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Manifold theory: An introduction for mathematical physicistsD Martin, Glasgow University, UK
This blend of local coordinate methods and intrinsic differential geometry enables workers to read and do calculations in relativity and high energy particle research. It provides foundations for study in gauge theory, differential geometry and differential topology.
Mathematical Reviews
Dr Martin’s very readable differential geometry text for graduate students in physics could also be used for
independent study.
American Mathematical Monthly
Accessible and clear, students will appreciate the numerous examples.
Zentralblatt fur Didaktik der Mathematik
- provides a comprehensive account of basic manifold theory for post-graduate students
- introduces the basic theory of differential geometry to students in theoretical physics and mathematics
- contains more than 130 exercises, with helpful hints and solutions
This account of basic manifold theory and global analysis, based on senior undergraduate and post-graduate courses at Glasgow University for students and researchers in theoretical physics, has been proven over many years. The treatment is rigorous yet less condensed than in books written primarily for pure mathematicians. Prerequisites include knowledge of basic linear algebra and topology. Topology is included in two appendices because many courses on mathematics for physics students do not include this subject.
ISBN 1 898563 84 5
ISBN-13: 978 1 898563 84 6
March 2002
424 pages 234 x 156mm paperback
£55.00 / US$95.00 / €65.00
Usually dispatched within 24 hours
About the author
Daniel Martin, Glasgow University, UK
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Contents
Vector spaces
- Change of basis
- Inner product spaces
- Affine spaces
- Minkowski space-time
- Exercises 1
Tensor algebra
- Tensors
- p-forms
- Tensors in an inner product space
- Exercises 2
Tensor algebra
- Tensors
- p-forms
- Tensors in an inner product space
- Exercises 2
Differential manifolds
- Notes on advanced calculus
- Differentiable manifolds
- The topology of a differentiable manifold
- Orientable manifolds
- Product manifolds
- Quotient manifolds
- Differentiable mappings
- Exercises 3
Differential manifolds
- Notes on advanced calculus
- Differentiable manifolds
- The topology of a differentiable manifold
- Orientable manifolds
- Product manifolds
- Quotient manifolds
- Differentiable mappings
- Exercises 3
Vector and tensor fields on a manifold
- Tangent vectors and spaces to a manifold
- The differential of a mapping
- Vector fields on a manifold
- Tensor fields on a manifold
- Exercises 4
Vector and tensor fields on a manifold
- Tangent vectors and spaces to a manifold
- The differential of a mapping
- Vector fields on a manifold
- Tensor fields on a manifold
- Exercises 4
Exterior differential forms
- Exterior forms on a manifold
- Exterior differentiation
- The interior product
- de Rham cohomology
- Application to vector calculus
- Maxwell’s equations of the electromagnetic field
- Exercises 5
Exterior differential forms
- Exterior forms on a manifold
- Exterior differentiation
- The interior product
- de Rham cohomology
- Application to vector calculus
- Maxwell’s equations of the electromagnetic field
- Exercises 5
Differentiation on a manifold
- Introduction
- The Lie derivative
- Affine connexions and covariant differentiation
- Parallel transport
- The curvature and torsion fields associated with an
- Curvature and torsion forms
- Exercises 6
Differentiation on a manifold
- Introduction
- The Lie derivative
- Affine connexions and covariant differentiation
- Parallel transport
- The curvature and torsion fields associated with an
- Curvature and torsion forms
- Exercises 6
Pseudo-Riemannian and Riemannian manifolds
- The metric tensor
- Products and warped products of manifolds
- Differentiation on a pseudo manifold
- Geodesics
- Riemannian curvature
- Computation of the curvature tensor
- Killing vector fields
- The dual of a tensor
- Exercises 7
Pseudo-Riemannian and Riemannian manifolds
- The metric tensor
- Products and warped products of manifolds
- Differentiation on a pseudo manifold
- Geodesics
- Riemannian curvature
- Computation of the curvature tensor
- Killing vector fields
- The dual of a tensor
- Exercises 7
Symplectic manifolds
- Symplectic manifolds
- Hamiltonian vector fields
- The symplectic group
- Exercises 8
Symplectic manifolds
- Symplectic manifolds
- Hamiltonian vector fields
- The symplectic group
- Exercises 8
Lie groups
- The Lie algebra of a Lie group
- The action of a Lie group on a manifold
- The exponential of a matrix
- The Lie algebras of some subgroups of GL(n, fR)
- The adjoint algebra of a Lie algebra
- Exercises 9
Lie groups
- The Lie algebra of a Lie group
- The action of a Lie group on a manifold
- The exponential of a matrix
- The Lie algebras of some subgroups of GL(n, fR)
- The adjoint algebra of a Lie algebra
- Exercises 9
Integration on a manifold
- Introduction
- Intergraction of an n-form over a manifold of dimension
- Intergraction of a function over a pseudo-Riemannian manifold
- Manifolds with boundary and Stokes’ theorem
- Manifolds with boundary
- Exercises 10
Integration on a manifold
- Introduction
- Intergraction of an n-form over a manifold of dimension
- Intergraction of a function over a pseudo-Riemannian manifold
- Manifolds with boundary and Stokes’ theorem
- Manifolds with boundary
- Exercises 10
Fibre bundles
- Fibre bundles
- Principal fibre bundles
- Constructions involving fibre bundles
- Connexions on a principal fibre bundle
- Curvative
- Parallelism and the holonomy group
- Linear connexions
- Note on Chern classes
- Exercises 11
Fibre bundles
- Fibre bundles
- Principal fibre bundles
- Constructions involving fibre bundles
- Connexions on a principal fibre bundle
- Curvative
- Parallelism and the holonomy group
- Linear connexions
- Note on Chern classes
- Exercises 11
Complex linear algebra and almost complex manifolds
Complex linear algebra
- Almost complex manifolds
- Hermitian manifolds
- Kâhlerian manifolds
- Covariant differentiation on a Hermitian manifold
- Exercises 12
Complex linear algebra and almost complex manifolds
Complex linear algebra
- Almost complex manifolds
- Hermitian manifolds
- Kâhlerian manifolds
- Covariant differentiation on a Hermitian manifold
- Exercises 12
Appendix 1 Anayltic topology
Appendix 2 Quaternions and Cayley numbers
Appendix 3 The semidirect product of two groups
Appendix 4 Homotopy review
Bibliography
Some answers, some hints and some fragmentary solutions to the exercises