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An Introduction to Manifolds

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined int...

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Main Author: Tu, Loring W. (Author, http://id.loc.gov/vocabulary/relators/aut)
Corporate Author: SpringerLink (Online service)
Language:English
Published: New York, NY : Springer New York : Imprint: Springer, 2008.
Edition:1st ed. 2008.
Series:Universitext,
Subjects:
Manifolds (Mathematics).
Complex manifolds.
Global analysis (Mathematics).
Differential geometry.
Manifolds and Cell Complexes (incl. Diff.Topology).
Global Analysis and Analysis on Manifolds.
Differential Geometry.
Online Access:https://doi.org/10.1007/978-0-387-48101-2
Table of Contents:
  • Euclidean Spaces
  • Smooth Functions on a Euclidean Space
  • Tangent Vectors in Rn as Derivations
  • Alternating k-Linear Functions
  • Differential Forms on Rn
  • Manifolds
  • Manifolds
  • Smooth Maps on a Manifold
  • Quotients
  • Lie Groups and Lie Algebras
  • The Tangent Space
  • Submanifolds
  • Categories and Functors
  • The Rank of a Smooth Map
  • The Tangent Bundle
  • Bump Functions and Partitions of Unity
  • Vector Fields
  • Lie Groups and Lie Algebras
  • Lie Groups
  • Lie Algebras
  • Differential Forms
  • Differential 1-Forms
  • Differential k-Forms
  • The Exterior Derivative
  • Integration
  • Orientations
  • Manifolds with Boundary
  • Integration on a Manifold
  • De Rham Theory
  • De Rham Cohomology
  • The Long Exact Sequence in Cohomology
  • The Mayer–Vietoris Sequence
  • Homotopy Invariance
  • Computation of de Rham Cohomology
  • Proof of Homotopy Invariance.

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