DOCSLIB.ORG

Differentiable Manifolds Carlos Sotillo Rodríguez

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Seminar: Introduction to Riemannian Geometry Differentiable Manifolds Carlos Sotillo Rodríguez In the following Chapter we are going to define the main concepts as well as the basic results. From now on, differentiable will always signify of class C 1 Definition 1: A differentiable manifold of dimension n is a set M and a family of injective mappings x : U Rn M (where U is an open set of Rn), such that: ® ® ½ ! ® [ 1. x®(U®) M ® Æ T 1 1 2. for any pair ®,¯ with x (U ) x (U ) W , the sets x¡ (W ) and x¡ (W ) are open ® ® ¯ ¯ Æ 6Æ ; ® ¯ n 1 sets of R and the mappings x¡ x are differentiable. ¯ ± ® 3. The family {(U®,x®)} is maximal relative to conditions 1 and 2. The pair (U ,x ) (or the mapping x ) with p x (U ) is called a parametrization (or system ® ® ® 2 ® ® of coordinates) of M at p, x®(u®) is then called a coordinated neighborhood at p. A family satisfying 1 and 2 is called a differentiable structure. The condition 3 is included for technical reasons. Indeed, given a differentiable structure, we can complete it to a maximal one. Therefore, with a certain abuse of lenguage, we can say that a differentiable manifold is a set with a differentiable structure. n m Definition 2: Let M1 and M2 (where n,m are the dimensions of M1,M2 respectively). A map- ping Á : M M is differentible at p M if given a parametrization y : V Rm M at '(p) 1 ! 2 2 1 ½ ! 2 there exists a parametrization x : U Rn M at p such that '(x(U)) y(V ) and the map- ½ ! 1 ½ ping 1 n m y¡ ' x : U R R ± ± ½ ! 1 is differentiable at x¡ (p). ' is differentiable on an open set of M1 if it is differentiable at every point of this open set (it follows from condition 2 that this definition does not depend on the 1 parametrization).The expression y¡ ' x is called the expression of ' in the parametriza- ± ± tions x and y. Definition 3: Let M be a differentiable manifold. A differentiable function ® :( ²,²) M ¡ ! is called a (differentiable) curve in M. Suppose that ®(0) p M and let D be the set of Æ 2 functions on M that are differentiable at p. The tangent vector to the curve ® at t 0 is a Æ function ®0(0) : D R given by ! ¯ d(f ®)¯ ®0(0)(f ) ± ¯ , f D Æ dt ¯t 0 2 Æ 1 A tangent vector at p is the tangent vector at t 0 of some curve ® :( ²,²) M with ®(0) p. Æ ¡ ! Æ The set of all tangent vectors to M at p will be indicated by Tp M. @ @ Further, the set Tp M is a vector space with assiciated basis {( )0, ,( )0}. It is immediate @x1 ¢¢¢ @xn that the basis depends on the parametrization, but not the linear structure. Proposition 4: Let M n and M m be differentiable manifolds and let ' : M M be a diffe- 1 2 1 ! 2 rentiable mapping. For every p M and for each v T M , choose a differentiable curve 2 1 2 p 1 ® :( ²,²) M with ®(0) p,®0(0) v. Take ¯ ' ®. The mapping d' : T M T M ¡ ! 1 Æ Æ Æ ± p p 1 ! '(p) 2 given by d' (v) ¯0(0) is a linear mapping that does not depend on the choice of ®. p Æ Definitions 5: The linear mapping d'p is called the differential of ' at p. Let M and M be differential manifolds. A mapping ' : M M is a diffeomorphism 1 2 1 ! 2 if it is differentiable, bijective and its inverse is differentiable. ' is said to be a local diffeomisphism at p M if there exist neighborhoods U of p and V of '(p) such that 2 1 ' : U V is a diffeomorphism. ! Theorem 6: Let ' : M n M n be a differentiable mapping and let p M be. d' : T M 1 ! 2 2 1 p p 1 ! T'(p)M2 is an isomorphism if and only if ' is a local diffeomorphism at p. Definition 7: Let M m and N n be differentiable manifolds. A differentiable mapping ' : M ! N is said to be an immersion if d' : T M T N is injective for all p M. If in addition, ' p p ! '(p) 2 is a homeomorphism onto '(M) N, where '(M) has the subspace topology induced from ½ N, we say that ' is an embedding. If M N and the inclusion i : M N is an embedding, we ½ ½ say that M is a submanifold of N. Proposition 8: Let ' : M n M m, n m, be an immersion of the differentiable manifold 1 ! 