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Notes for Math 230A, Differential Geometry

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Notes for Math 230A, Differential Geometry NOTES FOR MATH 230A, DIFFERENTIAL GEOMETRY AARON LANDESMAN CONTENTS 1. Introduction 4 2. 9/3/15 5 2.1. Logistics 5 2.2. Lecture begins 5 3. 9/8/15 7 3.1. Curvature of curves 8 3.2. Manifolds 8 3.3. Partitions of Unity 9 3.4. A is compact 10 3.5. A = [iAi with Ai compact and Ai ⊂ int(Ai+1) 10 3.6. A is open 11 3.7. A general 11 4. 9/10/15 11 4.1. Partitions of Unity, Hiro’s version 11 4.2. Submersions 13 5. 9/15/15 13 5.1. Tangent Spaces 13 5.2. Return to the submersion theorem 16 6. 9/17/15 16 6.1. Completing the submersion theorem 16 6.2. Lie Brackets 17 6.3. Constructing the Tangent bundle 18 7. 9/22/15 20 7.1. Constructing vector bundles 23 8. 9/24/15 25 8.1. Logistics 25 8.2. structure groups 25 8.3. Fiber bundles in general 26 8.4. Algebraic Prelude to differential forms 26 8.5. Differential Forms 27 9. 9/29/15 29 9.1. Integration 33 10. 10/1/15 34 10.1. Review 34 10.2. Flows and Lie Groups 34 10.3. Lie Derivatives 36 11. 10/6/15 39 1 2 AARON LANDESMAN 11.1. Key theorems to remember from this class, not proven until later today 39 11.2. Class as usual 39 12. 10/8/15 43 12.1. Overview 43 12.2. Today’s class 44 12.3. Riemannian Geometry on vector bundles 47 12.4. Connections 48 13. 10/20/2015 49 13.1. Key theorems for today 49 13.2. Class time 49 13.3. Connections 51 14. 10/22/15 54 14.1. Class time 55 14.2. Connections and Riemannian Geometry 56 15. 10/27/15 60 15.1. Overview 60 15.2. Parallel Transport 60 16. 10/29/15 65 16.1. Overview 65 16.2. Connections 66 16.3. The Fundamental Theorem of Riemannian Geometry 67 16.4. Geodesics 70 17. 11/2/15 71 17.1. Geodesics and coming attractions 71 17.2. Properties of the exponential map 74 18. 11/5/15 76 18.1. Review 76 18.2. Geodesics and length 77 19. 11/10/15 81 19.1. Preliminary questions 81 19.2. Hopf-Rinow 82 19.3. Curvature 83 19.4. Towards some properties and intuition on curvature tensors 85 20. 11/12/15 87 20.1. Types of curvatures 87 20.2. Review of Linear Algebra 88 20.3. Traces in Riemannian Geometry 89 20.4. Back to linear algebra 92 21. 11/17/15 93 21.1. Plan and Review 93 21.2. Scalar curvature 93 21.3. Normal Coordinates 97 21.4. Hodge Theory 99 22. 11/19/15 100 22.1. Questions and Overview 100 22.2. Gauss’ Theorema Egregium 100 22.3. Sectional Curvature and the Exp map 102 NOTES FOR MATH 230A, DIFFERENTIAL GEOMETRY 3 22.4. Hodge Theory 103 23. 11/24/15 105 23.1. Good covers, and finite dimensional cohomology 105 23.2. Return to Hodge Theory 107 23.3. Harmonic Forms and Poincare Duality 110 24. 12/1/15 113 24.1. Overview, with a twist on the lecturer 113 24.2. Special Relativity 113 24.3. The Differential Geometry Set Up 114 24.4. Toward Maxwell’s equations 115 25. 12/3/15 118 25.1. Overview 118 25.2. Principle G-bundles 118 25.3. Connections and curvature on principle G-bundles 119 25.4. An Algebraic characterization of connections on principle G-bundles120 25.5. Curvature as Integrability 124 4 AARON LANDESMAN 1. INTRODUCTION Hiro Tanaka taught a course (Math 230a) on Differential Geometry at Harvard in Fall 2015. These are my “live-TEXed“ notes from the course. Conventions are as follows: Each lecture gets its own “chapter,” and appears in the table of contents with the date. Of course, these notes are not a faithful representation of the course, either in the mathematics itself or in the quotes, jokes, and philosophical musings; in particular, the errors are my fault. By the same token, any virtues in the notes are to be credited to the lecturer and not the scribe. 1 Please email corrections to [email protected]. 1This introduction has been adapted from Akhil Matthew’s introduction to his notes, with his permission. NOTES FOR MATH 230A, DIFFERENTIAL GEOMETRY 5 2. 