CLASSICAL DIFFERENTIAL GEOMETRY Curves and Surfaces in Euclidean Space
Total Page:16
File Type:pdf, Size:1020Kb
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. -
The Gaussian Curvature 13/09/2007 Renzo Mattioli
The Gaussian Curvature 13/09/2007 Renzo Mattioli The Gaussian Curvature Map Renzo Mattioli (Optikon 2000, Roma*) *Author is a full time employee of Optikon 2000 Spa (Int.: C3) Historical background: Theorema Egregium by Carl Friderich Gauss Refractive On Line 2007 1 The Gaussian Curvature 13/09/2007 Renzo Mattioli Historical background: C.F. Gauss (1777-1855) was professor of mathematics in Göttingen (D). Has developed theories adopted in – Analysis – Number theory – Differential Geometry – Statistic (Guaussian distribution) – Physics (magnetism, electrostatics, optics, Astronomy) "Almost everything, which the mathematics of our century has brought forth in the way of original scientific ideas, attaches to the name of Gauss.“ Kronecker, L. Gaussian Curvature of a surface (Definition) Gaussian curvature is the product of steepest and flattest local curvatures, at each point. Negative GC Zero GC Positive GC Refractive On Line 2007 2 The Gaussian Curvature 13/09/2007 Renzo Mattioli Gauss’ Theorema Egregium (1828) Gaussian curvature (C1 x C2) of a flexible surface is invariant. (non-elastic isometric distortion) C1=0 C1=0 C2>0 C2=0 C1 x C2 = 0 C1 x C2 = 0 Gauss’ Theorema Egregium Gaussian curvature is an intrinsic property of surfaces C1 C2 C1 C2 Curvature Gaussian Curvature Gaussian C2 C2 C1 x C2 C1 x C2 C1 C1 Refractive On Line 2007 3 The Gaussian Curvature 13/09/2007 Renzo Mattioli Gauss’ Theorema Egregium • “A consequence of the Theorema Egregium is that the Earth cannot be displayed on a map without distortion. The Mercator projection… preserves angles but fails to preserve area.” (Wikipedia) The Gaussian Map in Corneal Topography Refractive On Line 2007 4 The Gaussian Curvature 13/09/2007 Renzo Mattioli Proposed by B.Barsky et al. -
A Comparison of Differential Calculus and Differential Geometry in Two
1 A Two-Dimensional Comparison of Differential Calculus and Differential Geometry Andrew Grossfield, Ph.D Vaughn College of Aeronautics and Technology Abstract and Introduction: Plane geometry is mainly the study of the properties of polygons and circles. Differential geometry is the study of curves that can be locally approximated by straight line segments. Differential calculus is the study of functions. These functions of calculus can be viewed as single-valued branches of curves in a coordinate system where the horizontal variable controls the vertical variable. In both studies the derivative multiplies incremental changes in the horizontal variable to yield incremental changes in the vertical variable and both studies possess the same rules of differentiation and integration. It seems that the two studies should be identical, that is, isomorphic. And, yet, students should be aware of important differences. In differential geometry, the horizontal and vertical units have the same dimensional units. In differential calculus the horizontal and vertical units are usually different, e.g., height vs. time. There are differences in the two studies with respect to the distance between points. In differential geometry, the Pythagorean slant distance formula prevails, while in the 2- dimensional plane of differential calculus there is no concept of slant distance. The derivative has a different meaning in each of the two subjects. In differential geometry, the slope of the tangent line determines the direction of the tangent line; that is, the angle with the horizontal axis. In differential calculus, there is no concept of direction; instead, the derivative describes a rate of change. In differential geometry the line described by the equation y = x subtends an angle, α, of 45° with the horizontal, but in calculus the linear relation, h = t, bears no concept of direction. -
Manifolds and Riemannian Geometry Steinmetz Symposium
Manifolds and Riemannian Geometry Steinmetz Symposium Daniel Resnick, Christina Tonnesen-Friedman (Advisor) 14 May 2021 2 What is Differential Geometry? Figure: Surfaces of different curvature. Differential geometry studies surfaces and how to distinguish between them| and other such objects unique to each type of surface. Daniel Resnick (Union College) Manifolds and Riemannian Geometry 14 May 2021 2 / 16 3 Defining a Manifold We begin by defining a manifold. Definition A topological space M is locally Euclidean of dimension n if every point p 2 M has a neighborhood U such that there is a homeomorphism φ from U to an open subset of Rn. We call the pair (U; φ : U ! Rn) a chart. Definition A topological space M is called a topological manifold if the space is Hausdorff, second countable, and locally Euclidean. Daniel Resnick (Union College) Manifolds and Riemannian Geometry 14 May 2021 3 / 16 4 The Riemann Metric Geometry deals with lengths, angles, and areas, and \measurements" of some kind, and we will encapsulate all of these into one object. Definition The tangent space Tp(M) is the vector space of every tangent vector to a point p 2 M. Definition (Important!) A Riemann metric on a manifold to each point p in M of an inner product h ; ip on the tangent space TpM; moreover, the assignment p 7! h ; i is required to be C 1 in the following sense: if X and Y are C 1 vector fields on M, then 1 p 7! hXp; Ypip is a C function on M.A Riemann manifold is a pair (M; h ; i) consisting of the manifold M together with the Riemann metric h ; i on M. -