2 · M into the differentiable manifold M . For every point p M , there exists a neighborhood 1 2 2 1 V M1 of p such that the restriction ' V : V M2 is an embedding. ½ j ! Definition + Example 9 (The tangent bundle): Let M n be a differentiable manifold and let TM : {(p,v): p M,v T M}. We are going to provide the set TM with a differentiable Æ 2 2 p structure. ® ® Let {(U®,x®)} be a maximal differentiable structure on M. Denote by (x1 , ,xn ) the coordi- @ @ ¢¢¢ nates of U® and by { @x® , , @x® } the associated bases to the tangent spaces of x®(U®). For 1 ¢¢¢ n every ®, define y : U Rn TM ® ® £ ! by n ® ® ® ® X @ n y®(x1 , ,xn ,u1, ,un) (x®(x1 , ,xn ), ui ® ), (u1, ,un) R ¢¢¢ ¢¢¢ Æ ¢¢¢ i 1 @xi ¢¢¢ 2 Æ n [ We are going to show that {(U® R , y®}) is a differentiable structure on TM. Since x®(U®) £ ® Æ 2 M and (dx ) (Rn) T M, q U , we have only to prove the condition 2. Now let ® q Æ x®(q) 2 ® \ (p,v) y (U Rn) y (U Rn) 2 ® ® £ ¯ ¯ £ Then (p,v) (x (q ),dx (v )) (x (q ),dx (v )), Æ ® ® ® ® Æ ¯ ¯ ¯ ¯ where q U ,q U ,v ,v Rn. Therefore, ® 2 ® ¯ 2 ¯ ® ¯ 2 1 1 1 1 y¡ y (q ,v ) y¡ (x (q ),dx (v )) ((x¡ x )(q ),d(x¡ x )(v )). ¯ ± ® ® ® Æ ¯ ® ® ® ® Æ ¯ ± ® ® ¯ ± ® ® 1 Since x¡ x is differentiable, d(x x ) is as well. It follows that condition 2 is verified. ¯ ± ® ¯ ± ® Example 10 (Regular surfaces in Rn): A subset M k Rn is a regular surface of dimension ½ k n if for every point p M k there exists a neighborhood V of p in Rn and a mapping · 2 x : U M V of an open set U Rk onto M V such that: ! \ ½ \ x is a differentiable homeomorphism. (dx) : Rk Rn is injective for all q U. q ! 2 It can be proved that if x : U M k and y : V M k are two parametrizations with x(U) ! 1 !1 1 \ y(V ) W , then the mapping h x¡ y : y¡ (W ) x¡ (W ) is a diffeomorphism. We give Æ 6Æ ; Æ ± ! a sketch of the proof: Let (u , ,u ) U and (v , ,v ) Rn, and write x in these coordina- 1 ¢¢¢ k 2 1 ¢¢¢ n 2 tes as x(u , ,u ) (v (u , ,u ), ,v (u , u )) 1 ¢¢¢ k Æ 1 1 ¢¢¢ k ¢¢¢ n 1 ¢¢¢ k From condition 2 we can suppose that @(v1, ,v ) ¢¢¢ k (q) 0 @(u , ,u ) 6Æ 1 ¢¢¢ k n k n Extend x to a mapping F : U R ¡ R given by £ ! F (u1, ,uk ,tk 1, ,tn) (v1(u1, ,uk ), ,vk (u1, ,uk ),vk 1(u1, ,uk ) ¢¢¢ Å ¢¢¢ Æ ¢¢¢ ¢¢¢ ¢¢¢ Å ¢¢¢ tk 1, ,vn(u1, ,uk ) tn) Å Å ¢¢¢ ¢¢¢ Å n k where (tk 1, ,tn) R ¡ . It is clear that F is differentiable and the restriction of F to U Å ¢¢¢ 2 £ {(0, ,0)} coincides with x. By a calculation, we obtain that ¢¢¢ @(v1, ,vk ) det(dFq ) ¢¢¢ (q) 0 Æ @(u , ,u ) 6Æ 1 ¢¢¢ k Now we are under the conditions of the inverse function theorem, which guarantees the exis- 1 tence of a neighborhood Q of x(q) where F ¡ exists and is differentiable. By the continuity of y, there exists a neighborhood R V of r such that y(R) Q. Note that the restriction of h 1 ½ ½ to R, h R F ¡ y R is a composition of differentiable mappings. Thus h is differentiable at r , j Æ 1 ± j 1 hence in y¡ (W ). A similar argument would prove that h¡ is differentiable as well, proving the assertion. Definition 11: Let M be a differentiable manifold. We say that M is orientable if M admits a differentiable structure {(U®,x®)} such that: 3 for every pair ®,¯ with x®(U®) x¯(U¯) W , the differential of the change of coor- 1 \ Æ 6Æ ; dinates x¡ x has positive determinant. ¯ ± ® In the opposite case, we say that M is non-orientable. If M is orientable, a choice of a diffe- rentiable structure satisfying th condition from above is called as orientation of M. Example 12 (Discontinuous action of a group): We say that a group G acts on a differentiable manifold M if there exists a mapping ' : G M M such that: £ ! 1. For each g G, the mapping ' : M M given by ' (p) '(g,p), p M, is a diffeo- 2 g ! g Æ 2 morphism, and ' Id . e Æ M 2. If g ,g G, ' ' ' . 1 2 2 g1g2 Æ g1 ± g2 Frequently, when dealing with a single action, we set g p : '(g,p). Æ We say that the action is properly discontinuous if every p M has a neighborhood U M 2 ½ such that U g(U) for all g e. When G acts on M, the action determines an equivalence \ Æ; 6Æ relation on M, in which p p if and only if p g p for some g G.
Recommended publications
  • An Introduction to Differentiable Manifolds

