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  • Manifolds and Differential Forms

    Reyer Sjamaar

    Department of Mathematics, Cornell University, Ithaca, New York 14853-4201

    E-mail address: sjamaar@math.cornell.eduURL:

  • Revised edition, 2017Copyright Reyer Sjamaar, 2001, 2015, 2017. Paper or electronic copies for

    personal use may be made without explicit permission from the author. All otherrights reserved.

  • Contents

    Preface v

    Chapter 1. Introduction 11.1. Manifolds 11.2. Equations 71.3. Parametrizations 91.4. Configuration spaces 10Exercises 14

    Chapter 2. Differential forms on Euclidean space 172.1. Elementary properties 172.2. The exterior derivative 202.3. Closed and exact forms 222.4. The Hodge star operator 242.5. div, grad and curl 25Exercises 27

    Chapter 3. Pulling back forms 313.1. Determinants 313.2. Pulling back forms 38Exercises 45

    Chapter 4. Integration of 1-forms 494.1. Definition and elementary properties of the integral 494.2. Integration of exact 1-forms 514.3. Angle functions and the winding number 54Exercises 58

    Chapter 5. Integration and Stokes theorem 635.1. Integration of forms over chains 635.2. The boundary of a chain 665.3. Cycles and boundaries 685.4. Stokes theorem 70Exercises 71

    Chapter 6. Manifolds 756.1. The definition 756.2. The regular value theorem 82Exercises 88

    Chapter 7. Differential forms on manifolds 91



    7.1. First definition 917.2. Second definition 92Exercises 99

    Chapter 8. Volume forms 1018.1. n-Dimensional volume in RN 1018.2. Orientations 1048.3. Volume forms 107Exercises 111

    Chapter 9. Integration and Stokes theorem for manifolds 1139.1. Manifolds with boundary 1139.2. Integration over orientable manifolds 1179.3. Gauss and Stokes 120Exercises 122

    Chapter 10. Applications to topology 12510.1. Brouwers fixed point theorem 12510.2. Homotopy 12610.3. Closed and exact forms re-examined 131Exercises 136

    Appendix A. Sets and functions 139A.1. Glossary 139A.2. General topology of Euclidean space 141Exercises 142

    Appendix B. Calculus review 145B.1. The fundamental theorem of calculus 145B.2. Derivatives 145B.3. The chain rule 148B.4. The implicit function theorem 149B.5. The substitution formula for integrals 151Exercises 151

    Appendix C. The Greek alphabet 155

    Bibliography 157

    Notation Index 159

    Index 161

  • Preface

    These are the lecture notes for Math 3210 (formerly named Math 321), Mani-folds and Differential Forms, as taught at Cornell University since the Fall of 2001.The course covers manifolds and differential forms for an audience of undergrad-uates who have taken a typical calculus sequence at a North American university,including basic linear algebra and multivariable calculus up to the integral theo-rems of Green, Gauss and Stokes. With a view to the fact that vector spaces arenowadays a standard item on the undergraduate menu, the text is not restricted tocurves and surfaces in three-dimensional space, but treats manifolds of arbitrarydimension. Some prerequisites are briefly reviewed within the text and in appen-dices. The selection of material is similar to that in Spivaks book [Spi71] and inFlanders book [Fla89], but the treatment is at a more elementary and informallevel appropriate for sophomores and juniors.

    A large portion of the text consists of problem sets placed at the end of eachchapter. The exercises range from easy substitution drills to fairly involved but, Ihope, interesting computations, as well as more theoretical or conceptual problems.More than once the text makes use of results obtained in the exercises.

    Because of its transitional nature between calculus and analysis, a text of thiskind has to walk a thin line between mathematical informality and rigour. I havetended to err on the side of caution by providing fairly detailed definitions andproofs. In class, depending on the aptitudes and preferences of the audience andalso on the available time, one can skip over many of the details without too muchloss of continuity. At any rate, most of the exercises do not require a great deal offormal logical skill and throughout I have tried to minimize the use of point-settopology.

    These notes, occasionally revised and updated, are available at

    Corrections, suggestions and comments sent to sjamaar@math.cornell.eduwill bereceived gratefully.

