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Geometric Manifolds
Wintersemester 2015/2016 University of Heidelberg Geometric Structures on Manifolds Geometric Manifolds by Stephan Schmitt Contents Introduction, first Definitions and Results 1 Manifolds - The Group way .................................... 1 Geometric Structures ........................................ 2 The Developing Map and Completeness 4 An introductory discussion of the torus ............................. 4 Definition of the Developing map ................................. 6 Developing map and Manifolds, Completeness 10 Developing Manifolds ....................................... 10 some completeness results ..................................... 10 Some selected results 11 Discrete Groups .......................................... 11 Stephan Schmitt INTRODUCTION, FIRST DEFINITIONS AND RESULTS Introduction, first Definitions and Results Manifolds - The Group way The keystone of working mathematically in Differential Geometry, is the basic notion of a Manifold, when we usually talk about Manifolds we mean a Topological Space that, at least locally, looks just like Euclidean Space. The usual formalization of that Concept is well known, we take charts to ’map out’ the Manifold, in this paper, for sake of Convenience we will take a slightly different approach to formalize the Concept of ’locally euclidean’, to formulate it, we need some tools, let us introduce them now: Definition 1.1. Pseudogroups A pseudogroup on a topological space X is a set G of homeomorphisms between open sets of X satisfying the following conditions: • The Domains of the elements g 2 G cover X • The restriction of an element g 2 G to any open set contained in its Domain is also in G. • The Composition g1 ◦ g2 of two elements of G, when defined, is in G • The inverse of an Element of G is in G. • The property of being in G is local, that is, if g : U ! V is a homeomorphism between open sets of X and U is covered by open sets Uα such that each restriction gjUα is in G, then g 2 G Definition 1.2. -
Black Hole Physics in Globally Hyperbolic Space-Times
Pram[n,a, Vol. 18, No. $, May 1982, pp. 385-396. O Printed in India. Black hole physics in globally hyperbolic space-times P S JOSHI and J V NARLIKAR Tata Institute of Fundamental Research, Bombay 400 005, India MS received 13 July 1981; revised 16 February 1982 Abstract. The usual definition of a black hole is modified to make it applicable in a globallyhyperbolic space-time. It is shown that in a closed globallyhyperbolic universe the surface area of a black hole must eventuallydecrease. The implications of this breakdown of the black hole area theorem are discussed in the context of thermodynamics and cosmology. A modifieddefinition of surface gravity is also given for non-stationaryuniverses. The limitations of these concepts are illustrated by the explicit example of the Kerr-Vaidya metric. Keywocds. Black holes; general relativity; cosmology, 1. Introduction The basic laws of black hole physics are formulated in asymptotically flat space- times. The cosmological considerations on the other hand lead one to believe that the universe may not be asymptotically fiat. A realistic discussion of black hole physics must not therefore depend critically on the assumption of an asymptotically flat space-time. Rather it should take account of the global properties found in most of the widely discussed cosmological models like the Friedmann models or the more general Robertson-Walker space times. Global hyperbolicity is one such important property shared by the above cosmo- logical models. This property is essentially a precise formulation of classical deter- minism in a space-time and it removes several physically unreasonable pathological space-times from a discussion of what the large scale structure of the universe should be like (Penrose 1972). -
Models of 2-Dimensional Hyperbolic Space and Relations Among Them; Hyperbolic Length, Lines, and Distances
Models of 2-dimensional hyperbolic space and relations among them; Hyperbolic length, lines, and distances Cheng Ka Long, Hui Kam Tong 1155109623, 1155109049 Course Teacher: Prof. Yi-Jen LEE Department of Mathematics, The Chinese University of Hong Kong MATH4900E Presentation 2, 5th October 2020 Outline Upper half-plane Model (Cheng) A Model for the Hyperbolic Plane The Riemann Sphere C Poincar´eDisc Model D (Hui) Basic properties of Poincar´eDisc Model Relation between D and other models Length and distance in the upper half-plane model (Cheng) Path integrals Distance in hyperbolic geometry Measurements in the Poincar´eDisc Model (Hui) M¨obiustransformations of D Hyperbolic length and distance in D Conclusion Boundary, Length, Orientation-preserving isometries, Geodesics and Angles Reference Upper half-plane model H Introduction to Upper half-plane model - continued Hyperbolic geometry Five Postulates of Hyperbolic geometry: 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. A circle may be described with any given point as its center and any distance as its radius. 4. All right angles are congruent. 5. For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. Some interesting facts about hyperbolic geometry 1. Rectangles don't exist in hyperbolic geometry. 2. In hyperbolic geometry, all triangles have angle sum < π 3. In hyperbolic geometry if two triangles are similar, they are congruent. -
USA Mathematical Talent Search Solutions to Problem 2/3/16
