MTH508/608 

SYLLABUS:

• 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 dimensionn, Stokes theorem, definition of de Rhamcohomology 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