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El. knyga: Introduction to Geometry and Topology

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This book provides an introduction to topology, differential topology, and differential geometry. It is based on manuscripts refined through use in a variety of lecture courses. The first chapter covers elementary results and concepts from point-set topology. An exception is the Jordan Curve Theorem, which is proved for polygonal paths and is intended to give students a first glimpse into the nature of deeper topological problems.The second chapter of the book introduces manifolds and Lie groups, and examines a wide assortment of examples. Further discussion explores tangent bundles, vector bundles, differentials, vector fields, and Lie brackets of vector fields. This discussion is deepened and expanded in the third chapter, which introduces the de Rham cohomology and the oriented integral and gives proofs of the Brouwer Fixed-Point Theorem, the Jordan-Brouwer Separation Theorem, and Stokes"s integral formula.The fourth and final chapter is devoted to the fundamentals of diffe

rential geometry and traces the development of ideas from curves to submanifolds of Euclidean spaces. Along the way, the book discusses connections and curvature--the central concepts of differential geometry. The discussion culminates with the Gauß equations and the version of Gauß"s theorema egregium for submanifolds of arbitrary dimension and codimension.This book is primarily aimed at advanced undergraduates in mathematics and physics and is intended as the template for a one- or two-semester bachelor"s course.

I. First Steps in the Topology.- II. Manifolds.- III. Differential Forms and Kohomology.- IV. Geometry of Submanifolds.- A. Alternating Multilinear Forms.- B. Kokette Complexes.- Bibliography.- Index.

Recenzijos

This book is an excellent companion to everyone involved in courses on Differential Topology and/or Differential Geometry. Quite efficiently it gets to several deep results avoiding diversions, still giving a sense of pace friendly to the reader. Otherwise it takes matters till the accesible point and proposes literature for further progress. Strongly recommendable. (Jesus M. Ruiz, European Mathematical Society, euro-math-soc.eu, February, 2019)

Mathematical exposition has a curatorial aspect. Along the path to the Stokes theorem, readers meet some undisguised algebraic topology and beyond it get a solid introduction to differential geometry, including the Riemann curvature tensor. Manywill wish they had used this book as students. Summing Up: Recommended. Lower-division undergraduates through faculty and professionals. (D. V. Feldman, Choice, Vol. 56 (6), February, 2019)

1 First Steps in Topology
1(26)
Werner Ballmann
1.1 Topological Spaces
1(4)
1.2 Continuous Maps
5(2)
1.3 Convergence and Hausdorff Spaces
7(1)
1.4 New from Old
8(2)
1.5 Connectedness and Path-Connectedness
10(4)
1.6 Compact Spaces
14(4)
1.7 The Jordan Curve Theorem
18(4)
1.8 Supplementary Literature
22(1)
1.9 Exercises
22(5)
2 Manifolds
27(42)
Werner Ballmann
2.1 Manifolds and Smooth Maps
27(13)
2.2 Tangent Vectors and Derivatives
40(9)
2.3 Submanifolds
49(5)
2.4 Tangent Bundles and Vector Fields
54(5)
2.5 Vector Bundles and Sections
59(4)
2.6 Supplementary Literature
63(1)
2.7 Exercises
63(6)
3 Differential Forms and Cohomology
69(34)
Werner Ballmann
3.1 Pfaffian Forms
69(3)
3.2 Differential Forms
72(4)
3.3 De Rham Cohomology
76(2)
3.4 The Poincare Lemma
78(4)
3.5 The Mayer-Vietoris Sequence and the Brouwer Fixed-Point Theorem
82(4)
3.6 Orientations and the Jordan-Brouwer Theorem
86(5)
3.7 The Oriented Integral and Stokes's Integral Formula
91(6)
3.8 Supplementary Literature
97(1)
3.9 Exercises
97(6)
4 The Geometry of Submanifolds
103(50)
Werner Ballmann
4.1 Curves
104(11)
4.2 Interior Geometry
115(15)
4.3 Exterior Geometry
130(11)
4.4 Gauß Equations and the Theorema Egregium
141(6)
4.5 Supplementary Literature
147(1)
4.6 Exercises
148(5)
A Alternating Multilinear Forms 153(6)
Werner Ballmann
B Cochain Complexes 159(4)
Werner Ballmann
Bibliography 163(2)
Index 165
Werner Ballmann is Professor of Differential Geometry at the University of Bonn and Director at the Max Planck Institute for Mathematics in Bonn.
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