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Differential geometry Description

Key Elements

Code

MATH 401

Formation

M1 Mathematics

Semester

1

Credits

6

Number of Teaching Hours

36

Number of Tutoring Sessions

24

Number of Laboratory Sessions

0

Content

Objective

Content

- Submanifold of a manifold: Immersion, submersion, the rank theorem, immersed parts, sub-tangent space of an immersed part, submanifolds. - Manifolds: Charts, Atlas, definition of a manifold, induced topology, differentiable map between 2 manifolds, germs. - Tangent spaces : Tangent vector, the manifold TM, vector fields, derivation on an algebra, bracket, the tangent map, the cotangent manifold T*M, fields of 1-form. - Differential equations and integral manifolds: System of differential equations, theorem of existence and uniqueness, flows, sub-integral manifolds, Frobenius theorem. - Differential Forms: Exterior Algebra, differential forms on a open of IRn. Lie derivation, Poincare's Lemma Integration on a manifold, Stokes' Theorem. - Tensor’s Algebra on a manifold: Tensors of type , tensor fields, alternating tensors, tensor products.