    An Introduction to Differentiable Manifolds

    Mathematics Letters 2016; 2(5): 32-35 http://www.sciencepublishinggroup.com/j/ml doi: 10.11648/j.ml.20160205.11 Conference Paper An Introduction to Differentiable Manifolds Kande Dickson Kinyua Department of Mathematics, Moi University, Eldoret, Kenya Email address: [email protected] To cite this article: Kande Dickson Kinyua. An Introduction to Differentiable Manifolds. Mathematics Letters. Vol. 2, No. 5, 2016, pp. 32-35. doi: 10.11648/j.ml.20160205.11 Received : September 7, 2016; Accepted : November 1, 2016; Published : November 23, 2016 Abstract: A manifold is a Hausdorff topological space with some neighborhood of a point that looks like an open set in a Euclidean space. The concept of Euclidean space to a topological space is extended via suitable choice of coordinates. Manifolds are important objects in mathematics, physics and control theory, because they allow more complicated structures to be expressed and understood in terms of the well–understood properties of simpler Euclidean spaces. A differentiable manifold is defined either as a set of points with neighborhoods homeomorphic with Euclidean space, Rn with coordinates in overlapping neighborhoods being related by a differentiable transformation or as a subset of R, defined near each point by expressing some of the coordinates in terms of the others by differentiable functions. This paper aims at making a step by step introduction to differential manifolds. Keywords: Submanifold, Differentiable Manifold, Morphism, Topological Space manifolds with the additional condition that the transition 1. Introduction maps are analytic. In other words, a differentiable (or, smooth) The concept of manifolds is central to many parts of manifold is a topological manifold with a globally defined geometry and modern mathematical physics because it differentiable (or, smooth) structure, [1], [3], [4].
  • DIFFERENTIAL GEOMETRY Contents 1. Introduction 2 2. Differentiation 3

