DOCSLIB.ORG

Differential Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Differential Geometry Differential Geometry Differentiable Manifolds Definition of topological manifold : It is a topological space (E; τ) so that 1. It is Hausdorff. 2. 8x 2 E there exists (U; ') with U open and x 2 U, such that ' : U ! '(U) is a homeomorphism. The pair (U; ') is called chart and the real numbers (x1; : : : ; xn) = '(x) are called local coordinates. 3. (E; τ) has a countable basis of open sets. Remark. Topological Manifolds are paracompact, i.e, every open cover has a locally finite refinement. Remark. Paracompactness implies having Partition of unity. Definition. A differentiable [C1;Ck;C!] structure on a topological ma- nifold M is a family of charts U = f(Uα;'αg so that S 1. Uα = M −1 2. If Uα \ Uβ 6= ; then 'β ◦ 'α : 'α(Uα \ Uβ) ! 'β(Uα \ Uβ) are differentiable [C1;Ck;C!] with differentiable [C1;Ck;C!] inverse. In this case we say that (Uα;'α) and (Uβ;'β) are compatible. 3. Completness property: If (V; ) is a chart which is compatible with every (Uα;'α) 2 U then (V; ) 2 U. Remark.It is not necessary to verify the third property because of the following proposition. Proposition. Let M be Hausdorff with countable basis of open sets. Let f(Vα; α): α 2 Ag be a covering of M by C1-compatible coordinate charts. Then 9! C1 structure containing the charts f(Vα; α): α 2 Ag: Examples of topological manifolds: 1. (Rn; Id) 2. If M is a C1 manifold and N ⊂ M is an open subset then N is a C1 manifold too. 1 3. By the previous example GL(n; R) is a manifold. 4. If M and N are C1 manifolds so is M × N. 5. S2 is a C! manifold. X Definition of open equivalence relation: Let π : X ! ∼ be the projection S −1 and [U] := x2U fπ (x)g. We will say that ∼ is open iff [U] is open for every U open. Proposition. 1. ∼ is open iff π : X ! X= ∼ is open. 2. If ∼ is open and X has a countable basis of open sets, then so does X= ∼. Definition. Let ∼ be an equiv. relation on X. The graph R of ∼ is the subset of X × X defined by R = f(x; y): x ∼ yg. Proposition. Let ∼ be an open eq. relation on a Hausdorff space. Then X= ∼ is Hausdorff iff R is closed in X × X. Applications. n+1 1. Pn(R) is a manifold because ∼ is open and its graph is closed .(Recall, R −0 is endowed with the equivalence relation x ∼ y iff 9t 6= 0 so that x = ty. n−1 R −0 Pn(R) := ∼ .) 2. G(n; k) is a C1 manifold of dimension k(n−k). ("G(n; k) = fk-dim’l subspaces n F (n;k) kn of R g", actually G(n; k) := ∼ where F (k; n) is a manifold in R of k- frames in Rn). Definition. Let M be a manifold and W ⊂ M open. We will say that f : W ! R is a differentiable [C1;Ck;C!] map if 8x 2 W 9(U; ') coordinate chart (with x 2 U) so that f ◦ '−1 : '(W \ U) ! R is C1. Definition. Let A ⊂ M. We say that f : A ! R is C1 if it has a C1 extension to an open set U ⊂ M such that A ⊂ U. Definition. Let M and N be differentiable [C1;Ck;C!] manifolds. We will say that f : M ! N is a differentiable [C1;Ck;C!] map if 8x 2 M 9(U; ') coordinate chart with x 2 U and (V; ) coordinate with f(x) 2 V so that f(U) ⊂ V and ◦f ◦'−1 : '(W \U) ! R is differentiable [C1;Ck;C!] as a map of euclidean spaces. 