9/3/15 2.1. Logistics. (1) Phil Tynan is the TF, who isn’t here (2) email: [email protected] (3) Hiro’s office is 341, office hours are Tuesday 1:30 - 2:30pm, and Wednesday 2-3pm. (4) Phil will have office hours 2-3pm on Thursdays, in office 536 and 532. (5) There will be homeworks, once a week, the first homework is due Sept 17. (6) When homework is graded, we will get a remark from Phil to see Hiro and Phil during office hours. You will not be numerically graded from week to week, but you have to come to them in person, so that we know what is going on. (7) There will be no midterm, but one take home final. Remark 2.1. There are two words in the title of the course, Differential and Ge- ometry. This is not Riemannian geometry and we’ll discuss the difference later. “Differential” connotates calculus. You can ask how to do calculus on shapes likes triangles and cubes. To understand calculus, we will learn about manifolds, and calculus on manifolds. To understand geometry, we will think of a space together with some structure (possibly some type of metric). Example 2.2. (1) Riemannian geometry (2) Symplectic geometry - use things like Hamiltonian to describe how vector spaces evolve. (3) Complex geometry - generalize complex analysis to shapes you can build with Cn or CW complex. (4) Kahler geometry (5) Calabi-Yau geometry - study supersymmetric string theory 2.2. Lecture begins. Consider a curve γ : R Rn, t 7 γ(t). Definition 2.3. For γ a curve, we define the length of γ to be ! ! jγ0(t)j dt, ZR where s q 0 0 2 0 0 jγ (t)j = γi(t) = hγ , γ i i . X Remark 2.4. The inner product from Definition 2.3 should really be thought of n n as an inner product on Tγ(t)R and not on R . Even though these objects are isomorphic, they should not be thought of as “the same.” Definition 2.5. Let U ⊂ Rn be an open set. A function f : U Rm is called • C0 if it is continuous • C1 if it has partial derivatives @f for i = 1, ::: , n which! are all C0 @xi • Cr if it has all partial derivatives of order at most r which are all C0 • C or smooth if f is Cr for all r. 1 6 AARON LANDESMAN Definition 2.6. Let U ⊂ Rn be an open set. Then, a Riemannian metric on U is a C function g : U Mn×n(R) (where the matrix represents an inner product on that space) such that 1 • g(x) is a symmetric! nondegenerate matrix. • g(x) is positive definite Example 2.7. (1) Set g(x) := In×n for all x. This is the standard Riemannian metric on Rn. m 1 (2) Fix a smooth map f : U R . Since f is C , it induces a map dfx : TxU =∼ n m n TxR Tf(x)R . In “standard basis” for TxR , we can write ! @fj ! dfx := @xi j=1,:::,m,i=1,:::,n If there is a Riemannian metric h on Rm, this induces a bilinear product on U: Given u, v 2 TxU, we send it to hu, vi := hdfx(u), dfx(v)i. This defines a Riemannian metric on U precisely when dfx is an injection. m Definition 2.8. A C map f : U R is an immersion if dfx is injective for all x 2 U. The induced Riemannian metric is denoted f∗h and is given by 1 ∗ f hx(u, v!) := h(dfx(u), dfx(v)) Remark 2.9. Caution: Immersions need not be injective. For example, one can send two points to the same point. Alternatively, one can take the universal cover R S1. Definition 2.10. Let g be a Riemannian metric on U ⊂ Rn, the volume of (U, g) is ! p Vol(U, g) := deg gdx1 ··· dxn ZU Next, we describe when we should be able to think of two open sets with a Riemannian metric as equivalent. Definition 2.11. A diffeomorphism if f is a bijection, f is C and f-1 is C . Definition 2.12. Fix (U, g) and (V, h) to be two open sets each1 with a Riemannian1 metric. An isometry from (U, g) to (V, h) is a smooth diffeomorphism f : U V such that f∗h = g. Remark 2.13. Why is there a square root in the volume function? When one! tries to evaluate the volume function, we get two contributions from dfx, so we have to take a square root. Remark 2.14. What is the connection between giving a matrix and giving an inner product? If the function g, viewed as a matrix, defines an inner product gij := t hei, eji, where ei is the ith standard basis vector. Then, g(u, v) := u · g · v. So far, there is an obvious constraint, that we’ve only been dealing with open sets in Rn. We would like the notion of manifolds, which are more general spaces in which one can do differential geometry.
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