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Indian Institute of Science Education and Research Bhopal

MTH 608: Introduction to Differentiable Manifolds and Lie Groups (4)

Pre-requisites: MTH 303 Real Analysis I, MTH 304 Metric Spaces and Topology, MTH 306 Ordinary Differential Equations, MTH 403 Real Analysis II

Differentiable manifolds: definition and examples, differentiable functions, existence of partitions of unity, tangent vectors and tangent space at a point, tangent bundle, differential of a smooth map, inverse function theorem, implicit function theorem,  immersions, submanifolds, submersions, Sard’s theorem, Whitney embedding theorem

Vector fields: vector fields, statement of the existence theorem for ordinary differential equations, one parameter and local one-parameter groups acting on a manifold, the Lie derivative and the Lie algebra of vector fields, distributions and the Frobenius theorem

Lie groups: definition and examples, action of a Lie group on a manifold, definition of Lie algebra, the exponential map, Lie subgroups and closed subgroups, homogeneous manifolds: definition and examples

Tensor fields and differential forms: cotangent vectors and the cotangent space at a point, cotangent bundle, covector fields or 1-forms on a manifold, tensors on a vector space, tensor product, symmetric and alternating tensors, the exterior algebra, tensor fields and differential forms on a manifold, the exterior algebra on a manifold

Integration: orientation of a manifold, a quick review of Riemann integration in Euclidean spaces, differentiable simplex in a manifold, singular chains, integration of forms over singular chains in a manifold, manifolds with boundary, integration of n-forms over regular domains in an oriented manifold of dimension n, Stokes theorem, definition of de Rham cohomology of a manifold, statement of de Rham theorem, Poincare lemma

Suggested Books:

Texts:

• J. Lee, Introduction to smooth manifolds, Springer, 2002
• W. Boothby, An Introduction to differentiable manifolds and Riemannian geometry, Academic Press, 2002
• F. Warner, Foundations of differentiable manifolds and Lie groups, Springer, GTM 94, 1983
• M. Spivak, A comprehensive introduction to differential geometry, Vol. 1, Publish or Perish, 1999

References:

• G. de Rham, Differentiable manifolds: forms, currents and harmonic forms, Springer, 1984
• V. Guillemin and A. Pollack., Differential topology, AMS Chelsea, 2010
• J. Milnor, Topology from the differentiable viewpoint, Princeton University Press, 1997
• J. Munkres, Analysis on manifolds, Westview Press, 199
• 01254217
• C. Chevalley, Theory of Lie groups, Princeton University Press, 1999
• R. Abraham, J. Marsden, T. Ratiu, Manifolds, tensor analysis, and applications, Springer, 1988

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