2 Definition. Let M and N be C1 manifolds and f : M ! N be a C1 map. We define the rank of f as the rank of ◦ f ◦ '−1 where (U; ') and (V; ) are charts as h @yi i −1 1 n above. That is, (rank f)x := rank @xj j'(x) where ◦ f ◦ ' = (y ; : : : ; y ). Definition. Let f; g be C1 in a neighb. of x 2 M. We will say that f ∼ g if 9U open so that f(y) = g(y) 8y 2 U. The class [f] is called germ of C1 function at x. The set of germs at x will be denoted by C1(x; R). Definition. Let x 2 M and X : C1(x; R) ! R. If for every chart (U; ') about x we have that there exist a1; : : : ; an 2 R so that n X @ −1 X([f]) = ai (f ◦ ' ) ; @x '(x) i=1 i we will say that X is a tangent vector at x . Observation. If the equation holds for some chart (U; ') about x, then it holds for every C1-compatible chart overlapping at x. Properties of tangent vectors: 1. X([f] + [g]) = X([f]) + X([g]) 2. X([λf]) = λX([f]) 8λ 2 R 3. X([f][g]) = X([f])[g] + [f]X([g]) Definition. The tangent space to M at x is the vector space of tangent vectors based at x. We will denote it by TxM. Remark. If n = dimM then dimTxM = n. @ Definition. Given a coordinate chart (U; ') about x 2 M the basis @xi x ; 1 ≤ i ≤ n is called the natural basis of TxM associated to this chart. @ @f◦'−1 Here, @xi jx [f] = @xi j'(x). Definition. Given f : M ! N a C1 map and x 2 M we define the push-forward of f at x as follows: f∗ jx: TxM ! Tf(x)N (f∗ jx X)([g]) := X([g ◦ f]) @ −1 @ Remark. With this definition we have @xi jx= '∗ ( @xi j'(x)). 3 Remark. Let ' and be charts about x. Let ◦ '−1(x1; : : : ; xn) = (y1(x1; : : : ; xn); : : : ; yn(x1; : : : ; xn)): Then @ X @yk @ = ('(x)) : @xi x @xi @yk x k Definition. We define the cotangent bundle to x at M to be the dual space ∗ of TxM. We denote it by Tx M. i @ Remark. Let fdx jx; 1 ≤ i ≤ ng be the dual basis of @xi jx; 1 ≤ i ≤ n , then we ∗ will say that it is the natural basis of Tx M. Remark. Let ' and be charts about x. Let ' ◦ −1(y1; : : : ; yn) = (x1(y1; : : : ; yn); : : : ; xn(y1; : : : ; yn)): Then X @xi dxi = ( (x))dyk : x @yk x k Definition. The tangent bundle of M, denoted by TM is defined as follows: a TM := TxM: x2M Proposition. Let π : TM ! M be defined by π(X) = x if X 2 TxM. Then there is a unique topology on TM so that for each coordinate chart (U; ') on M the set U~ = π−1(U) is open and the map '~ : U~ ! '(U) × Rn defined by n ! X i @ 1 n '~ v := ('(x); v ; : : : ; v ) @xi i=1 x is a homeomorphism. With this topology, TM is a topological manifold of dimension 2n and the coordi- nate charts (U;~ '~) define a differentiable [C1;Ck;C!] structure on TM relative to which π is an open mapping. Definition. The cotangent bundle of M, denoted by T ∗M is defined as follows: ∗ a ∗ T M := Tx M: x2M 4 1 Definition. A C map f : M ! N is said to be an inmersion if f∗ is injective at every point. Definition. An inmersed submanifold of a manifold N is a pair (M; f) where f : M ! N is an injective inmersion. Definition. An embedded submanifold of a manifold N is an inmersed sub- manifold (M; f) such that f : M ! f(M) is a homeomorphism. Theorem [Inverse function Theorem]. Let M and N be C1 manifolds of dimension 1 n. Let f : M ! N be a C map. Suppose that xo 2 M is such that f∗ jxo : Txo M ! Tf(xo)N is an isomorphism. Then there exists an open neighb. U of xo so that 1. f j U is injective. 2. f(U) is open in N. 3. f −1 : f(U) ! U is C1. Theorem [Implicit function Theorem]. Let M and N be C1 manifolds with 1 −1 dimM > dimN and let f : M ! N be C . Let yo 2 f(M) and let Mo = f (yo). Suppose that 8x 2 Mo the map f∗ jx is surjective. Then Mo can be endowed with 1 a C structure relative to which the inclusion map { : Mo ! M is an embedding. Furthermore, dimMo = dimM − dimN. Applications. 1. Sn is a C1 manifold of dimension n. 2. SL(n; R) = fX 2 GL(n; R) : det X = 1g is a C1 manifold of dimension n2 − 1. t 1 3. O(n; R) = fX 2 GL(n; R): X X = Idng is a C manifold of dimension n(n − 1)=2. Differential forms Definition. Let V be a real n-dimensional space and let V ∗ be its dual space. We define the space of alternating k-forms as follows: k ∗ Λ (V ) = f! : V ⊕ · · · ⊕ V (k times) ! R : ! is linear and alternatingg: k ∗ n!k! Observe that dimΛ (V ) = (n−k)! . 