DOCSLIB.ORG

MAT 531 Geometry/Topology II Introduction to Smooth Manifolds

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MAT 531 Geometry/Topology II Introduction to Smooth Manifolds MAT 531 Geometry/Topology II Introduction to Smooth Manifolds Claude LeBrun Stony Brook University April 9, 2020 1 Dual of a vector space: 2 Dual of a vector space: Let V be a real, finite-dimensional vector space. 3 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be 4 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: 5 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: ∗ Proposition. V is finite-dimensional vector space, too, and 6 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: ∗ Proposition. V is finite-dimensional vector space, too, and ∗ dimV = dimV: 7 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: ∗ Proposition. V is finite-dimensional vector space, too, and ∗ dimV = dimV: ∗ ∼ In particular, V = V as vector spaces. 8 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: ∗ Proposition. V is finite-dimensional vector space, too, and ∗ dimV = dimV: ∗ ∼ In particular, V = V as vector spaces. But there is no natural, preferred isomorphism! 9 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: ∗ Proposition. V is finite-dimensional vector space, too, and ∗ dimV = dimV: ∗ ∼ In particular, V = V as vector spaces. But there is no natural, preferred isomorphism! Let's not confuse one twin for the other! 10 In fact, 9 contravariant functor from category 11 In fact, 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) 12 In fact, 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) to itself, given by ∗. V V 13 In fact, 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself, given by V V . Namely, if A : V ! W any linear map, 14 In fact, 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself, given by V V . Namely, if A : V ! W any linear map, ∗ ∗ ∗ 9 adjoint map A : W ! V , defined by 15 In fact, 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself, given by V V . Namely, if A : V ! W any linear map, ∗ ∗ ∗ 9 adjoint map A : W ! V , defined by (A∗')(v) = '(A(v)): 16 In fact, 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself, given by V V . Namely, if A : V ! W any linear map, ∗ ∗ ∗ 9 adjoint map A : W ! V , defined by (A∗')(v) = '(A(v)): ∗ ∗ A ∗ V − W A V −! W 17 In fact, 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself, given by V V . Namely, if A : V ! W any linear map, ∗ ∗ ∗ 9 adjoint map A : W ! V , defined by (A∗')(v) = '(A(v)): ∗ ∗ A ∗ V − W A V −! W &# R 18 ∼ n Example. Let V = R be the set of column vectors of length n: 19 ∼ n Example. Let V = R be the set of column vectors of length n: 8 2 a1 3 9 > > < 6 a2 7 = V = 6 . 7 > 4 . 5 > :> an ;> 20 ∼ n Example. Let V = R be the set of column vectors of length n: 8 2 a1 3 9 > > < 6 a2 7 = V = 6 . 7 > 4 . 5 > :> an ;> ∗ ∼ n Then V = R can be identified with the set of row vectors of length n 21 ∼ n Example. Let V = R be the set of column vectors of length n: 8 2 a1 3 9 > > < 6 a2 7 = V = 6 . 7 > 4 . 5 > :> an ;> ∗ ∼ n Then V = R can be identified with the set of row vectors of length n ∗ V = b1 b2 ··· bn 22 ∼ n Example. Let V = R be the set of column vectors of length n: 8 2 a1 3 9 > > < 6 a2 7 = V = 6 . 7 > 4 . 