Lecture Notes for Differential Geometry, MATH 624, Iowa State University Domenico D’Alessandro∗ Copyright by Domenico D’Alessandro, 2020 December 22, 2020 ∗Department of Mathematics, Iowa State University, Ames, Iowa, U.S.A. Electronic address: da- [email protected] 1 Contents I Calculus On Manifolds 5 1 Manifolds 6 1.1 Examples and further definitions . .7 1.1.1 Manifolds of dimension m=1 . .7 1.1.2 Surfaces . .8 1.1.3 n-dimensional Spheres . .9 1.1.4 Product manifolds . 10 1.1.5 Projective spaces . 10 1.1.6 Grassmann Manifolds . 11 1.2 Maps between manifolds . 14 1.3 Exercises . 17 2 Tangent and cotangent spaces 18 2.1 Tangent vector and tangent spaces . 18 2.2 Co-tangent vectors and co-tangent space . 21 2.3 Induced maps: Push-forward . 22 2.3.1 Computational Example . 24 2.4 Induced maps: Pull-back . 26 2.4.1 Computational Example . 27 2.5 Inverse functions theorem; Submanifolds . 28 2.6 Exercises . 30 3 Tensors and Tensor Fields 31 3.1 Tensors . 31 3.2 Vector fields and tensor fields . 32 3.2.1 f−related vector fields and tensor fields . 34 3.3 Exercises . 36 4 Integral curves and flows 38 4.1 Relation with ODE’s. The problem ‘upstairs’ and ‘downstairs’ . 38 4.2 Definition and properties of the flow . 39 4.3 Exercises . 43 5 Lie Derivative 44 5.1 Lie derivative of a vector field . 44 5.2 Lie derivatives of co-vector fields and general tensor fields . 50 5.3 Exercises . 52 6 Differential Forms Part I: Algebra on Tensors 53 6.1 Preliminaries: Permutations acting on tensors . 53 6.2 Differential forms and exterior product . 55 r 6.3 Characterization of the vector spaces Ωp(M) ..................... 58 6.4 Exercises . 60 2 7 Differential Forms Part II: Fields and the Exterior Derivative 61 7.1 Fields . 61 7.2 The exterior derivative . 61 7.2.1 Independence of coordinates . 62 7.3 Properties of the exterior derivative . 63 7.3.1 Examples . 64 7.3.2 Closed and Exact Forms . 65 7.4 Interior product . 66 7.4.1 Properties of the interior product . 67 7.5 Exercises . 69 8 Integration of differential forms on manifolds part I: Preliminary Concepts 70 8.1 Orientation on Manifolds . 70 8.2 Partition of Unity . 73 8.3 Orientation and existence of a nowhere vanishing form . 76 8.4 Simplexes . 78 8.5 Singular r-chains, boundaries and cycles . 81 8.6 Exercises . 84 9 Integration of differential forms on manifolds part II: Stokes theorem 85 9.1 Integration of differential r-forms over r−chains; Stokes theorem . 85 9.2 Integration of Differential forms on regular domains and the second version of Stokes’ Theorem . 91 9.2.1 Regular Domains . 91 9.2.2 Orientation and induced orientation . 93 9.2.3 Integration of differential forms over regular domains . 94 9.2.4 The second version of Stokes theorem . 97 9.3 De Rham Theorem and Poincare’ Lemma . 99 9.3.1 Consequences of the De Rham Theorem . 100 9.3.2 Poincare’ Lemma . 100 9.4 Exercises . 101 10 Lie groups Part I; Basic Concepts 102 10.1 Basic Definitions and Examples . 102 10.2 Lie subgroups and coset spaces . 103 10.3 Invariant vector fields and Lie algebras . 104 10.3.1 The Lie algebra of a Lie group . 106 10.3.2 Lie algebra of a Lie subgroup . 107 10.4 The Lie algebras of matrix Lie groups . 108 11 Exercises 110 12 Lie groups Part II 111 13 Fiber bundles; Part I 112 14 Fiber bundles; Part II 113 3 Other resources: 1. D. Martin, Manifold Theory; an introduction for mathematical physicists, Woodhead Pub- lishing, Cambridge, UK, 2012 2. J.M. Lee, Introduction to Smooth Manifolds, 2-nd Edition, Graduate Texts in Mathematics, 218, Springer, 2012. 3. W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, Pure and Applied Mathematics Series, Vol. 120, 1986. 4. M. Nakahara, Geometry, Topology and Physics (Graduate Student Series in Physics) 2nd Edition, Taylor and Francis Group, New York, 2003. 5. M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1, Publish or Perish; 3rd edition (January 1, 1999) 6. F. Warner, , Graduate Texts in Mathematics (Book 94), Springer 1983. 