DOCSLIB.ORG

CHAPTER 2 MANIFOLDS in This Chapter, We Address the Basic Notions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
CHAPTER 2 MANIFOLDS in This Chapter, We Address the Basic Notions CHAPTER 2 MANIFOLDS In this chapter, we address the basic notions: What is a manifold and what is a map between manifolds. Several examples are given. An n dimensional manifold is a topological space that appears to be Rn near a point, i.e., locally like Rn. Since these topological spaces appear to be locally like Rn, we may hope to develop tools similar to those used to study Rn in order to study manifolds. The term manifold comes from \many fold," and it refers to the many dimensions of space that a manifold may describe. Later in this section we will also need structure to discuss C 1 functions on manifolds. Our notion of smooth will be C1, i.e., continuous partial derivatives of all orders. The terms C1 and smooth are usually synonymous, however, in this chapter and the next, we will use C1 to describe maps between real vector spaces (as in advanced Calculus) and smooth for maps between manifolds. We use this distinction since many confusing compositions occur and the reader is assumed to be familiar with C 1 maps between real vector spaces while results on smooth maps between manifolds must be proven. We begin with the notion of a topological manifold. Definition 2.1***. A topological manifold of dimension n is a second countable Haus- dorff space M for which each point has a neighborhood homeomorphic to an open set in Rn. This notion appears to capture the topological ideal of locally looking like Rn, but in order to do Calculus we will need more structure. The term n-manifold is usually written for n dimensional manifold, and the dimension n is often suppressed. Definition 2.2***. Let U ⊂ M be a connected open set in a topological n-manifold M, and φ : U ! Rn be a homeomorphism to its image φ(U), an open set in Rn. n The pair (U; φ) is called a coordinate system or chart. If x0 2 U and φ(x0) = 0 2 R , then the coordinate system is centered at x0. Call the map φ a coordinate map. Definition 2.3***. Suppose M is a topological manifold. For k 2 N; k = 0, or k = 1, k a C atlas is a set of charts f(Ui; φi) j i 2 Ig such that (1) i2I Ui = M, and −1 k (2) φSi ◦ φj is C on its domain for all i; j 2 I. The k denotes the degree of differentiability. If k = 0, then maps are just continuous. k n Definition 2.4***. If A = f(Ui; φi)ji 2 Ig is a C atlas for an n-manifold M and f : U ! Rn is a homeomorphism onto its image with U ⊂ M open, then (U; f) is −1 k −1 k compatible with A if φi ◦ f : f(U \ Ui) ! φ(U \ Ui) is C and f ◦ φi is C for all i 2 I. copyright c 2002 Larry Smolinsky Typeset by AMS-TEX 1 2 CHAPTER 2 MANIFOLDS k n Theorem 2.5***. If A = f(Ui; φi)ji 2 Ig is a C atlas for M then A is contained in a unique maximal atlas for M where atlases are ordered as sets by containment. Proof. Let n M =f(Vα; fα)jVα ⊂ M is open; fα : Vα ! R is a homeomorphism onto its image, and (Vα; fα) is compatible with Ag: Since M contains all compatible charts it is the unique maximal atlas if it is an atlas. We now show that it is an atlas. If (Vα; fα) and (Vβ; fβ) are in M then we must show that −1 k fβ ◦ fα : fα(Vα \ Vβ) ! fβ(Vα \ Vβ) is C . Suppose m 2 Vα \ Vβ. Take i 2 I such that −1 −1 −1 m 2 Ui. Then on fα(Vα \ Vβ \ Ui), fβ ◦ fα = (fβ ◦ φi ) ◦ (φi ◦ fα ). Since fβ and fα are −1 −1 k n n compatible with A, fβ ◦ φi and φi ◦ fα are C on open sets of R to open sets of R . −1 k Therefore fβ ◦ fα is C on its domain. Definition 2.6***. A maximal Ck atlas is called a Ck differential structure. Definition 2.7***. A Ck n-manifold is a topological n-manifold M along with a Ck differential