bonnet theorem. Read B&L 8.4; S3.1. Homework due Tuesday May 10: B&L: 8.1.8, 8.4.5, 8.4.6 S3.1: 2, 4, 5, 8, 9 Project assignment: Submit final version of paper electronically to me BY FRIDAY MAY 13. May 10: Hyperbolic geometry. Read B&L 8.5; S3.2. No homework this week. Third exam: May 12 (in class) Take-home exam: due May 19 (at presentation of final projects) Instructors for MAT 362 Differential Geometry, Spring 2011 Joshua Bowman (main instructor) Office: Math Tower 3-114 Office hours: Monday 4:00-5:00, Friday 9:30-10:30 Email: joshua dot bowman at gmail dot com Lloyd Smith (grader Feb.
    [Show full text]
  • 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 )= + .
    [Show full text]
  • 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.
    [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]
  • 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 .
    [Show full text]
  • Basics of the Differential Geometry of Surfaces
    Chapter 20 Basics of the Differential Geometry of Surfaces 20.1 Introduction The purpose of this chapter is to introduce the reader to some elementary concepts of the differential geometry of surfaces. Our goal is rather modest: We simply want to introduce the concepts needed to understand the notion of Gaussian curvature, mean curvature, principal curvatures, and geodesic lines. Almost all of the material presented in this chapter is based on lectures given by Eugenio Calabi in an upper undergraduate differential geometry course offered in the fall of 1994. Most of the topics covered in this course have been included, except a presentation of the global Gauss–Bonnet–Hopf theorem, some material on special coordinate systems, and Hilbert’s theorem on surfaces of constant negative curvature. What is a surface? A precise answer cannot really be given without introducing the concept of a manifold. An informal answer is to say that a surface is a set of points in R3 such that for every point p on the surface there is a small (perhaps very small) neighborhood U of p that is continuously deformable into a little flat open disk. Thus, a surface should really have some topology. Also,locally,unlessthe point p is “singular,” the surface looks like a plane. Properties of surfaces can be classified into local properties and global prop- erties.Intheolderliterature,thestudyoflocalpropertieswascalled geometry in the small,andthestudyofglobalpropertieswascalledgeometry in the large.Lo- cal properties are the properties that hold in a small neighborhood of a point on a surface. Curvature is a local property. Local properties canbestudiedmoreconve- niently by assuming that the surface is parametrized locally.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Geometry and Entropies in a Fixed Conformal Class on Surfaces
    GEOMETRY AND ENTROPIES IN A FIXED CONFORMAL CLASS ON SURFACES THOMAS BARTHELME´ AND ALENA ERCHENKO Abstract. We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a thick-thin decomposition, and precompactness for the considered class of metrics. Also, we extend some of the results to metrics of fixed total area in a fixed conformal class with no focal points and with some integral bounds on the positive part of the Gaussian curvature. 1. Introduction When M is a fixed surface, there has been a long history of studying how the geometric or dynamical data (e.g., the Laplace spectrum, systole, entropies or Lyapunov exponents of the geodesic flow) varies when one varies the metric on M, possibly inside a particular class. In [BE20], we studied these questions in a class of metrics that seemed to have been overlooked: the family of non-positively curved metrics within a fixed conformal class. In this article, we prove several conjectures made in [BE20], as well as give a fairly complete, albeit coarse, picture of the geometry of non-positively curved metrics within a fixed conformal class. Since Gromov’s famous systolic inequality [Gro83], there has been a lot of interest in upper bounds on the systole (see for instance [Gut10]). In general, there is no positive lower bound on the systole. However, we prove here that non-positively curved metrics in a fixed conformal class do admit such a lower bound.
    [Show full text]
  • On the Quantization Problem in Curved Space
    Wright State University CORE Scholar Browse all Theses and Dissertations Theses and Dissertations 2012 On the Quantization Problem in Curved Space Benjamin Bernard Wright State University Follow this and additional works at: https://corescholar.libraries.wright.edu/etd_all Part of the Physics Commons Repository Citation Bernard, Benjamin, "On the Quantization Problem in Curved Space" (2012). Browse all Theses and Dissertations. 1094. https://corescholar.libraries.wright.edu/etd_all/1094 This Thesis is brought to you for free and open access by the Theses and Dissertations at CORE Scholar. It has been accepted for inclusion in Browse all Theses and Dissertations by an authorized administrator of CORE Scholar. For more information, please contact [email protected]. ON THE QUANTIZATION PROBLEM IN CURVED SPACE A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science by BENJAMIN JOSEPH BERNARD B.S., University of Cincinnati, 1999 2012 Wright State University WRIGHT STATE UNIVERSITY GRADUATE SCHOOL July 3, 2012 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVISION BY Benjamin Joseph Bernard ENTITLED On the Quantization Problem in Curved Space BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science. Lok C. Lew Yan Voon, Ph.D., Thesis Director Doug Petkie, Ph.D. Chair, Department of Physics Committee on Final Examination Lok C. Lew Yan Voon, Ph.D. Morten Willatzen, Ph.D. Gary Farlow, Ph.D. Andrew Hsu, Ph.D. Dean, Graduate School ABSTRACT Bernard, Benjamin Joseph. M.S., Department of Physics, Wright State Univer- sity, 2012. On the Quantization Problem in Curved Space The nonrelativistic quantum mechanics of particles constrained to curved surfaces is studied.
    [Show full text]
  • 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.
    [Show full text]
  • Arxiv:1707.09532V1 [Math.DG] 29 Jul 2017
    TRACTORS AND TRACTRICES IN RIEMANNIAN MANIFOLDS JESPER J. MADSEN AND STEEN MARKVORSEN Abstract. We generalize the notion of planar bicycle tracks { a.k.a. one-trailer systems { to so-called tractor/tractrix systems in general Riemannian manifolds and prove explicit expressions for the length of the ensuing tractrices and for the area of the do- mains that are swept out by any given tractor/tractrix system. These expressions are sensitive to the curvatures of the ambient Riemannian manifold, and we prove explicit estimates for them based on Rauch's and Toponogov's comparison theorems. More- over, the general length shortening property of tractor/tractrix sys- tems is used to generate geodesics in homotopy classes of curves in the ambient manifold. 1. Introduction The classical tractrix curve appears in virtually every textbook on differential geometry of curves and surfaces { see e.g. [6], [16], [31, p. 239], [21, p. 67], and figure 9 below. The tractrix has a long and fasci- nating history beginning with works of Huygens, Leibniz, Newton, and Euler, see [3]. Not to mention the celebrated watch track experiment by Claude Perrault, long before the invention of the bicycle { see [12], [7], [11]. We quote from [3, p. 1065]:\At a meeting in Paris in 1693 Claude Perrault laid his watch on the table, with the long chain drawn out in a straight line ([4, vol. 3]). He showed that when he moved the end of the chain along a straight line, keeping the chain taut, the watch was dragged along a certain curve. This was one of the early demonstra- tions of the tractrix." arXiv:1707.09532v1 [math.DG] 29 Jul 2017 Moreover, this classical tractrix gives rise to interesting classical sur- faces as well: When rotated around the axis along which the tractrix is pulled, any regular segment of the tractrix curve generates a pseudo- sphere of constant negative Gauss curvature, and the involute of the full classical tractrix curve, including its cusp singularity, is a catenary, which itself, when rotated around the axis, generates a catenoid, i.e.
    [Show full text]