USA Mathematical Talent Search Solutions to Problem 2/3/16 www.usamts.org 2/3/16. Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle’s area is numerically equal to 6 times its perimeter. Credit This is a slight modification of a problem provided by Suresh T. Thakar of India. The original problem asked for five isosceles triangles with integer sides such that the area is numerically 12 times the perimeter. Comments Many students simply set up an equation using Heron’s formula and then turned to a calculator or a computer for a solution. Below are presented more elegant solutions. Zachary Abel shows how to reduce this problem to finding Pythagorean triples which have 12 among the side lengths. Adam Hesterberg gives us a solution using Heron’s Create PDF with GO2PDFformula. for free, if Finally,you wish to remove Kristin this line, click Cordwell here to buy Virtual shows PDF Printer how to take an intelligent trial-and-error approach to construct the solutions. Solutions edited by Richard Rusczyk. Solution 1 by: Zachary Abel (11/TX) In 4ABC with AB = AC, we use the common notations r = inradius, s = semiperimeter, p = perimeter, K = area, a = BC, and b = AC. The diagram shows triangle ABC with its incircle centered at I and tangent to BC and AC at M and N respectively. A x h N 12 I a/2 12 B M a/2 C The area of the triangle is given by rs = K = 6p = 12s, which implies r = 12. -
Imaginary Crystals Made Real
Imaginary crystals made real Simone Taioli,1, 2 Ruggero Gabbrielli,1 Stefano Simonucci,3 Nicola Maria Pugno,4, 5, 6 and Alfredo Iorio1 1Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic 2European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*), Bruno Kessler Foundation & Trento Institute for Fundamental Physics and Applications (TIFPA-INFN), Trento, Italy 3Department of Physics, University of Camerino, Italy & Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Italy 4Laboratory of Bio-inspired & Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Italy 5School of Engineering and Materials Science, Queen Mary University of London, UK 6Center for Materials and Microsystems, Bruno Kessler Foundation, Trento, Italy (Dated: August 20, 2018) We realize Lobachevsky geometry in a simulation lab, by producing a carbon-based mechanically stable molecular structure, arranged in the shape of a Beltrami pseudosphere. We find that this structure: i) corresponds to a non-Euclidean crystallographic group, namely a loxodromic subgroup of SL(2; Z); ii) has an unavoidable singular boundary, that we fully take into account. Our approach, substantiated by extensive numerical simulations of Beltrami pseudospheres of different size, might be applied to other surfaces of constant negative Gaussian curvature, and points to a general pro- cedure to generate them. Our results also pave the way to test certain scenarios of the physics of curved spacetimes. Lobachevsky used to call his Non-Euclidean geometry \imaginary geometry" [1]. Beltrami showed that this geom- etry can be realized in our Euclidean 3-space, through surfaces of constant negative Gaussian curvature K [2]. -
Areas of Polygons and Circles
Chapter 8 Areas of Polygons and Circles Copyright © Cengage Learning. All rights reserved. Perimeter and Area of 8.2 Polygons Copyright © Cengage Learning. All rights reserved. Perimeter and Area of Polygons Definition The perimeter of a polygon is the sum of the lengths of all sides of the polygon. Table 8.1 summarizes perimeter formulas for types of triangles. 3 Perimeter and Area of Polygons Table 8.2 summarizes formulas for the perimeters of selected types of quadrilaterals. However, it is more important to understand the concept of perimeter than to memorize formulas. 4 Example 1 Find the perimeter of ABC shown in Figure 8.17 if: a) AB = 5 in., AC = 6 in., and BC = 7 in. b) AD = 8 cm, BC = 6 cm, and Solution: a) PABC = AB + AC + BC Figure 8.17 = 5 + 6 + 7 = 18 in. 5 Example 1 – Solution cont’d b) With , ABC is isosceles. Then is the bisector of If BC = 6, it follows that DC = 3. Using the Pythagorean Theorem, we have 2 2 2 (AD) + (DC) = (AC) 2 2 2 8 + 3 = (AC) 6 Example 1 – Solution cont’d 64 + 9 = (AC)2 AC = Now Note: Because x + x = 2x, we have 7 HERON’S FORMULA 8 Heron’s Formula If the lengths of the sides of a triangle are known, the formula generally used to calculate the area is Heron’s Formula. One of the numbers found in this formula is the semiperimeter of a triangle, which is defined as one-half the perimeter. For the triangle that has sides of lengths a, b, and c, the semiperimeter is s = (a + b + c). -
Figures Circumscribing Circles Tom M
Figures Circumscribing Circles Tom M. Apostol and Mamikon A. Mnatsakanian 1. INTRODUCTION. The centroid of the boundary of an arbitrarytriangle need not be at the same point as the centroid of its interior. But we have discovered that the two centroids are always collinear with the center of the inscribed circle, at distances in the ratio 3 : 2 from the center. We thought this charming fact must surely be known, but could find no mention of it in the literature. This paper generalizes this elegant and surprising result to any polygon that circumscribes a circle (Theorem 6). A key ingredient of the proof is a link to Archimedes' striking discovery concerning the area of a circular disk [4, p. 91], which for our purposes we prefer to state as follows: Theorem 1 (Archimedes). The area of a circular disk is equal to the product of its semiperimeter and its radius. Expressed as a formula, this becomes A = Pr, (1) where A is the area, P is the perimeter, and r is the radius of the disk. First we extend (1) to a large class of plane figures circumscribing a circle that we call circumgons, defined in section 2. They include arbitrarytriangles, all regular polygons, some irregularpolygons, and other figures composed of line segments and circular arcs. Examples are shown in Figures 1 through 4. Section 3 treats circum- gonal rings, plane regions lying between two similar circumgons. These rings have a constant width that replaces the radius in the corresponding extension of (1). We also show that all rings of constant width are necessarily circumgonal rings. -
