Description This book provides an Introduction to topology, differential topology, and differential geometry.
About the Author Werner Ballmann is Professor of Differential Geometry at the University of Bonn and Director at the Max Planck Institute for Mathematics in Bonn..
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.
The discussion culminates with the Gau equations and the version of Gau \'s theorema egregium for submanifolds of arbitrary dimension and codimension.
Along the way, the book discusses connections and curvature--the central concepts of differential geometry.
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.
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.
Further discussion explores tangent bundles, vector bundles, differentials, vector fields, and Lie brackets of vector fields.
The second chapter of the book introduces manifolds and Lie groups, and examines a wide assortment of examples.
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 first chapter covers elementary results and concepts from point-set topology.
It is based on manuscripts refined through use in a variety of lecture courses.
Description This book provides an Introduction to topology, differential topology, and differential geometry
