PMATH 465/665: Differential geometry
PMATH 465/665: Differential geometry
Lectures: MWF 10:30--11:20 (MC 4040).
Office hours: W 13:00--15:00, F 13:00--14:30.
Course information:
Outline.
Course web page:
http//:uwace.uwaterloo.ca.
Description from the course calendar:
An introduction to differentiable manifolds. The tangent and cotangent
bundles. Vector fields and differential forms. The Lie bracket and Lie
derivative of vector fields. Exterior differentiation, integration of
differential forms, and Stokes's Theorem. Riemannian manifolds, affine
connections, and the Riemann curvature tensor.
Outline of topics:
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Topological manifolds; smooth manifolds;
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Smooth functions and smooth maps; partitions of unity;
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Tangent vectors; the tangent bundle; vector fields on manifolds;
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Covectors; the cotangent bundle; conservative vector fields;
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Submersions, immersions, and embeddings; submanifolds;
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Tensors; Riemannian metrics; differential forms;
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Orientations of manifolds; integration on manifolds; Stokes's Theorem;
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The de Rham cohomology groups;
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Integral curves; global flows and complete vector fields;
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Lie derivatives; commuting vector fields;
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Integrable submanifolds; tangent distributions; the Frobenius Theorem;
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Time permitting: affine connections; the Riemann curvature
tensor; vector bundles.
Prerequisites:
AMATH 333/PMATH 365 -- Elementary differential geometry.
Textbook:
John M. Lee, Introduction to smooth manifolds, Springer-Verlag.
Other references:
- John M. Lee, Introduction to topological manifolds.
- Frank W. Warner, Foundations of differentiable manifolds and Lie
groups.
- William M. Boothby, An introduction to differentiable manifolds
and Riemannian geometry.
- John M. Lee, Riemannian manifolds: an introduction to
curvature.
- Manfredo P. do Carmo, Differential forms and applications.
- James R. Munkres, Analysis on manifolds.
- James R. Munkres, Topology.
- Manfredo P. do Carmo, Differential geometry of curves and
surfaces.
- Chris J. Isham, Modern differential geometry for physicists.
Assignments:
Handouts