5 > :> an ;> ∗ ∼ n Then V = R can be identified with the set of row vectors of length n ∗ V = b1 b2 ··· bn because any linear map V ! R takes the form 23 ∼ n Example. Let V = R be the set of column vectors of length n: 8 2 a1 3 9 > > < 6 a2 7 = V = 6 . 7 > 4 . 5 > :> an ;> ∗ ∼ n Then V = R can be identified with the set of row vectors of length n ∗ V = b1 b2 ··· bn because any linear map V ! R takes the form 2 a1 3 2 a1 3 2 2 6 a 7 6 a 7 ~ 6 7 7−! b1 b2 ··· bn 6 7 9!b: 4 . 5 4 . 5 an an 24 ∼ n Example. Let V = R be the set of column vectors of length n: 8 2 a1 3 9 > > < 6 a2 7 = V = 6 . 7 > 4 . 5 > :> an ;> ∗ ∼ n Then V = R can be identified with the set of row vectors of length n ∗ V = b1 b2 ··· bn because any linear map V ! R takes the form 2 a1 3 2 a1 3 2 2 6 a 7 6 a 7 ~ 6 7 7−! b1 b2 ··· bn 6 7 9!b: 4 . 5 4 . 5 an an 25 ∼ n ∼ m Example. If V = R , W = R , and 26 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W 27 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 28 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 2 a1 3 2 a1 3 2 A ··· A 3 a2 11 1n a2 6 7 7−! . 6 7 6 . 7 4 5 6 . 7 4 5 A ··· A 4 5 an m1 mn an 29 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 2 a1 3 2 a1 3 6 a2 7 6 a2 7 6 7 7−! 6 7 4 . 5 A 4 . 5 an an 30 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 2 a1 3 2 a1 3 2 A ··· A 3 a2 11 1n a2 6 7 7−! . 6 7 6 . 7 4 5 6 . 7 4 5 A ··· A 4 5 an m1 mn an 31 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 2 a1 3 2 a1 3 2 A ··· A 3 a2 11 1n a2 6 7 7−! . 6 7 6 . 7 4 5 6 . 7 4 5 A ··· A 4 5 an m1 mn an then ∗ ∗ ∗ A : W ! V 32 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 2 a1 3 2 a1 3 2 A ··· A 3 a2 11 1n a2 6 7 7−! . 6 7 6 . 7 4 5 6 . 7 4 5 A ··· A 4 5 an m1 mn an then ∗ ∗ ∗ A : W ! V given by right-multiplication 33 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 2 a1 3 2 a1 3 2 A ··· A 3 a2 11 1n a2 6 7 7−! . 6 7 6 . 7 4 5 6 . 7 4 5 A ··· A 4 5 an m1 mn an then ∗ ∗ ∗ A : W ! V given by right-multiplication 2 3 A11 ··· A1n . b1 b2 ··· bm 7−! b1 b2 ··· bm 4 . 5 Am1 ··· Amn 34 ∼ n ∼ m Example. If V = R , W = R , and A : V ! W given by left-multiplication 2 a1 3 2 a1 3 6 a2 7 6 a2 7 6 7 7−! 6 7 4 . 5 A 4 . 5 an an then ∗ ∗ ∗ A : W ! V given by right-multiplication b1 b2 ··· bm 7−! b1 b2 ··· bm A 35 Example. If we \artificially” identify column vec- tors with row vectors 2 3 b1 t 6 b2 7 b1 b2 ··· bm = 6 . 7 4 . 5 bm then the action of A∗ becomes 2 3 2 3 b1 2 3 b1 A11 ··· Am1 6 b2 7 . 6 b2 7 6 . 7 7! 4 . 5 6 . 7 4 . 5 4 . 5 A1n ··· Amn bm bm so adjoint A∗ sometimes called transpose of A. 36 Example. If we \artificially” identify column vec- tors with row vectors 2 3 b1 t 6 b2 7 b1 b2 ··· bm = 6 . 7 4 . 5 bm then the action of A∗ becomes 2 3 2 3 b1 b1 6 b2 7 t 6 b2 7 6 . 7 7! 6 . 7 4 . 5 A 4 . 5 bm bm so adjoint A∗ sometimes called transpose of A. 37 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself given by V V . If A : V ! W any linear map, ∗ ∗ ∗ 9 adjoint map A : W ! V , defined by (A∗')(v) = '(A(v)): ∗ ∗ A ∗ V − W A V −! W 38 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself given by V V . If B : V ! W any linear map, ∗ ∗ ∗ 9 adjoint map B : W ! V , defined by I∗ = I: ∗ I ∗ V − V I V −! V 39 9 contravariant functor from category (fvector spaces over Rg; fR-linear mapsg) ∗ to itself given by V V .