structure S. By Theorem 2.5***, a single atlas is enough to determine the differential structure. The reader should note that this definition for a C0 structure agrees with the definition of a topological manifold. A C1 n-manifold is also called a smooth manifold. The word \manifold," without other adjectives, will denote a smooth manifold as these will be the subject of the remainder of this manuscript. Usually the notation for the structure S is suppressed. However, the phrase \a manifold M" supposes that there is an unnamed differential structure S in the background. In particular this means that if A is an atlas, then A ⊂ S; and, if (U; φ) is a chart, then (U; φ) 2 S. The differential structure contains all compatible charts. For example, if (U; φ) 2 S and V ⊂ U is open, then (V; φjV ) 2 S. Example 2.8.a***. A real n-dimensional vector space is an n-manifold. Let V be an n-dimensional vector space. Pick an ordered basis v1; · · · ; vn and define n n charts (V; fv1;··· ;vn ), where fv1;··· ;vn ( aivi) = (a1; · · · ; an) is an isomorphism of V to R . Xi=1 The image is all of Rn, which is an open subset of itself. These charts are compatible since −1 n fw1;··· ;wn fv1;··· ;vn is a linear automorphism on R . Recall that every real vector space is isomorphic to Rn, however Rn comes equiped with a standard ordered basis or set of coordinates. A real vector space may appear in several guises. We now mention a few instances of vector spaces that relate to matrices. Example 2.8.b***. Let the set of n × n real matrices be denoted Matn×n. The set 2 Matn×n is a real vector space of dimension n . m Let e1; e2; · · · ; em be the standard ordered basis for R . If eij is the matrix with 1 in the i-th row and j-th column and zero elsewhere, then feij j 1 ≤ i; j ≤ ng is a basis. n2 The isomorphism of Matn×n to R is determined by the linear map f(eij) = en(i−1)+j 2 for 1 ≤ i; j ≤ n, and allows one to think of the matrix entries as coordinates in Rn . In coordinates, if A = (aij), then f(A) = (a11; a12; · · · ; a1n; a21; · · · ; ann). CHAPTER 2 MANIFOLDS 3 t Example 2.8.c***. The symmetric matrices Symn×n = fA 2 Matn×n j A = Ag form a n(n + 1) vector subspace of dimension . 2 The entries on and above the diagonal can be arbitrary. Below the diagonal, the entries are determined by symmetry, i.e., aij = aji. t Example 2.8.d***. The skew symmetric matrices Skewn×n = fA 2 Matn×n j A = −Ag n(n − 1) form a vector subspace of dimension . 2 The entries on the diagonal must be zero, since aij = −aji. The entries above the diagonal can be arbitrary and the entries below are determined by the antisymmetry, i.e., aij = −aji. Example 2.9a***. The sphere Sn = fx 2 Rn+1 j jxj = 1g is an n-manifold. We construct an atlas f(U1; φ1); (U2; φ2)g with the aid of a standard well-known map n n called stereographic projection. Let U1 = S nf(0; · · · ; 0; 1)g and U2 = S nf(0; · · · ; 0; −1)g. n x1 xn Note that U [ U = S . Let φ (x ; x ; · · · ; xn ) = ( ; · · · ; ). The map 1 2 1 1 2 +1 1−xn+1 1−xn+1 n −1 n φ1 : U1 ! R is called stereographic projection. The inverse map φ1 : R ! U1 is defined by −1 2y1 2y2 2yn 2 φ1 (y1; · · · ; yn) = n 2 ; n 2 ; · · · ; n 2 ; 1 − n 2 : i=1 yi + 1 i=1 yi + 1 i=1 yi + 1 i=1 yi + 1 P P P P −1 Both φ1 and φ1 are continuous and hence φ1 is a homeomorphism. The second coordinate chart (U2; φ2), stereographic projection from the south pole, is given by φ2 = −φ1 ◦ (−1) where (-1) is multiplication by −1 on the sphere. Since multiplication by -1 is a homeomorphism of the sphere to itself (its inverse is itself), the n map φ2 : U2 ! R is a homeomorphism. −1 1 Checking the compatability conditions, we have φ2◦φ1 (y1; · · · ; yn) = n 2 (y1; · · · ; yn) i=1 yi −1 −1 n