M574, Syllabus, Description and Links
MATH 574 
Geometry of Manifolds 
3 hours 
Differentiable manifolds, Tensor fields and operations. Differential forms and de Rham's Theorem. Principal fiber bundles, holonomy groups. Curvature form and structural equations. Bianchi's identity. Covariant differentiation, Geodesics, normal coordinates. Riemannian connection. Spaces of constant curvature. Schurs Theorem. 
Recommended Books:
1) I. Chavel, Riemannian Geometry: A Modern Introduction, Cambridge University Press, Cambridge, 1993.
2) F.W. Warner, Foundations of differentiable manifolds and Lie groups, Scot Foresman and Company, 1971
M574, Course Description
Topic

No. of weeks 
Differentiable manifolds, examples, smooth maps, submersions, implicit function theorem, Tangent and cotangent bundles.

3 
Tensor fields tensor products, tensor bundles, Lie derivative of tensor fields.

3 
Differential forms, exterior products, differential and codifferential operators, interior product on forms, closed and exact forms, deRham’s cohomology groups and deRham’s Theorem.

3 
Riemannian manifold, Riemannian connection, Principal fibre bundle, connection in fibre bundle, curvature form and structural equations, Bianchi identities.

3 
Geodesics, normal coordinates, Spaces of constant curvature, Examples of Riemannian manifolds of constant curvature, Schur’s theorem.

3 
Links:
lie.html
Lie_group