Load more

Recommended publications
  • A Guide to Symplectic Geometry
    OSU — SYMPLECTIC GEOMETRY CRASH COURSE IVO TEREK A GUIDE TO SYMPLECTIC GEOMETRY IVO TEREK* These are lecture notes for the SYMPLECTIC GEOMETRY CRASH COURSE held at The Ohio State University during the summer term of 2021, as our first attempt for a series of mini-courses run by graduate students for graduate students. Due to time and space constraints, many things will have to be omitted, but this should serve as a quick introduction to the subject, as courses on Symplectic Geometry are not currently offered at OSU. There will be many exercises scattered throughout these notes, most of them routine ones or just really remarks, not only useful to give the reader a working knowledge about the basic definitions and results, but also to serve as a self-study guide. And as far as references go, arXiv.org links as well as links for authors’ webpages were provided whenever possible. Columbus, May 2021 *[email protected] Page i OSU — SYMPLECTIC GEOMETRY CRASH COURSE IVO TEREK Contents 1 Symplectic Linear Algebra1 1.1 Symplectic spaces and their subspaces....................1 1.2 Symplectomorphisms..............................6 1.3 Local linear forms................................ 11 2 Symplectic Manifolds 13 2.1 Definitions and examples........................... 13 2.2 Symplectomorphisms (redux)......................... 17 2.3 Hamiltonian fields............................... 21 2.4 Submanifolds and local forms......................... 30 3 Hamiltonian Actions 39 3.1 Poisson Manifolds................................ 39 3.2 Group actions on manifolds.......................... 46 3.3 Moment maps and Noether’s Theorem................... 53 3.4 Marsden-Weinstein reduction......................... 63 Where to go from here? 74 References 78 Index 82 Page ii OSU — SYMPLECTIC GEOMETRY CRASH COURSE IVO TEREK 1 Symplectic Linear Algebra 1.1 Symplectic spaces and their subspaces There is nothing more natural than starting a text on Symplecic Geometry1 with the definition of a symplectic vector space.
    [Show full text]
  • Book: Lectures on Differential Geometry
    Lectures on Differential geometry John W. Barrett 1 October 5, 2017 1Copyright c John W. Barrett 2006-2014 ii Contents Preface .................... vii 1 Differential forms 1 1.1 Differential forms in Rn ........... 1 1.2 Theexteriorderivative . 3 2 Integration 7 2.1 Integrationandorientation . 7 2.2 Pull-backs................... 9 2.3 Integrationonachain . 11 2.4 Changeofvariablestheorem. 11 3 Manifolds 15 3.1 Surfaces .................... 15 3.2 Topologicalmanifolds . 19 3.3 Smoothmanifolds . 22 iii iv CONTENTS 3.4 Smoothmapsofmanifolds. 23 4 Tangent vectors 27 4.1 Vectorsasderivatives . 27 4.2 Tangentvectorsonmanifolds . 30 4.3 Thetangentspace . 32 4.4 Push-forwards of tangent vectors . 33 5 Topology 37 5.1 Opensubsets ................. 37 5.2 Topologicalspaces . 40 5.3 Thedefinitionofamanifold . 42 6 Vector Fields 45 6.1 Vectorsfieldsasderivatives . 45 6.2 Velocityvectorfields . 47 6.3 Push-forwardsofvectorfields . 50 7 Examples of manifolds 55 7.1 Submanifolds . 55 7.2 Quotients ................... 59 7.2.1 Projectivespace . 62 7.3 Products.................... 65 8 Forms on manifolds 69 8.1 Thedefinition. 69 CONTENTS v 8.2 dθ ....................... 72 8.3 One-formsandtangentvectors . 73 8.4 Pairingwithvectorfields . 76 8.5 Closedandexactforms . 77 9 Lie Groups 81 9.1 Groups..................... 81 9.2 Liegroups................... 83 9.3 Homomorphisms . 86 9.4 Therotationgroup . 87 9.5 Complexmatrixgroups . 88 10 Tensors 93 10.1 Thecotangentspace . 93 10.2 Thetensorproduct. 95 10.3 Tensorfields. 97 10.3.1 Contraction . 98 10.3.2 Einstein summation convention . 100 10.3.3 Differential forms as tensor fields . 100 11 The metric 105 11.1 Thepull-backmetric . 107 11.2 Thesignature . 108 12 The Lie derivative 115 12.1 Commutator of vector fields .