and φ2 ◦ φ1 = φ1 ◦ φ2 . Hence, S is shown to be an n-manifold. P Example 2.9b***. Another atlas for the sphere Sn. We use 2(n + 1) coordinate charts to construct this atlas. For each i 2 f1; · · · ; n + 1g let Ui;+ = f(x1; · · · ; xn+1) j xi > 0g and Ui;− = f(x1; · · · ; xn+1) j xi > 0g. Define φi;+ : n n Ui;+ ! R by φi;+(x1; · · · ; xn+1) = (x1; · · · ; xi−1; xi+1; · · · ; xn+1) and φi;+ : Ui;− ! R by φi;−(x1; · · · ; xn+1) = (x1; · · · ; xi−1; xi+1; · · · ; xn+1). The coordinate xi is a function of x1; · · · ; xi−1; xi+1; · · · ; xn+1 on the sets Ui;+ and Ui;−. The atlases in Examples 2.9a*** and 2.9b*** are compatible and give the same differ- ential structure. See Exercise 1***. n m Example 2.10***. Suppose U1 ⊂ R and U2 ⊂ R are open sets.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Examples of Manifolds
    Examples of Manifolds Example 1 (Open Subset of IRn) Any open subset, O, of IRn is a manifold of dimension n. One possible atlas is A = (O, ϕid) , where ϕid is the identity map. That is, ϕid(x) = x. n Of course one possible choice of O is IR itself. Example 2 (The Circle) The circle S1 = (x,y) ∈ IR2 x2 + y2 = 1 is a manifold of dimension one. One possible atlas is A = {(U , ϕ ), (U , ϕ )} where 1 1 1 2 1 y U1 = S \{(−1, 0)} ϕ1(x,y) = arctan x with − π < ϕ1(x,y) <π ϕ1 1 y U2 = S \{(1, 0)} ϕ2(x,y) = arctan x with 0 < ϕ2(x,y) < 2π U1 n n n+1 2 2 Example 3 (S ) The n–sphere S = x =(x1, ··· ,xn+1) ∈ IR x1 +···+xn+1 =1 n A U , ϕ , V ,ψ i n is a manifold of dimension . One possible atlas is 1 = ( i i) ( i i) 1 ≤ ≤ +1 where, for each 1 ≤ i ≤ n + 1, n Ui = (x1, ··· ,xn+1) ∈ S xi > 0 ϕi(x1, ··· ,xn+1)=(x1, ··· ,xi−1,xi+1, ··· ,xn+1) n Vi = (x1, ··· ,xn+1) ∈ S xi < 0 ψi(x1, ··· ,xn+1)=(x1, ··· ,xi−1,xi+1, ··· ,xn+1) n So both ϕi and ψi project onto IR , viewed as the hyperplane xi = 0. Another possible atlas is n n A2 = S \{(0, ··· , 0, 1)}, ϕ , S \{(0, ··· , 0, −1)},ψ where 2x1 2xn ϕ(x , ··· ,xn ) = , ··· , 1 +1 1−xn+1 1−xn+1 2x1 2xn ψ(x , ··· ,xn ) = , ··· , 1 +1 1+xn+1 1+xn+1 are the stereographic projections from the north and south poles, respectively.
    [Show full text]
  • An Introduction to Topology the Classification Theorem for Surfaces by E
    An Introduction to Topology An Introduction to Topology The Classification theorem for Surfaces By E. C. Zeeman Introduction. The classification theorem is a beautiful example of geometric topology. Although it was discovered in the last century*, yet it manages to convey the spirit of present day research. The proof that we give here is elementary, and its is hoped more intuitive than that found in most textbooks, but in none the less rigorous. It is designed for readers who have never done any topology before. It is the sort of mathematics that could be taught in schools both to foster geometric intuition, and to counteract the present day alarming tendency to drop geometry. It is profound, and yet preserves a sense of fun. In Appendix 1 we explain how a deeper result can be proved if one has available the more sophisticated tools of analytic topology and algebraic topology. Examples. Before starting the theorem let us look at a few examples of surfaces. In any branch of mathematics it is always a good thing to start with examples, because they are the source of our intuition. All the following pictures are of surfaces in 3-dimensions. In example 1 by the word “sphere” we mean just the surface of the sphere, and not the inside. In fact in all the examples we mean just the surface and not the solid inside. 1. Sphere. 2. Torus (or inner tube). 3. Knotted torus. 4. Sphere with knotted torus bored through it. * Zeeman wrote this article in the mid-twentieth century. 1 An Introduction to Topology 5.