David Hilbert's Contributions to Logical Theory
David Hilbert’s contributions to logical theory CURTIS FRANKS 1. A mathematician’s cast of mind Charles Sanders Peirce famously declared that “no two things could be more directly opposite than the cast of mind of the logician and that of the mathematician” (Peirce 1976, p. 595), and one who would take his word for it could only ascribe to David Hilbert that mindset opposed to the thought of his contemporaries, Frege, Gentzen, Godel,¨ Heyting, Łukasiewicz, and Skolem. They were the logicians par excellence of a generation that saw Hilbert seated at the helm of German mathematical research. Of Hilbert’s numerous scientific achievements, not one properly belongs to the domain of logic. In fact several of the great logical discoveries of the 20th century revealed deep errors in Hilbert’s intuitions—exemplifying, one might say, Peirce’s bald generalization. Yet to Peirce’s addendum that “[i]t is almost inconceivable that a man should be great in both ways” (Ibid.), Hilbert stands as perhaps history’s principle counter-example. It is to Hilbert that we owe the fundamental ideas and goals (indeed, even the name) of proof theory, the first systematic development and application of the methods (even if the field would be named only half a century later) of model theory, and the statement of the first definitive problem in recursion theory. And he did more. Beyond giving shape to the various sub-disciplines of modern logic, Hilbert brought them each under the umbrella of mainstream mathematical activity, so that for the first time in history teams of researchers shared a common sense of logic’s open problems, key concepts, and central techniques. -
Understand the Principles and Properties of Axiomatic (Synthetic
Michael Bonomi Understand the principles and properties of axiomatic (synthetic) geometries (0016) Euclidean Geometry: To understand this part of the CST I decided to start off with the geometry we know the most and that is Euclidean: − Euclidean geometry is a geometry that is based on axioms and postulates − Axioms are accepted assumptions without proofs − In Euclidean geometry there are 5 axioms which the rest of geometry is based on − Everybody had no problems with them except for the 5 axiom the parallel postulate − This axiom was that there is only one unique line through a point that is parallel to another line − Most of the geometry can be proven without the parallel postulate − If you do not assume this postulate, then you can only prove that the angle measurements of right triangle are ≤ 180° Hyperbolic Geometry: − We will look at the Poincare model − This model consists of points on the interior of a circle with a radius of one − The lines consist of arcs and intersect our circle at 90° − Angles are defined by angles between the tangent lines drawn between the curves at the point of intersection − If two lines do not intersect within the circle, then they are parallel − Two points on a line in hyperbolic geometry is a line segment − The angle measure of a triangle in hyperbolic geometry < 180° Projective Geometry: − This is the geometry that deals with projecting images from one plane to another this can be like projecting a shadow − This picture shows the basics of Projective geometry − The geometry does not preserve length -
Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg
Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg By “elementary” plane geometry I mean the geometry of lines and circles—straight- edge and compass constructions—in both Euclidean and non-Euclidean planes. An axiomatic description of it is in Sections 1.1, 1.2, and 1.6. This survey highlights some foundational history and some interesting recent discoveries that deserve to be better known, such as the hierarchies of axiom systems, Aristotle’s axiom as a “missing link,” Bolyai’s discovery—proved and generalized by William Jagy—of the relationship of “circle-squaring” in a hyperbolic plane to Fermat primes, the undecidability, incom- pleteness, and consistency of elementary Euclidean geometry, and much more. A main theme is what Hilbert called “the purity of methods of proof,” exemplified in his and his early twentieth century successors’ works on foundations of geometry. 1. AXIOMATIC DEVELOPMENT 1.0. Viewpoint. Euclid’s Elements was the first axiomatic presentation of mathemat- ics, based on his five postulates plus his “common notions.” It wasn’t until the end of the nineteenth century that rigorous revisions of Euclid’s axiomatics were presented, filling in the many gaps in his definitions and proofs. The revision with the great- est influence was that by David Hilbert starting in 1899, which will be discussed below. Hilbert not only made Euclid’s geometry rigorous, he investigated the min- imal assumptions needed to prove Euclid’s results, he showed the independence of some of his own axioms from the others, he presented unusual models to show certain statements unprovable from others, and in subsequent editions he explored in his ap- pendices many other interesting topics, including his foundation for plane hyperbolic geometry without bringing in real numbers. -
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.