    DIFFERENTIAL GEOMETRY Contents 1. Introduction 2 2. Differentiation 3

    DIFFERENTIAL GEOMETRY FINNUR LARUSSON´ Lecture notes for an honours course at the University of Adelaide Contents 1. Introduction 2 2. Differentiation 3 2.1. Review of the basics 3 2.2. The inverse function theorem 4 3. Smooth manifolds 7 3.1. Charts and atlases 7 3.2. Submanifolds and embeddings 8 3.3. Partitions of unity and Whitney’s embedding theorem 10 4. Tangent spaces 11 4.1. Germs, derivations, and equivalence classes of paths 11 4.2. The derivative of a smooth map 14 5. Differential forms and integration on manifolds 16 5.1. Introduction 16 5.2. A little multilinear algebra 17 5.3. Differential forms and the exterior derivative 18 5.4. Integration of differential forms on oriented manifolds 20 6. Stokes’ theorem 22 6.1. Manifolds with boundary 22 6.2. Statement and proof of Stokes’ theorem 24 6.3. Topological applications of Stokes’ theorem 26 7. Cohomology 28 7.1. De Rham cohomology 28 7.2. Cohomology calculations 30 7.3. Cechˇ cohomology and de Rham’s theorem 34 8. Exercises 36 9. References 42 Last change: 26 September 2008. These notes were originally written in 2007. They have been classroom-tested twice. Address: School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia. E-mail address: [email protected] Copyright c Finnur L´arusson 2007. 1 1. Introduction The goal of this course is to acquire familiarity with the concept of a smooth manifold. Roughly speaking, a smooth manifold is a space on which we can do calculus. Manifolds arise in various areas of mathematics; they have a rich and deep theory with many applications, for example in physics.
  • Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md

    Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md

    International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 157 ISSN 2229-5518 Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md. Abdul Halim Sajal Saha Md Shafiqul Islam Abstract-In the paper some aspects of Riemannian manifolds, pseudo-Riemannian manifolds, Lorentz manifolds, Riemannian metrics, affine connections, parallel transport, curvature tensors, torsion tensors, killing vector fields, conformal killing vector fields are focused. The purpose of this paper is to develop the theory of manifolds equipped with Riemannian metric. I have developed some theorems on torsion and Riemannian curvature tensors using affine connection. A Theorem 1.20 named “Fundamental Theorem of Pseudo-Riemannian Geometry” has been established on Riemannian geometry using tensors with metric. The main tools used in the theorem of pseudo Riemannian are tensors fields defined on a Riemannian manifold. Keywords: Riemannian manifolds, pseudo-Riemannian manifolds, Lorentz manifolds, Riemannian metrics, affine connections, parallel transport, curvature tensors, torsion tensors, killing vector fields, conformal killing vector fields. —————————— —————————— I. Introduction (c) { } is a family of open sets which covers , that is, 푖 = . Riemannian manifold is a pair ( , g) consisting of smooth 푈 푀 manifold and Riemannian metric g. A manifold may carry a (d) ⋃ is푈 푖푖 a homeomorphism푀 from onto an open subset of 푀 ′ further structure if it is endowed with a metric tensor, which is a 푖 . 푖 푖 휑 푈 푈 natural generation푀 of the inner product between two vectors in 푛 ℝ to an arbitrary manifold. Riemannian metrics, affine (e) Given and such that , the map = connections,푛 parallel transport, curvature tensors, torsion tensors, ( ( ) killingℝ vector fields and conformal killing vector fields play from푖 푗 ) to 푖 푗 is infinitely푖푗 −1 푈 푈 푈 ∩ 푈 ≠ ∅ 휓 important role to develop the theorem of Riemannian manifolds.
  • Differentiable Manifolds

    Differentiable Manifolds

    Gerardo F. Torres del Castillo Differentiable Manifolds ATheoreticalPhysicsApproach Gerardo F. Torres del Castillo Instituto de Ciencias Universidad Autónoma de Puebla Ciudad Universitaria 72570 Puebla, Puebla, Mexico [email protected] ISBN 978-0-8176-8270-5 e-ISBN 978-0-8176-8271-2 DOI 10.1007/978-0-8176-8271-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011939950 Mathematics Subject Classification (2010): 22E70, 34C14, 53B20, 58A15, 70H05 © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com) Preface The aim of this book is to present in an elementary manner the basic notions related with differentiable manifolds and some of their applications, especially in physics. The book is aimed at advanced undergraduate and graduate students in physics and mathematics, assuming a working knowledge of calculus in several variables, linear algebra, and differential equations.
  • Differential Geometry: Curvature and Holonomy Austin Christian