5 Remark. ! is linear and alternating if !(v1; : : : ; vn) is linear in each argument and π !(vπ(1); : : : ; vπ(k)) = (−1) !(v1; : : : ; vn): Definition. Let Λ0(V ∗) := R. We define the grassman algebra of V ∗ as follows n M Λ∗(V ∗) := Λk(V ∗): k=0 Observe that with this definition we have dimΛ∗(V ∗) = 2n. Λ∗(V ∗) will be endowed with a graded product operation as follows: ^ :Λk(V ∗) × Λl(V ∗) ! Λk+l(V ∗)(!; τ) 7! ! ^ τ; where 1 X ! ^ τ(v ; : : : ; v ) := (−1)π!(v ; : : : ; v )τ(v ; : : : ; v ): 1 k+l (k + l)! π(1) π(k) π(k+1) π(k+l) π2Sk+l Remark.
Load more

Recommended publications
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • LECTURE 1. Differentiable Manifolds, Differentiable Maps
    LECTURE 1. Differentiable manifolds, differentiable maps Def: topological m-manifold. Locally euclidean, Hausdorff, second-countable space. At each point we have a local chart. (U; h), where h : U ! Rm is a topological embedding (homeomorphism onto its image) and h(U) is open in Rm. Def: differentiable structure. An atlas of class Cr (r ≥ 1 or r = 1 ) for a topological m-manifold M is a collection U of local charts (U; h) satisfying: (i) The domains U of the charts in U define an open cover of M; (ii) If two domains of charts (U; h); (V; k) in U overlap (U \V 6= ;) , then the transition map: k ◦ h−1 : h(U \ V ) ! k(U \ V ) is a Cr diffeomorphism of open sets in Rm. (iii) The atlas U is maximal for property (ii). Def: differentiable map between differentiable manifolds. f : M m ! N n (continuous) is differentiable at p if for any local charts (U; h); (V; k) at p, resp. f(p) with f(U) ⊂ V the map k ◦ f ◦ h−1 = F : h(U) ! k(V ) is m differentiable at x0 = h(p) (as a map from an open subset of R to an open subset of Rn.) f : M ! N (continuous) is of class Cr if for any charts (U; h); (V; k) on M resp. N with f(U) ⊂ V , the composition k ◦ f ◦ h−1 : h(U) ! k(V ) is of class Cr (between open subsets of Rm, resp. Rn.) f : M ! N of class Cr is an immersion if, for any local charts (U; h); (V; k) as above (for M, resp.
    [Show full text]
  • DIFFERENTIABLE MANIFOLDS Course C3.1B 2012 Nigel Hitchin
    DIFFERENTIABLE MANIFOLDS Course C3.1b 2012 Nigel Hitchin [email protected] 1 Contents 1 Introduction 4 2 Manifolds 6 2.1 Coordinate charts . .6 2.2 The definition of a manifold . .9 2.3 Further examples of manifolds . 11 2.4 Maps between manifolds . 13 3 Tangent vectors and cotangent vectors 14 3.1 Existence of smooth functions . 14 3.2 The derivative of a function . 16 3.3 Derivatives of smooth maps . 20 4 Vector fields 22 4.1 The tangent bundle . 22 4.2 Vector fields as derivations . 26 4.3 One-parameter groups of diffeomorphisms . 28 4.4 The Lie bracket revisited . 32 5 Tensor products 33 5.1 The exterior algebra . 34 6 Differential forms 38 6.1 The bundle of p-forms . 38 6.2 Partitions of unity . 39 6.3 Working with differential forms . 41 6.4 The exterior derivative . 43 6.5 The Lie derivative of a differential form . 47 6.6 de Rham cohomology . 50 2 7 Integration of forms 57 7.1 Orientation . 57 7.2 Stokes' theorem . 62 8 The degree of a smooth map 68 8.1 de Rham cohomology in the top dimension . 68 9 Riemannian metrics 76 9.1 The metric tensor . 76 9.2 The geodesic flow . 80 10 APPENDIX: Technical results 87 10.1 The inverse function theorem . 87 10.2 Existence of solutions of ordinary differential equations . 89 10.3 Smooth dependence . 90 10.4 Partitions of unity on general manifolds . 93 10.5 Sard's theorem (special case) . 94 3 1 Introduction This is an introductory course on differentiable manifolds.
