**MTH 508: 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, 1997 - 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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