    Differential Geometry: Curvature and Holonomy Austin Christian

    University of Texas at Tyler Scholar Works at UT Tyler Math Theses Math Spring 5-5-2015 Differential Geometry: Curvature and Holonomy Austin Christian Follow this and additional works at: https://scholarworks.uttyler.edu/math_grad Part of the Mathematics Commons Recommended Citation Christian, Austin, "Differential Geometry: Curvature and Holonomy" (2015). Math Theses. Paper 5. http://hdl.handle.net/10950/266 This Thesis is brought to you for free and open access by the Math at Scholar Works at UT Tyler. It has been accepted for inclusion in Math Theses by an authorized administrator of Scholar Works at UT Tyler. For more information, please contact [email protected]. DIFFERENTIAL GEOMETRY: CURVATURE AND HOLONOMY by AUSTIN CHRISTIAN A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics David Milan, Ph.D., Committee Chair College of Arts and Sciences The University of Texas at Tyler May 2015 c Copyright by Austin Christian 2015 All rights reserved Acknowledgments There are a number of people that have contributed to this project, whether or not they were aware of their contribution. For taking me on as a student and learning differential geometry with me, I am deeply indebted to my advisor, David Milan. Without himself being a geometer, he has helped me to develop an invaluable intuition for the field, and the freedom he has afforded me to study things that I find interesting has given me ample room to grow. For introducing me to differential geometry in the first place, I owe a great deal of thanks to my undergraduate advisor, Robert Huff; our many fruitful conversations, mathematical and otherwise, con- tinue to affect my approach to mathematics.
  • Tight Immersions of Simplicial Surfaces in Three Space

    Tight Immersions of Simplicial Surfaces in Three Space

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Pergamon Topology Vol. 35, No. 4, pp. 863-873, 19% Copyright Q 1996 Elsevier Sctence Ltd Printed in Great Britain. All tights reserved CO4&9383/96 $15.00 + 0.00 0040-9383(95)00047-X TIGHT IMMERSIONS OF SIMPLICIAL SURFACES IN THREE SPACE DAVIDE P. CERVONE (Received for publication 13 July 1995) 1. INTRODUCTION IN 1960,Kuiper [12,13] initiated the study of tight immersions of surfaces in three space, and showed that every compact surface admits a tight immersion except for the Klein bottle, the real projective plane, and possibly the real projective plane with one handle. The fate of the latter surface went unresolved for 30 years until Haab recently proved [9] that there is no smooth tight immersion of the projective plane with one handle. In light of this, a recently described simplicial tight immersion of this surface [6] came as quite a surprise, and its discovery completes Kuiper’s initial survey. Pinkall [17] broadened the question in 1985 by looking for tight immersions in each equivalence class of immersed surfaces under image homotopy. He first produced a complete description of these classes, and proceeded to generate tight examples in all but finitely many of them. In this paper, we improve Pinkall’s result by giving tight simplicial examples in all but two of the immersion classes under image homotopy for which tight immersions are possible, and in particular, we point out and resolve an error in one of his examples that would have left an infinite number of immersion classes with no tight examples.
  • Tensor Calculus and Differential Geometry

    Tensor Calculus and Differential Geometry

    Course Notes Tensor Calculus and Differential Geometry 2WAH0 Luc Florack March 10, 2021 Cover illustration: papyrus fragment from Euclid’s Elements of Geometry, Book II [8]. Contents Preface iii Notation 1 1 Prerequisites from Linear Algebra 3 2 Tensor Calculus 7 2.1 Vector Spaces and Bases . .7 2.2 Dual Vector Spaces and Dual Bases . .8 2.3 The Kronecker Tensor . 10 2.4 Inner Products . 11 2.5 Reciprocal Bases . 14 2.6 Bases, Dual Bases, Reciprocal Bases: Mutual Relations . 16 2.7 Examples of Vectors and Covectors . 17 2.8 Tensors . 18 2.8.1 Tensors in all Generality . 18 2.8.2 Tensors Subject to Symmetries . 22 2.8.3 Symmetry and Antisymmetry Preserving Product Operators . 24 2.8.4 Vector Spaces with an Oriented Volume . 31 2.8.5 Tensors on an Inner Product Space . 34 2.8.6 Tensor Transformations . 36 2.8.6.1 “Absolute Tensors” . 37 CONTENTS i 2.8.6.2 “Relative Tensors” . 38 2.8.6.3 “Pseudo Tensors” . 41 2.8.7 Contractions . 43 2.9 The Hodge Star Operator . 43 3 Differential Geometry 47 3.1 Euclidean Space: Cartesian and Curvilinear Coordinates . 47 3.2 Differentiable Manifolds . 48 3.3 Tangent Vectors . 49 3.4 Tangent and Cotangent Bundle . 50 3.5 Exterior Derivative . 51 3.6 Affine Connection . 52 3.7 Lie Derivative . 55 3.8 Torsion . 55 3.9 Levi-Civita Connection . 56 3.10 Geodesics . 57 3.11 Curvature . 58 3.12 Push-Forward and Pull-Back . 59 3.13 Examples . 60 3.13.1 Polar Coordinates in the Euclidean Plane .
  • Lecture 10: Tubular Neighborhood Theorem