    [Show full text]
  • INTRODUCTION to ALGEBRAIC GEOMETRY 1. Preliminary Of
    INTRODUCTION TO ALGEBRAIC GEOMETRY WEI-PING LI 1. Preliminary of Calculus on Manifolds 1.1. Tangent Vectors. What are tangent vectors we encounter in Calculus? 2 0 (1) Given a parametrised curve α(t) = x(t); y(t) in R , α (t) = x0(t); y0(t) is a tangent vector of the curve. (2) Given a surface given by a parameterisation x(u; v) = x(u; v); y(u; v); z(u; v); @x @x n = × is a normal vector of the surface. Any vector @u @v perpendicular to n is a tangent vector of the surface at the corresponding point. (3) Let v = (a; b; c) be a unit tangent vector of R3 at a point p 2 R3, f(x; y; z) be a differentiable function in an open neighbourhood of p, we can have the directional derivative of f in the direction v: @f @f @f D f = a (p) + b (p) + c (p): (1.1) v @x @y @z In fact, given any tangent vector v = (a; b; c), not necessarily a unit vector, we still can define an operator on the set of functions which are differentiable in open neighbourhood of p as in (1.1) Thus we can take the viewpoint that each tangent vector of R3 at p is an operator on the set of differential functions at p, i.e. @ @ @ v = (a; b; v) ! a + b + c j ; @x @y @z p or simply @ @ @ v = (a; b; c) ! a + b + c (1.2) @x @y @z 3 with the evaluation at p understood.
    [Show full text]
  • Optimization Algorithms on Matrix Manifolds
    00˙AMS September 23, 2007 © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. Index 0x, 55 of a topology, 192 C1, 196 bijection, 193 C∞, 19 blind source separation, 13 ∇2, 109 bracket (Lie), 97 F, 33, 37 BSS, 13 GL, 23 Grass(p, n), 32 Cauchy decrease, 142 JF , 71 Cauchy point, 142 On, 27 Cayley transform, 59 PU,V , 122 chain rule, 195 Px, 47 characteristic polynomial, 6 ⊥ chart Px , 47 Rn×p, 189 around a point, 20 n×p R∗ /GLp, 31 of a manifold, 20 n×p of a set, 18 R∗ , 23 Christoffel symbols, 94 Ssym, 26 − Sn 1, 27 closed set, 192 cocktail party problem, 13 S , 42 skew column space, 6 St(p, n), 26 commutator, 189 X, 37 compact, 27, 193 X(M), 94 complete, 56, 102 ∂ , 35 i conjugate directions, 180 p-plane, 31 connected, 21 S , 58 sym+ connection S (n), 58 upp+ affine, 94 ≃, 30 canonical, 94 skew, 48, 81 Levi-Civita, 97 span, 30 Riemannian, 97 sym, 48, 81 symmetric, 97 tr, 7 continuous, 194 vec, 23 continuously differentiable, 196 convergence, 63 acceleration, 102 cubic, 70 accumulation point, 64, 192 linear, 69 adjoint, 191 order of, 70 algebraic multiplicity, 6 quadratic, 70 arithmetic operation, 59 superlinear, 70 Armijo point, 62 convergent sequence, 192 asymptotically stable point, 67 convex set, 198 atlas, 19 coordinate domain, 20 compatible, 20 coordinate neighborhood, 20 complete, 19 coordinate representation, 24 maximal, 19 coordinate slice, 25 atlas topology, 20 coordinates, 18 cotangent bundle, 108 basis, 6 cotangent space, 108 For general queries, contact [email protected] 00˙AMS September 23, 2007 © Copyright, Princeton University Press.