Riemann's Contribution to Differential Geometry
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Historia Mathematics 9 (1982) l-18 RIEMANN'S CONTRIBUTION TO DIFFERENTIAL GEOMETRY BY ESTHER PORTNOY UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN, URBANA, IL 61801 SUMMARIES In order to make a reasonable assessment of the significance of Riemann's role in the history of dif- ferential geometry, not unduly influenced by his rep- utation as a great mathematician, we must examine the contents of his geometric writings and consider the response of other mathematicians in the years immedi- ately following their publication. Pour juger adkquatement le role de Riemann dans le developpement de la geometric differentielle sans etre influence outre mesure par sa reputation de trks grand mathematicien, nous devons &udier le contenu de ses travaux en geometric et prendre en consideration les reactions des autres mathematiciens au tours de trois an&es qui suivirent leur publication. Urn Riemann's Einfluss auf die Entwicklung der Differentialgeometrie richtig einzuschZtzen, ohne sich von seinem Ruf als bedeutender Mathematiker iiberm;issig beeindrucken zu lassen, ist es notwendig den Inhalt seiner geometrischen Schriften und die Haltung zeitgen&sischer Mathematiker unmittelbar nach ihrer Verijffentlichung zu untersuchen. On June 10, 1854, Georg Friedrich Bernhard Riemann read his probationary lecture, "iber die Hypothesen welche der Geometrie zu Grunde liegen," before the Philosophical Faculty at Gdttingen ill. His biographer, Dedekind [1892, 5491, reported that Riemann had worked hard to make the lecture understandable to nonmathematicians in the audience, and that the result was a masterpiece of presentation, in which the ideas were set forth clearly without the aid of analytic techniques. -
Carl Friedrich Gauss
GENERAL I ARTICLE Carl Friedrich Gauss D Surya Ramana D Surya Ramana is a Carl Friedrich Gauss, whose mathematical work in the 19th junior research fellow century earned him the title of Prince among Mathematicians ,1 at the Institute of was born into the family of Gebhard Dietrich Gauss, a bricklayer Mathematical Sciences, at the German town of Brunswick, on 30 April 1777. Chennai. He works in the area of analytic number theory. Gauss was a child prodigy whose calculating prowess showed itself fairly early in his life. Stories of his childhood are full of anecdotes involving his abilities that first amazed his mother, then his teachers at school and much more recently his biographers to the present day. For instance, when he was barely ten years old he surprised his teachers by calculating the sum of the first hundred natural numbers using the fact that they So varied was the mathe f<lrmed an arithmetical progression. matical work of Gauss that this article is limited in scope by While his talents impressed his mother Dorothea, they caused the author's own special a great deal of anxiety to his father. Apparently afraid of losing interests. a hand that would otherwise ply some useful trade, he very reluctantly permitted his son to enroll in a gymnasium (high school) at Brunswick. Gauss studied here from 1788-1791. Perhaps conscious of the fact that his father would not sponsor his studies anymore, Gauss began to look for a patron who would finance his education beyond the gymnasium. Encouraged by his teachers and his mother, Gauss appeared in the court of Duke Carl Wilhelm Ferdinand of Brunswick to demonstrate his computational abilities. -
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. -
Relativistic Spacetime Structure
Relativistic Spacetime Structure Samuel C. Fletcher∗ Department of Philosophy University of Minnesota, Twin Cities & Munich Center for Mathematical Philosophy Ludwig Maximilian University of Munich August 12, 2019 Abstract I survey from a modern perspective what spacetime structure there is according to the general theory of relativity, and what of it determines what else. I describe in some detail both the “standard” and various alternative answers to these questions. Besides bringing many underexplored topics to the attention of philosophers of physics and of science, metaphysicians of science, and foundationally minded physicists, I also aim to cast other, more familiar ones in a new light. 1 Introduction and Scope In the broadest sense, spacetime structure consists in the totality of relations between events and processes described in a spacetime theory, including distance, duration, motion, and (more gener- ally) change. A spacetime theory can attribute more or less such structure, and some parts of that structure may determine other parts. The nature of these structures and their relations of determi- nation bear on the interpretation of the theory—what the world would be like if the theory were true (North, 2009). For example, the structures of spacetime might be taken as its ontological or conceptual posits, and the determination relations might indicate which of these structures is more fundamental (North, 2018). Different perspectives on these questions might also reveal structural similarities with other spacetime theories, providing the resources to articulate how the picture of the world that that theory provides is different (if at all) from what came before, and might be different from what is yet to come.1 ∗Juliusz Doboszewski, Laurenz Hudetz, Eleanor Knox, J. -