    [Show full text]
  • Affine Connections and Second-Order Affine Structures
    Affine connections and second-order affine structures Filip Bár Dedicated to my good friend Tom Rewwer on the occasion of his 35th birthday. Abstract Smooth manifolds have been always understood intuitively as spaces with an affine geometry on the infinitesimal scale. In Synthetic Differential Geometry this can be made precise by showing that a smooth manifold carries a natural struc- ture of an infinitesimally affine space. This structure is comprised of two pieces of data: a sequence of symmetric and reflexive relations defining the tuples of mu- tual infinitesimally close points, called an infinitesimal structure, and an action of affine combinations on these tuples. For smooth manifolds the only natural infin- itesimal structure that has been considered so far is the one generated by the first neighbourhood of the diagonal. In this paper we construct natural infinitesimal structures for higher-order neighbourhoods of the diagonal and show that on any manifold any symmetric affine connection extends to a second-order infinitesimally affine structure. Introduction A deeply rooted intuition about smooth manifolds is that of spaces that become linear spaces in the infinitesimal neighbourhood of each point. On the infinitesimal scale the geometry underlying a manifold is thus affine geometry. To make this intuition precise requires a good theory of infinitesimals as well as defining precisely what it means for two points on a manifold to be infinitesimally close. As regards infinitesimals we make use of Synthetic Differential Geometry (SDG) and adopt the neighbourhoods of the diagonal from Algebraic Geometry to define when two points are infinitesimally close. The key observations on how to proceed have been made by Kock in [8]: 1) The first neighbourhood arXiv:1809.05944v2 [math.DG] 29 Aug 2020 of the diagonal exists on formal manifolds and can be understood as a symmetric, reflexive relation on points, saying when two points are infinitesimal neighbours, and 2) we can form affine combinations of points that are mutual neighbours.
    [Show full text]
  • 21. Orthonormal Bases
    21. Orthonormal Bases The canonical/standard basis 011 001 001 B C B C B C B0C B1C B0C e1 = B.C ; e2 = B.C ; : : : ; en = B.C B.C B.C B.C @.A @.A @.A 0 0 1 has many useful properties. • Each of the standard basis vectors has unit length: q p T jjeijj = ei ei = ei ei = 1: • The standard basis vectors are orthogonal (in other words, at right angles or perpendicular). T ei ej = ei ej = 0 when i 6= j This is summarized by ( 1 i = j eT e = δ = ; i j ij 0 i 6= j where δij is the Kronecker delta. Notice that the Kronecker delta gives the entries of the identity matrix. Given column vectors v and w, we have seen that the dot product v w is the same as the matrix multiplication vT w. This is the inner product on n T R . We can also form the outer product vw , which gives a square matrix. 1 The outer product on the standard basis vectors is interesting. Set T Π1 = e1e1 011 B C B0C = B.C 1 0 ::: 0 B.C @.A 0 01 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 0 . T Πn = enen 001 B C B0C = B.C 0 0 ::: 1 B.C @.A 1 00 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 1 In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else.
