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Books > Science & Mathematics > Mathematics > Geometry > Analytic geometry
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Introduction to Geometry and Topology (Paperback, 1st ed. 2018)
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Introduction to Geometry and Topology (Paperback, 1st ed. 2018)
Series: Compact Textbooks in Mathematics
Expected to ship within 10 - 15 working days
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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 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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