DIFFERENTIAL GEOMETRY
                                                     RUI LOJA FERNANDES
                                                           Contents
                           Preface                                                                   4
                           Part 1.    Basic Concepts                                                 6
                             Lecture 0.   Manifolds as subsets of Euclidean space                    8
                             Lecture 1.   Abstract Manifolds                                        13
                             Lecture 2.   Manifolds with Boundary                                   19
                             Lecture 3.   Partitions of Unity                                       23
                             Lecture 4.   Tangent Space and the Differential                         27
                             Lecture 5.   Immersions, Submersions and Submanifolds                  38
                             Lecture 6.   Embeddings and Whitney’s Theorem                          46
                             Lecture 7.   Foliations                                                55
                             Lecture 8.   Quotients                                                 63
                           Part 2.    Lie Theory                                                    72
                             Lecture 9.   Vector Fields and Flows                                   73
                             Lecture 10.   Lie Bracket and Lie Derivative                           81
                             Lecture 11.   Distributions and the Frobenius Theorem                  86
                             Lecture 12.   Lie Groups and Lie Algebras                              90
                             Lecture 13.   Integrations of Lie Algebras and the Exponential         97
                             Lecture 14.   Transformation Groups                                   105
                           Part 3.    Differential Forms                                            114
                             Lecture 15.   Differential Forms and Tensor Fields                     116
                             Lecture 16.   Differential and Cartan Calculus                         125
                             Lecture 17.   Integration on Manifolds                                132
                             Lecture 18.   de Rham Cohomology                                      139
                             Lecture 19.   The de Rham Theorem                                     144
                             Lecture 20.   Homotopy Invariance and Mayer-Vietoris Sequence         151
                             Lecture 21.   Computations in Cohomology                              161
                             Lecture 22.   The Degree and the Index                                171
                           Part 4.    Fiber Bundles                                                179
                             Lecture 23.   Vector Bundles                                          181
                             Lecture 24.   The Thom Class and the Euler Class                      190
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                             Lecture 25.   Pull-backs and the Classification of Vector Bundles     197
                             Lecture 26.   Connections and Parallel Transport                     205
                             Lecture 27.   Curvature and Holonomy                                 212
                             Lecture 28.   Characteristic Classes                                 216
                             Lecture 29.   Fiber Bundles                                          225
                             Lecture 30.   Principal Fiber Bundles                                232
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                                         Preface
                    These are lecture notes for the courses “Differentiable Manifolds I” and
                   “Differentiable Manifolds II”, that I am lecturing at UIUC. This course is
                   usually taken by graduate students in Mathematics in their first or second
                   year of studies. The background for this course is a basic knowledge of
                   analysis, algebra and topology.
                    Mymainaiminwritinguptheselecturesnotesistoofferawrittenversion
                   of the lectures. This should give a chance to students to concentrate more
                   on the class, without worrying about taking notes. It offers also a guide
                   for what material was covered in class. These notes do not replace the
                   recommended texts for this course, quite the contrary: I hope they will be
                   a stimulus for the students to consult those works. In fact, some of these
                   notes follow the material in theses texts.
                    These notes are organized into “Lectures”. Each of these lectures should
                   correspondapproximately to 1 hour and 30 minutes of classroom time. How-
                   ever, some lectures do include more material than others, which correspond
                   to different rhythms in class. The exercises at the end of each lecture are
                   a very important part of the course, since one learns a good deal about
                   mathematics by solving exercises. Moreover, sometimes the exercises con-
                   tain results that were mentioned in class, but not proved, and which are used
                   in later lectures. The students should also keep in mind that the exercises
                   are not homogeneous: this is in line with the fact that in mathematics when
                   one faces for the first time a problem, one usually does not know if it has
                   an easy solution, a hard solution or if it is an open problem.
                    These notes are a modified version of similar lectures in portuguese that
                   I have used at IST-Lisbon. For the portuguese version I have profited from
                   commentsfromAnaRitaPires,Georgios Kydonakis, Miguel Negr˜ao, Miguel
                   Olmos, Ricardo Inglˆes, Ricardo Joel, Jos´e Nat´ario and Roger Picken. Since
                   this is the first english version of these notes, they contain too many typos
                   and mistakes. I will be grateful for any corrections and suggestions for
                   improvement that are sent to me.
                                     Rui Loja Fernandes
                                     ruiloja@illinois.edu
                                     Department of Mathematics, UIUC
                                     Urbana IL, 2013
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