    Lecture 10: Tubular Neighborhood Theorem

    LECTURE 10: TUBULAR NEIGHBORHOOD THEOREM 1. Generalized Inverse Function Theorem We will start with Theorem 1.1 (Generalized Inverse Function Theorem, compact version). Let f : M ! N be a smooth map that is one-to-one on a compact submanifold X of M. Moreover, suppose dfx : TxM ! Tf(x)N is a linear diffeomorphism for each x 2 X. Then f maps a neighborhood U of X in M diffeomorphically onto a neighborhood V of f(X) in N. Note that if we take X = fxg, i.e. a one-point set, then is the inverse function theorem we discussed in Lecture 2 and Lecture 6. According to that theorem, one can easily construct neighborhoods U of X and V of f(X) so that f is a local diffeomor- phism from U to V . To get a global diffeomorphism from a local diffeomorphism, we will need the following useful lemma: Lemma 1.2. Suppose f : M ! N is a local diffeomorphism near every x 2 M. If f is invertible, then f is a global diffeomorphism. Proof. We need to show f −1 is smooth. Fix any y = f(x). The smoothness of f −1 at y depends only on the behaviour of f −1 near y. Since f is a diffeomorphism from a −1 neighborhood of x onto a neighborhood of y, f is smooth at y. Proof of Generalized IFT, compact version. According to Lemma 1.2, it is enough to prove that f is one-to-one in a neighborhood of X. We may embed M into RK , and consider the \"-neighborhood" of X in M: X" = fx 2 M j d(x; X) < "g; where d(·; ·) is the Euclidean distance.
  • DISCRETE DIFFERENTIAL GEOMETRY: an APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 LECTURE 10: INTRODUCTION to CURVES

    DISCRETE DIFFERENTIAL GEOMETRY: an APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 LECTURE 10: INTRODUCTION to CURVES

    DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 LECTURE 10: INTRODUCTION TO CURVES DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 Curves & Surfaces •Much of the geometry we encounter in life well-described by curves and surfaces* (Curves) *Or solids… but the boundary of a solid is a surface! (Surfaces) Much Ado About Manifolds • In general, differential geometry studies n-dimensional manifolds; we’ll focus on low dimensions: curves (n=1), surfaces (n=2), and volumes (n=3) • Why? Geometry we encounter in everyday life/applications • Low-dimensional manifolds are not “baby stuff!” • n=1: unknot recognition (open as of 2021) • n=2: Willmore conjecture (2012 for genus 1) • n=3: Geometrization conjecture (2003, $1 million) • Serious intuition gained by studying low-dimensional manifolds • Conversely, problems involving very high-dimensional manifolds (e.g., data analysis/machine learning) involve less “deep” geometry than you might imagine! • fiber bundles, Lie groups, curvature flows, spinors, symplectic structure, ... • Moreover, curves and surfaces are beautiful! (And in some cases, high- dimensional manifolds have less interesting structure…) Smooth Descriptions of Curves & Surfaces •Many ways to express the geometry of a curve or surface: •height function over tangent plane •local parameterization •Christoffel symbols — coordinates/indices •differential forms — “coordinate free” •moving frames — change in adapted frame •Riemann surfaces (local); Quaternionic functions
  • Lecture 2 Tangent Space, Differential Forms, Riemannian Manifolds