    [Show full text]
  • SYMPLECTIC GEOMETRY and MECHANICS a Useful Reference Is
    GEOMETRIC QUANTIZATION I: SYMPLECTIC GEOMETRY AND MECHANICS A useful reference is Simms and Woodhouse, Lectures on Geometric Quantiza- tion, available online. 1. Symplectic Geometry. 1.1. Linear Algebra. A pre-symplectic form ω on a (real) vector space V is a skew bilinear form ω : V ⊗ V → R. For any W ⊂ V write W ⊥ = {v ∈ V | ω(v, w) = 0 ∀w ∈ W }. (V, ω) is symplectic if ω is non-degenerate: ker ω := V ⊥ = 0. Theorem 1.2. If (V, ω) is symplectic, there is a “canonical” basis P1,...,Pn,Q1,...,Qn such that ω(Pi,Pj) = 0 = ω(Qi,Qj) and ω(Pi,Qj) = δij. Pi and Qi are said to be conjugate. The theorem shows that all symplectic vector spaces of the same (always even) dimension are isomorphic. 1.3. Geometry. A symplectic manifold is one with a non-degenerate 2-form ω; this makes each tangent space TmM into a symplectic vector space with form ωm. We should also assume that ω is closed: dω = 0. We can also consider pre-symplectic manifolds, i.e. ones with a closed 2-form ω, which may be degenerate. In this case we also assume that dim ker ωm is constant; this means that the various ker ωm fit tog ether into a sub-bundle ker ω of TM. Example 1.1. If V is a symplectic vector space, then each TmV = V . Thus the symplectic form on V (as a vector space) determines a symplectic form on V (as a manifold). This is locally the only example: Theorem 1.4.
    [Show full text]
  • M7210 Lecture 7. Vector Spaces Part 4, Dual Spaces Wednesday September 5, 2012 Assume V Is an N-Dimensional Vector Space Over a field F
    M7210 Lecture 7. Vector Spaces Part 4, Dual Spaces Wednesday September 5, 2012 Assume V is an n-dimensional vector space over a field F. Definition. The dual space of V , denoted V ′ is the set of all linear maps from V to F. Comment. F is an ambiguous object. It may be regarded: (a) as a field, (b) vector space over F, (c) as a vector space equipped with basis, {1}. It is always important to bear in mind which of the three objects we are thinking about. In the definition of V ′, we use (c). The distinctions are often pushed into the background, but let us keep them in clear view for the remainder of this paragraph. Suppose V has basis E = {e1,e2,...,en}. ′ If w ∈ V , then for each ei, there is a unique scalar ω(ei) such that w(ei) = ω(ei)1. Then w = ω(e1) ω(e2) · · · ω(en) . (1) {1}E The columns (of height 1) on the right hand side are the coefficients required to write the images of the basis elements in E in terms of the basis {1}. (The definition of the matrix of a linear map that we gave in the last lecture demands that each entry in the matrix be an element of the scalar field, not of a vector space.) The notation w(ei) = ω(ei)1 is a reminder that w(ei) is an element of the vector space F, while ω(ei) is an element of the field F. But scalar multiplication of elements of the vector space F by elements of the field F is good old fashioned multiplication ∗ : F × F → F.
    [Show full text]
  • Manifolds, Tangent Vectors and Covectors
    Physics 250 Fall 2015 Notes 1 Manifolds, Tangent Vectors and Covectors 1. Introduction Most of the “spaces” used in physical applications are technically differentiable man- ifolds, and this will be true also for most of the spaces we use in the rest of this course. After a while we will drop the qualifier “differentiable” and it will be understood that all manifolds we refer to are differentiable. We will build up the definition in steps. A differentiable manifold is basically a topological manifold that has “coordinate sys- tems” imposed on it. Recall that a topological manifold is a topological space that is Hausdorff and locally homeomorphic to Rn. The number n is the dimension of the mani- fold. On a topological manifold, we can talk about the continuity of functions, for example, of functions such as f : M → R (a “scalar field”), but we cannot talk about the derivatives of such functions. To talk about derivatives, we need coordinates. 2. Charts and Coordinates Generally speaking it is impossible to cover a manifold with a single coordinate system, so we work in “patches,” technically charts. Given a topological manifold M of dimension m, a chart on M is a pair (U, φ), where U ⊂ M is an open set and φ : U → V ⊂ Rm is a homeomorphism. See Fig. 1. Since φ is a homeomorphism, V is also open (in Rm), and φ−1 : V → U exists. If p ∈ U is a point in the domain of φ, then φ(p) = (x1,...,xm) is the set of coordinates of p with respect to the given chart.