    Ithaca, New York, December 2017




    We start with an informal, intuitive introduction to manifolds and how theyarise in mathematical nature. Most of this material will be examined more thor-oughly in later chapters.

    1.1. Manifolds

    Recall that Euclidean n-space Rn is the set of all column vectors with n realentries






    which we shall call points or n-vectors and denote by lower case boldface letters. InR2 or R3 we often write





    , resp. x *.,



    For reasons having to do with matrix multiplication, column vectors are not to beconfused with row vectors (x1 x2 xn ). Nevertheless, to save space we shallfrequently write a column vector x as an n-tuple

    x (x1 , x2 , . . . , xn )

    with the entries separated by commas.A manifold is a certain type of subset of Rn . A precise definition will follow

    in Chapter 6, but one important consequence of the definition is that at each of itspoints a manifold has a well-defined tangent space, which is a linear subspace ofRn . This fact enables us to apply the methods of calculus and linear algebra to thestudy of manifolds. The dimension of a manifold is by definition the dimension ofany of its tangent spaces. The dimension of a manifold in Rn can be no higher thann.

    Dimension 1. A one-dimensional manifold is, loosely speaking, a curve with-out kinks or self-intersections. Instead of the tangent space at a point one usuallyspeaks of the tangent line. A curve in R2 is called a plane curve and a curve in R3

    is a space curve, but you can have curves in any Rn . Curves can be closed (asin the first picture below), unbounded (as indicated by the arrows in the secondpicture), or have one or two endpoints (the third picture shows a curve with anendpoint, indicated by a black dot; the white dot at the other end indicates that



    that point does not belong to the curve; the curve peters out without coming toan endpoint). Endpoints are also called boundary points.

    A circle with one point deleted is also an example of a manifold. Think of a tornelastic band.

    By straightening out the elastic band we see that this manifold is really the sameas an open interval.

    The four plane curves below are not manifolds. The teardrop has a kink, where twodistinct tangent lines occur instead of a single well-defined tangent line; the five-fold loop has five points of self-intersection, at each of which there are two distincttangent lines. The bow tie and the five-pointed star have well-defined tangent lineseverywhere. Still they are not manifolds: the bow tie has a self-intersection and thecusps of the star have a jagged appearance which is proscribed by the definition ofa manifold (which we have not yet given). The points where these curves fail to bemanifolds are called singularities. The good points are called smooth.

    Singularities can sometimes be resolved. For instance, the self-intersections ofthe Archimedean spiral, which is given in polar coordinates by r is a constant times

  • 1.1. MANIFOLDS 3

    , where r is allowed to be negative,

    can be removed by uncoiling the spiral and wrapping it around a cone. You canconvince yourself that the resulting space curve has no singularities by peeking atit along the direction of the x-axis or the y-axis. What you will see are the smoothcurves shown in the (y , z)-plane and the (x , z)-plane.




    Singularities are very interesting, but in this course we shall focus on gaining athorough understanding of the smooth points.

    Dimension 2. A two-dimensional manifold is a smooth surface without self-intersections. It may have a boundary,which is always a one-dimensional manifold.You can have two-dimensional manifolds in the plane R2, but they are relativelyboring. Examples are: an arbitrary open subset of R2, such as an open square, or


    a closed subset with a smooth boundary.

    A closed square is not a manifold, because the corners are not smooth.1

    Two-dimensional manifolds in three-dimensional space include a sphere (the sur-face of a ball), a paraboloid and a torus (the surface of a doughnut).




    The famous Mbius band is made by pasting together the two ends of a rectangularstrip of paper giving one end a half twist. The boundary of the band consists oftwo boundary edges of the rectangle tied together and is therefore a single closed

    1To be strictly accurate, the closed square is a topological manifold with boundary, but not a smooth

    manifold with boundary. In these notes we will consider only smooth manifolds.

  • 1.1. MANIFOLDS 5


    Out of the Mbius band we can create a manifold without boundary by closing itup along the boundary edge. This can be done in two different ways. Accordingto the direction in which we glue the edge to itself, we obtain the Klein bottle or theprojective plane. A simple way to represent these three surfaces is by the followinggluing diagrams. The labels tell you which edges to glue together and the arrowstell you in which direction


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