    [Show full text]
  • Algebraic Equations for Nonsmoothable 8-Manifolds
    PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. NICHOLAAS H. KUIPER Algebraic equations for nonsmoothable 8-manifolds Publications mathématiques de l’I.H.É.S., tome 33 (1967), p. 139-155 <http://www.numdam.org/item?id=PMIHES_1967__33__139_0> © Publications mathématiques de l’I.H.É.S., 1967, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ALGEBRAIC EQUATIONS FOR NONSMOOTHABLE 8-MANIFOLDS by NICOLAAS H. KUIPER (1) SUMMARY The singularities of Brieskorn and Hirzebruch are used in order to obtain examples of algebraic varieties of complex dimension four in P^C), which are homeo- morphic to closed combinatorial 8-manifolds, but not homeomorphic to any differentiable manifold. Analogous nonorientable real algebraic varieties of dimension 8 in P^R) are also given. The main theorem states that every closed combinatorial 8-manifbld is homeomorphic to a Nash-component with at most one singularity of some real algebraic variety. This generalizes the theorem of Nash for differentiable manifolds. § i. Introduction. The theorem of Wall. From the smoothing theory of Thorn [i], Munkres [2] and others and the knowledge of the groups of differential structures on spheres due to Kervaire, Milnor [3], Smale and Gerf [4] follows a.o.
    [Show full text]
  • Vectors, Matrices and Coordinate Transformations
    S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between them, physical laws can often be written in a simple form. Since we will making extensive use of vectors in Dynamics, we will summarize some of their important properties. Vectors For our purposes we will think of a vector as a mathematical representation of a physical entity which has both magnitude and direction in a 3D space. Examples of physical vectors are forces, moments, and velocities. Geometrically, a vector can be represented as arrows. The length of the arrow represents its magnitude. Unless indicated otherwise, we shall assume that parallel translation does not change a vector, and we shall call the vectors satisfying this property, free vectors. Thus, two vectors are equal if and only if they are parallel, point in the same direction, and have equal length. Vectors are usually typed in boldface and scalar quantities appear in lightface italic type, e.g. the vector quantity A has magnitude, or modulus, A = |A|. In handwritten text, vectors are often expressed using the −→ arrow, or underbar notation, e.g. A , A. Vector Algebra Here, we introduce a few useful operations which are defined for free vectors. Multiplication by a scalar If we multiply a vector A by a scalar α, the result is a vector B = αA, which has magnitude B = |α|A. The vector B, is parallel to A and points in the same direction if α > 0.
    [Show full text]
  • Multivector Differentiation and Linear Algebra 0.5Cm 17Th Santaló
    Multivector differentiation and Linear Algebra 17th Santalo´ Summer School 2016, Santander Joan Lasenby Signal Processing Group, Engineering Department, Cambridge, UK and Trinity College Cambridge [email protected], www-sigproc.eng.cam.ac.uk/ s jl 23 August 2016 1 / 78 Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. 2 / 78 Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. 3 / 78 Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. 4 / 78 Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... 5 / 78 Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary 6 / 78 We now want to generalise this idea to enable us to find the derivative of F(X), in the A ‘direction’ – where X is a general mixed grade multivector (so F(X) is a general multivector valued function of X). Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. Then, provided A has same grades as X, it makes sense to define: F(X + tA) − F(X) A ∗ ¶XF(X) = lim t!0 t The Multivector Derivative Recall our definition of the directional derivative in the a direction F(x + ea) − F(x) a·r F(x) = lim e!0 e 7 / 78 Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi.
    [Show full text]
  • 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]
  • 4. Groups of Permutations 1
    4. Groups of permutations 1 4. Groups of permutations Consider a set of n distinguishable objects, fB1;B2;B3;:::;Bng. These may be arranged in n! different ways, called permutations of the set. Permu- tations can also be thought of as transformations of a given ordering of the set into other orderings. A convenient notation for specifying a given permutation operation is 1 2 3 : : : n ! , a1 a2 a3 : : : an where the numbers fa1; a2; a3; : : : ; ang are the numbers f1; 2; 3; : : : ; ng in some order. This operation can be interpreted in two different ways, as follows. Interpretation 1: The object in position 1 in the initial ordering is moved to position a1, the object in position 2 to position a2,. , the object in position n to position an. In this interpretation, the numbers in the two rows of the permutation symbol refer to the positions of objects in the ordered set. Interpretation 2: The object labeled 1 is replaced by the object labeled a1, the object labeled 2 by the object labeled a2,. , the object labeled n by the object labeled an. In this interpretation, the numbers in the two rows of the permutation symbol refer to the labels of the objects in the ABCD ! set. The labels need not be numerical { for instance, DCAB is a well-defined permutation which changes BDCA, for example, into CBAD. Either of these interpretations is acceptable, but one interpretation must be used consistently in any application. The particular application may dictate which is the appropriate interpretation to use. Note that, in either interpre- tation, the order of the columns in the permutation symbol is irrelevant { the columns may be written in any order without affecting the result, provided each column is kept intact.