    [Show full text]
  • Chapter 5 Manifolds, Tangent Spaces, Cotangent
    Chapter 5 Manifolds, Tangent Spaces, Cotangent Spaces, Submanifolds, Manifolds With Boundary 5.1 Charts and Manifolds In Chapter 1 we defined the notion of a manifold embed- ded in some ambient space, RN . In order to maximize the range of applications of the the- ory of manifolds it is necessary to generalize the concept of a manifold to spaces that are not a priori embedded in some RN . The basic idea is still that, whatever a manifold is, it is atopologicalspacethatcanbecoveredbyacollectionof open subsets, U↵,whereeachU↵ is isomorphic to some “standard model,” e.g.,someopensubsetofEuclidean space, Rn. 293 294 CHAPTER 5. MANIFOLDS, TANGENT SPACES, COTANGENT SPACES Of course, manifolds would be very dull without functions defined on them and between them. This is a general fact learned from experience: Geom- etry arises not just from spaces but from spaces and interesting classes of functions between them. In particular, we still would like to “do calculus” on our manifold and have good notions of curves, tangent vec- tors, di↵erential forms, etc. The small drawback with the more general approach is that the definition of a tangent vector is more abstract. We can still define the notion of a curve on a manifold, but such a curve does not live in any given Rn,soitit not possible to define tangent vectors in a simple-minded way using derivatives. 5.1. CHARTS AND MANIFOLDS 295 Instead, we have to resort to the notion of chart. This is not such a strange idea. For example, a geography atlas gives a set of maps of various portions of the earth and this provides a very good description of what the earth is, without actually imagining the earth embedded in 3-space.
    [Show full text]
  • 2 0.2. What Is Done in This Chapter? 4 1
    CHAPTER III.1. DEFORMATION THEORY Contents Introduction 2 0.1. What does deformation theory do? 2 0.2. What is done in this chapter? 4 1. Push-outs of schemes 9 1.1. Push-outs in the category of affine schemes 9 1.2. The case of closed embeddings 9 1.3. The push-out of a nil-isomorphism 10 1.4. Behavior of quasi-coherent sheaves 11 2. (pro)-cotangent and tangent spaces 14 2.1. Split square-zero extensions 14 2.2. The condition of admitting a (pro)-cotangent space at a point 15 2.3. The condition of admitting (pro)-cotangent spaces 17 2.4. The relative situation 17 2.5. Describing the pro-cotangent space as a limit 18 3. Properties of (pro)-cotangent spaces 20 3.1. Connectivity conditions 20 3.2. Pro-cotangent vs cotangent 22 3.3. The convergence condition 22 3.4. The almost finite type condition 23 3.5. Prestacks locally almost of finite type 25 4. The (pro)-cotangent complex 25 4.1. Functoriality of (pro)-cotangent spaces 25 4.2. Conditions on the (pro)-cotangent complex 27 4.3. The pro-cotangent complex as an object of a category 28 4.4. The tangent complex 29 4.5. The (co)differential 30 4.6. The value of the (pro)-cotangent complex on a non-affine scheme 30 5. Digression: square-zero extensions 32 5.1. The notion of square-zero extension 32 5.2. Functoriality of square-zero extensions 33 5.3. Pull-back of square-zero extensions 35 5.4.
    [Show full text]
  • Maxwell's Equations in Terms of Differential Forms
    Maxwell's Equations in Terms of Differential Forms Solomon Akaraka Owerre ([email protected]) African Institute for Mathematical Sciences (AIMS) Supervised by: Dr. Bruce Bartlett Stellenbosch University, South Africa 20 May 2010 Submitted in partial fulfillment of a postgraduate diploma at AIMS Abstract Maxwell's equations, which depict classical electromagnetic theory, are pulled apart and brought together into a modern language of differential geometry. A background of vector fields and differential forms on a manifold is introduced, as well as the Hodge star operator, which eventually lead to the success of rewriting Maxwell's equations in terms of differential forms. In order to appreciate the beauty of differential forms, we first review these equations in covariant form which are shown afterwards to be consistent with the differential forms when expressed explicitly in terms of components. Declaration I, the undersigned, hereby declare that the work contained in this essay is my original work, and that any work done by others or by myself previously has been acknowledged and referenced accordingly. Solomon Akaraka Owerre, 20 May 2010 i Contents Abstract i 1 Introduction 1 1.1 Introduction to Maxwell's Equations............................1 1.2 Minkowski Spacetime....................................1 1.3 Covariant Form of Maxwell's Equations..........................3 1.4 Gauge Transformation....................................6 2 Vector Fields and Differential Forms7 2.1 Vector Fields.........................................7 2.1.1 Tangent Vectors...................................8 2.2 The Space of 1-Forms....................................9 2.2.1 Cotangent Vectors.................................. 11 2.3 Change of Coordinates................................... 11 2.4 The Space of p-Forms................................... 13 2.5 Exterior Derivatives..................................... 15 3 The Metric and Star Operator 16 3.1 The Metric.........................................
