EAN: 9783034809825

COMPACT TEXTBOOKS IN MATHEMATICS INTRODUCTION...

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 differential 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.

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