4 Part I Calculus On Manifolds 5 1 Manifolds A manifold is an object that locally looks like RI m. More in detail: Definition 1.0.1: Manifolds an m−dimensional manifold M is a topological space together with a family of pairs (Uj; φj) called charts such that Uj are open sets in M and φj are homeomorphishms 0 m φj : Uj ! Uj ⊆ RI . Moreover, the Uj’s cover all of M, i.e., [ Uj = M: (1.0.1) j Additionally, the maps φj are assumed to satisfy the smooth compatibility condi- T −1 tion, which means that, for any pair j; k such that Uj Uk 6= ;, the map φk ◦ φj : T T 1 m m φj(Uj Uk) ! φk(Uj Uk) is in C in the usual sense of calculus for maps RI ! RI . −1 The maps φk ◦ φj are referred to as transition functions. (cf. Figure 1) Two more conditions are usually assumed and we will do so as well: 1) M is assumed to be second countable, i.e., there exists a countable base in its topol- ogy.a 2) M is Hausdorff, that is, every two distinct points in M have disjoint neighborhoods. aA countable family of open sets such that every open set can be written as the union of sets in this family. Definition 1.0.2: Coordinates The collection of all charts f(Uj; φj)g is called an atlas for M. Uj is called a coordinate neighborhood while φj is called a coordinate system or coordinate function. Since its image is in RI m , it is often denoted by using the coordinates (x1; : : : ; xm) in RI m, 1 m i.e., for p 2 M, φj(p) := (x (p); : : : ; x (p)). The number m is called the dimension of the manifold, and the manifold M is often denoted by Mm to emphasize its dimension. Figure 1 describes the definition of a manifold and in particular the smooth compatibility condition. 6 Figure 1: Definition of Manifold and the Smooth Compatibility Condition . 1.1 Examples and further definitions 1.1.1 Manifolds of dimension m=1 Up to homeomorphisms , there are only two possible manifolds of dimension 1, RI and S1. RI can be given the trivial structure of a manifold by using the atlas fUj; φjg where Uj’s are the open sets in the standard topology of RI , and all the φj’s can be taken equal to the identity. 1 In practice, we only need one chart ( RI ; id). For S , let U1 be the open set consisting of all 1 of S except the point (1; 0) and φ1 map every point (cos(θ); sin(θ)) to θ 2 (0; 2π). Moreover, consider U2 the circle except the point (−1; 0) and φ2 mapping every point (cos(u); sin(u)) to −1 S 1 u 2 (−π; π). φ2 ◦ φ1 is defined on (0; π) (π; 2π) and it is in C since u = θ for θ 2 (0; π) and −1 1 u = θ − 2π for θ 2 (π; 2π). Analogously, one can see that φ1 ◦ φ2 is in C . 7 Figure 2: Manifold structure and transition function for S1. 1.1.2 Surfaces Much of our intuition about manifolds comes from curves and surfaces in RI 3 , which we studied in multivariable calculus. There, Definition 1.1.1: Surface a surface is described by a function z = f(x; y) , an open set D ⊆ RI 2 or, more in general, by parametric coordinates x = x(u; v), y = y(u; v), z = z(u; v), with (u; v) in an open set D ⊆ RI 2 , assuming that the coordinate functions are smooth as functions from RI 2 to RI 3, and havea smooth inverse such an inverse can be taken as the coordinate function φ of a unique chart defining the manifold structure of the surface. Closed surfaces such as cylinders and spheres can be still described by parametric surfaces locally. Therefore, they still can be given the structure of a manifold, but the corresponding atlas contains more than one chart. The situation is similar to the one for the circle Sm described above. 8 1.1.3 n-dimensional Spheres Definition 1.1.2: Sn The n−dimensional sphere Sn is the set of points in RI n+1 with coordinates (x0; x1; : : : ; xn) such that n X (xj)2 = 1; (1.1.1) j=0 n+1 with the subset topology induced from RI . Consider the charts (Uj+; φj+) and (Uj−; φj−), for j = 0; 1; : : : ; n, where the coordinate neighborhoods Uj+’s and Uj−’s are defined as 0 1 n n j 0 1 n n j Uj+ := f(x ; x ; : : : ; x ) 2 S j x > 0g;Uj− := f(x ; x ; : : : ; x ) 2 S j x < 0g; n and the coordinates functions, φj± : Uj± ! RI , are defined as, 0 1 n 0 1 j−1 j+1 n φj± (x ; x ; : : : ; x ) = (x ; x ; : : : ; x ; x ; : : : ; x ): The set of coordinate charts (Uj±; φj±) is an atlas since the Uj±, j = 0; 1; 2; : : : ; n form a n 0 1 n cover for S (if (x ; x ; : : : ; x ) does not belong to any of the Uj±, then all the components, 0 1 n x ; x ; : : : ; x , must be zero which contradicts (1.1.1)).