    [Show full text]
  • Continuity Equations, Pdes, Probabilities, and Gradient Flows
    Continuity equations, PDEs, probabilities, and gradient flows∗ Xiaohui Chen This version: July 5, 2020 Contents 1 Continuity equation2 2 Heat equation3 2.1 Heat equation as a PDE in space and time.....................3 2.2 Heat equation as a (continuous) Markov semi-group of operators........6 2.3 Heat equation as a (negative) gradient flow in Wasserstein space........7 3 Fokker-Planck equations: diffusions with drift 10 3.1 Mehler flow...................................... 12 3.2 Fokker-Planck equation as a stochastic differential equation........... 13 3.3 Rate of convergence for Wasserstein gradient flows of the Fokker-Planck equation 15 3.4 Problem set...................................... 20 4 Nonlinear parabolic equations: interacting diffusions 21 4.1 McKean-Vlasov equation.............................. 21 4.2 Mean field asymptotics................................ 22 4.3 Application: landscape of two-layer neural networks............... 23 5 Diffusions on manifold 24 5.1 Heat equation on manifold.............................. 24 5.2 Application: manifold clustering.......................... 25 6 Nonlinear geometric equations 26 6.1 Mean curvature flow................................. 26 6.2 Problem set...................................... 27 A From SDE to PDE, and back 27 B Logarithmic Sobolev inequalities and relatives 29 B.1 Gaussian Sobolev inequalties............................ 29 B.2 Euclidean Sobolev inequalties............................ 30 ∗Working draft in progress and comments are welcome (email: [email protected]). 1 C Riemannian geometry: some basics 30 C.1 Smooth manifolds.................................. 31 C.2 Tensors........................................ 31 C.3 Tangent and cotangent spaces............................ 32 C.4 Tangent bundles and tensor fields.......................... 36 C.5 Connections and curvatures............................. 38 C.6 Riemannian manifolds and geodesics........................ 41 C.7 Volume forms..................................... 45 1 Continuity equation n Let Ω ⊂ R be a spatial domain.
    [Show full text]
  • 1 Poisson Manifolds
    Classical Mechanics, Lecture 8 February 4, 2008 lecture by John Baez notes by Alex Hoffnung 1 Poisson Manifolds We have described Poisson brackets for functions on R2n - the phase space for a system whose config- uration space is Rn. Now let's generalize this to systems whose configuration space is any manifold, M. Here we will see that the phase space is the \cotangent bundle" T ∗M and this is a \Poisson manifold" - a manifold such that the commutative algebra of smooth real-valued functions on it, C1(T ∗M) is equipped with Poisson bracket ; making it into a Poisson algebra - the Poisson algebra of \observables" for our system. {· ·} Example: a particle on a sphere S2. (picture of a sphere M = S2 with point q M) 2 2 2 2 The position and momentum of this particle give a point in T ∗S : q S ; p Tq∗M, so (q; p) T ∗S . Recall that a manifold M is a topological space such that every poin2 t q 2M has a \neighb2orhood 2 that looks like Rn." In other words, there is an open set U M with q U and a bijection ⊂ 2 n φ: U R : ! n Indeed we have a collection of these (Ui; φi: Ui R ) and they are compatible: (picture of charts overlapping) ! 1 that is, φj φi− is smooth (infinitely differentiable) where defined. A collection of this sort is an ◦ n atlas, and the functions φi: Ui R are called charts. We will usually use a maximal atlas, i.e. one containing all charts that are! compatible with all charts in the atlas.
    [Show full text]