    Lecture 2 Tangent Space, Differential Forms, Riemannian Manifolds

    Lecture 2 tangent space, differential forms, Riemannian manifolds differentiable manifolds A manifold is a set that locally look like Rn. For example, a two-dimensional sphere S2 can be covered by two subspaces, one can be the northen hemisphere extended slightly below the equator and another can be the southern hemisphere extended slightly above the equator. Each patch can be mapped smoothly into an open set of R2. In general, a manifold M consists of a family of open sets Ui which covers M, i.e. iUi = M, n ∪ and, for each Ui, there is a continuous invertible map ϕi : Ui R . To be precise, to define → what we mean by a continuous map, we has to define M as a topological space first. This requires a certain set of properties for open sets of M. We will discuss this in a couple of weeks. For now, we assume we know what continuous maps mean for M. If you need to know now, look at one of the standard textbooks (e.g., Nakahara). Each (Ui, ϕi) is called a coordinate chart. Their collection (Ui, ϕi) is called an atlas. { } The map has to be one-to-one, so that there is an inverse map from the image ϕi(Ui) to −1 Ui. If Ui and Uj intersects, we can define a map ϕi ϕj from ϕj(Ui Uj)) to ϕi(Ui Uj). ◦ n ∩ ∩ Since ϕj(Ui Uj)) to ϕi(Ui Uj) are both subspaces of R , we express the map in terms of n ∩ ∩ functions and ask if they are differentiable.
  • MA4E0 Lie Groups

    MA4E0 Lie Groups

    MA4E0 Lie Groups December 5, 2012 Contents 0 Topological Groups 1 1 Manifolds 2 2 Lie Groups 3 3 The Lie Algebra 6 4 The Tangent space 8 5 The Lie Algebra of a Lie Group 10 6 Integrating Vector fields and the exponential map 13 7 Lie Subgroups 17 8 Continuous homomorphisms are smooth 22 9 Distributions and Frobenius Theorem 23 10 Lie Group topics 24 These notes are based on the 2012 MA4E0 Lie Groups course, taught by John Rawnsley, typeset by Matthew Egginton. No guarantee is given that they are accurate or applicable, but hopefully they will assist your study. Please report any errors, factual or typographical, to [email protected] i MA4E0 Lie Groups Lecture Notes Autumn 2012 0 Topological Groups Let G be a group and then write the group structure in terms of maps; multiplication becomes m : G × G ! G −1 defined by m(g1; g2) = g1g2 and inversion becomes i : G ! G defined by i(g) = g . If we suppose that there is a topology on G as a set given by a subset T ⊂ P (G) with the usual rules. We give G × G the product topology. Then we require that m and i are continuous maps. Then G with this topology is a topological group Examples include 1. G any group with the discrete topology 2. Rn with the Euclidean topology and m(x; y) = x + y and i(x) = −x 1 2 2 2 3. S = f(x; y) 2 R : x + y = 1g = fz 2 C : jzj = 1g with m(z1; z2) = z1z2 and i(z) =z ¯.
  • The Ricci Curvature of Rotationally Symmetric Metrics

    The Ricci Curvature of Rotationally Symmetric Metrics

    The Ricci curvature of rotationally symmetric metrics Anna Kervison Supervisor: Dr Artem Pulemotov The University of Queensland 1 1 Introduction Riemannian geometry is a branch of differential non-Euclidean geometry developed by Bernhard Riemann, used to describe curved space. In Riemannian geometry, a manifold is a topological space that locally resembles Euclidean space. This means that at any point on the manifold, there exists a neighbourhood around that point that appears ‘flat’and could be mapped into the Euclidean plane. For example, circles are one-dimensional manifolds but a figure eight is not as it cannot be pro- jected into the Euclidean plane at the intersection. Surfaces such as the sphere and the torus are examples of two-dimensional manifolds. The shape of a manifold is defined by the Riemannian metric, which is a measure of the length of tangent vectors and curves in the manifold. It can be thought of as locally a matrix valued function. The Ricci curvature is one of the most sig- nificant geometric characteristics of a Riemannian metric. It provides a measure of the curvature of the manifold in much the same way the second derivative of a single valued function provides a measure of the curvature of a graph. Determining the Ricci curvature of a metric is difficult, as it is computed from a lengthy ex- pression involving the derivatives of components of the metric up to order two. In fact, without additional simplifications, the formula for the Ricci curvature given by this definition is essentially unmanageable. Rn is one of the simplest examples of a manifold.