    [Show full text]
  • Graphs and Patterns in Mathematics and Theoretical Physics, Volume 73
    http://dx.doi.org/10.1090/pspum/073 Graphs and Patterns in Mathematics and Theoretical Physics This page intentionally left blank Proceedings of Symposia in PURE MATHEMATICS Volume 73 Graphs and Patterns in Mathematics and Theoretical Physics Proceedings of the Conference on Graphs and Patterns in Mathematics and Theoretical Physics Dedicated to Dennis Sullivan's 60th birthday June 14-21, 2001 Stony Brook University, Stony Brook, New York Mikhail Lyubich Leon Takhtajan Editors Proceedings of the conference on Graphs and Patterns in Mathematics and Theoretical Physics held at Stony Brook University, Stony Brook, New York, June 14-21, 2001. 2000 Mathematics Subject Classification. Primary 81Txx, 57-XX 18-XX 53Dxx 55-XX 37-XX 17Bxx. Library of Congress Cataloging-in-Publication Data Stony Brook Conference on Graphs and Patterns in Mathematics and Theoretical Physics (2001 : Stony Brook University) Graphs and Patterns in mathematics and theoretical physics : proceedings of the Stony Brook Conference on Graphs and Patterns in Mathematics and Theoretical Physics, June 14-21, 2001, Stony Brook University, Stony Brook, NY / Mikhail Lyubich, Leon Takhtajan, editors. p. cm. — (Proceedings of symposia in pure mathematics ; v. 73) Includes bibliographical references. ISBN 0-8218-3666-8 (alk. paper) 1. Graph Theory. 2. Mathematics-Graphic methods. 3. Physics-Graphic methods. 4. Man• ifolds (Mathematics). I. Lyubich, Mikhail, 1959- II. Takhtadzhyan, L. A. (Leon Armenovich) III. Title. IV. Series. QA166.S79 2001 511/.5-dc22 2004062363 Copying and reprinting. Material in this book may be reproduced by any means for edu• cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg• ment of the source is given.
    [Show full text]
  • EXOTIC SPHERES and CURVATURE 1. Introduction Exotic
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 45, Number 4, October 2008, Pages 595–616 S 0273-0979(08)01213-5 Article electronically published on July 1, 2008 EXOTIC SPHERES AND CURVATURE M. JOACHIM AND D. J. WRAITH Abstract. Since their discovery by Milnor in 1956, exotic spheres have pro- vided a fascinating object of study for geometers. In this article we survey what is known about the curvature of exotic spheres. 1. Introduction Exotic spheres are manifolds which are homeomorphic but not diffeomorphic to a standard sphere. In this introduction our aims are twofold: First, to give a brief account of the discovery of exotic spheres and to make some general remarks about the structure of these objects as smooth manifolds. Second, to outline the basics of curvature for Riemannian manifolds which we will need later on. In subsequent sections, we will explore the interaction between topology and geometry for exotic spheres. We will use the term differentiable to mean differentiable of class C∞,and all diffeomorphisms will be assumed to be smooth. As every graduate student knows, a smooth manifold is a topological manifold that is equipped with a smooth (differentiable) structure, that is, a smooth maximal atlas. Recall that an atlas is a collection of charts (homeomorphisms from open neighbourhoods in the manifold onto open subsets of some Euclidean space), the domains of which cover the manifold. Where the chart domains overlap, we impose a smooth compatibility condition for the charts [doC, chapter 0] if we wish our manifold to be smooth. Such an atlas can then be extended to a maximal smooth atlas by including all possible charts which satisfy the compatibility condition with the original maps.
    [Show full text]