AN INTRODUCTION to the CURVATURE of SURFACES by PHILIP ANTHONY BARILE a Thesis Submitted to the Graduate School-Camden Rutgers
AN INTRODUCTION TO THE CURVATURE OF SURFACES By PHILIP ANTHONY BARILE A thesis submitted to the Graduate School-Camden Rutgers, The State University Of New Jersey in partial fulfillment of the requirements for the degree of Master of Science Graduate Program in Mathematics written under the direction of Haydee Herrera and approved by Camden, NJ January 2009 ABSTRACT OF THE THESIS An Introduction to the Curvature of Surfaces by PHILIP ANTHONY BARILE Thesis Director: Haydee Herrera Curvature is fundamental to the study of differential geometry. It describes different geometrical and topological properties of a surface in R3. Two types of curvature are discussed in this paper: intrinsic and extrinsic. Numerous examples are given which motivate definitions, properties and theorems concerning curvature. ii 1 1 Introduction For surfaces in R3, there are several different ways to measure curvature. Some curvature, like normal curvature, has the property such that it depends on how we embed the surface in R3. Normal curvature is extrinsic; that is, it could not be measured by being on the surface. On the other hand, another measurement of curvature, namely Gauss curvature, does not depend on how we embed the surface in R3. Gauss curvature is intrinsic; that is, it can be measured from on the surface. In order to engage in a discussion about curvature of surfaces, we must introduce some important concepts such as regular surfaces, the tangent plane, the first and second fundamental form, and the Gauss Map. Sections 2,3 and 4 introduce these preliminaries, however, their importance should not be understated as they lay the groundwork for more subtle and advanced topics in differential geometry. -
A CONCISE MINI HISTORY of GEOMETRY 1. Origin And
Kragujevac Journal of Mathematics Volume 38(1) (2014), Pages 5{21. A CONCISE MINI HISTORY OF GEOMETRY LEOPOLD VERSTRAELEN 1. Origin and development in Old Greece Mathematics was the crowning and lasting achievement of the ancient Greek cul- ture. To more or less extent, arithmetical and geometrical problems had been ex- plored already before, in several previous civilisations at various parts of the world, within a kind of practical mathematical scientific context. The knowledge which in particular as such first had been acquired in Mesopotamia and later on in Egypt, and the philosophical reflections on its meaning and its nature by \the Old Greeks", resulted in the sublime creation of mathematics as a characteristically abstract and deductive science. The name for this science, \mathematics", stems from the Greek language, and basically means \knowledge and understanding", and became of use in most other languages as well; realising however that, as a matter of fact, it is really an art to reach new knowledge and better understanding, the Dutch term for mathematics, \wiskunde", in translation: \the art to achieve wisdom", might be even more appropriate. For specimens of the human kind, \nature" essentially stands for their organised thoughts about sensations and perceptions of \their worlds outside and inside" and \doing mathematics" basically stands for their thoughtful living in \the universe" of their idealisations and abstractions of these sensations and perceptions. Or, as Stewart stated in the revised book \What is Mathematics?" of Courant and Robbins: \Mathematics links the abstract world of mental concepts to the real world of physical things without being located completely in either". -
Differential Geometry Lecture 17: Geodesics and the Exponential
Differential geometry Lecture 17: Geodesics and the exponential map David Lindemann University of Hamburg Department of Mathematics Analysis and Differential Geometry & RTG 1670 3. July 2020 David Lindemann DG lecture 17 3. July 2020 1 / 44 1 Geodesics 2 The exponential map 3 Geodesics as critical points of the energy functional 4 Riemannian normal coordinates 5 Some global Riemannian geometry David Lindemann DG lecture 17 3. July 2020 2 / 44 Recap of lecture 16: constructed covariant derivatives along curves defined parallel transport studied the relation between a given connection in the tangent bundle and its parallel transport maps introduced torsion tensor and metric connections, studied geometric interpretation defined the Levi-Civita connection of a pseudo-Riemannian manifold David Lindemann DG lecture 17 3. July 2020 3 / 44 Geodesics Recall the definition of the acceleration of a smooth curve n 00 n γ : I ! R , that is γ 2 Γγ (T R ). Question: Is there a coordinate-free analogue of this construc- tion involving connections? Answer: Yes, uses covariant derivative along curves. Definition Let M be a smooth manifold, r a connection in TM ! M, 0 and γ : I ! M a smooth curve. Then rγ0 γ 2 Γγ (TM) is called the acceleration of γ (with respect to r). Of particular interest is the case if the acceleration of a curve vanishes, that is if its velocity vector field is parallel: Definition A smooth curve γ : I ! M is called geodesic with respect to a 0 given connection r in TM ! M if rγ0 γ = 0. David Lindemann DG lecture 17 3. -
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.