    [Show full text]
  • INTRODUCTION to MANIFOLDS Contents 1. Differentiable Curves 1
    INTRODUCTION TO MANIFOLDS TSOGTGEREL GANTUMUR Abstract. We discuss manifolds embedded in Rn and the implicit function theorem. Contents 1. Differentiable curves 1 2. Manifolds 3 3. The implicit function theorem 6 4. The preimage theorem 10 5. Tangent spaces 12 1. Differentiable curves Intuitively, a differentiable curves is a curve with the property that the tangent line at each of its points can be defined. To motivate our definition, let us look at some examples. Example 1.1. (a) Consider the function γ : (0; π) ! R2 defined by ( ) cos t γ(t) = : (1) sin t As t varies in the interval (0; π), the point γ(t) traces out the curve C = [γ] ≡ fγ(t): t 2 (0; π)g ⊂ R2; (2) which is a semicircle (without its endpoints). We call γ a parameterization of C. If γ(t) represents the coordinates of a particle in the plane at time t, then the \instantaneous velocity vector" of the particle at the time moment t is given by ( ) 0 − sin t γ (t) = : (3) cos t Obviously, γ0(t) =6 0 for all t 2 (0; π), and γ is smooth (i.e., infinitely often differentiable). The direction of the velocity vector γ0(t) defines the direction of the line tangent to C at the point p = γ(t). (b) Under the substitution t = s2, we obtain a different parameterization of C, given by ( ) cos s2 η(s) ≡ γ(s2) = : (4) sin s2 p Note that the parameter s must take values in (0; π). (c) Now consider the function ξ : [0; π] ! R2 defined by ( ) cos t ξ(t) = : (5) sin t Date: 2016/11/12 10:49 (EDT).
    [Show full text]
  • Introduction to Manifolds
    Introduction to manifolds M. L¨ubke April 17, 2018 1 Contents 1 Submanifolds of Euclidean space 4 1.1 Submanifolds of Rn ..................................... 4 1.2 Tangent spaces . 11 2 Differentiable manifolds 14 2.1 Manifolds . 14 2.2 Differentiable maps . 18 2.3 Tangent spaces . 23 2.4 The tangent map . 32 2.5 Submanifolds . 35 3 Differential forms 39 3.1 The exterior algebra of a manifold . 39 3.2 The exterior differential . 44 4 Integration on manifolds 47 4.1 Orientations on a manifold . 47 4.2 The integral . 51 4.3 Subsets with smooth boundary . 56 4.4 The Theorem of Stokes . 58 4.5 The Integral Theorem of Greene . 63 4.6 The Fixed Point Theorem of Brouwer . 64 5 Appendix: (Multi)Linear algebra 67 5.1 Duality . 67 5.2 Exterior powers . 70 5.3 Orientation of vector spaces . 74 5.4 Tensor products . 75 6 Vector bundles and connections 77 6.1 Vector bundles . 77 6.2 Connections in vector bundles . 85 2 In this reader, by differentiable we always mean ∞. C We will use the following notation. n m Let U R be open and f = (f1, . , fm): U R be a differentiable map. Then the Jacobian ∈ −→ (matrix) of f at x U is ∈ ∂f1 (x) ... ∂f1 (x) ∂f ∂x1 ∂xn i . .. Df(x) := (x) = . . ∂xj i=1,...,m j=1,...,n ∂fm (x) ... ∂fm (x) ∂x1 ∂xn We also recall the following results which should be well known from calculus. Theorem 0.0.1 (Implicit Function Theorem) n m n+m Consider R R = R with coordinates (x, y) = (x1, .
    [Show full text]