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M-574, 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

 

M-574,  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 co-differential  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_group

        lie.html                                                                                                                                                                                

 MATH 574

M-574FINALEXM1430.pdfM-574FINALEXM1430شريف صادق دشموخ
M-574midterm(1430).pdfM-574midterm(1430)شريف صادق دشموخ
RiemanniangeomGudmudsson.pdfRiemanniangeomGudmudssonشريف صادق دشموخ
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