Middle East Technical University 2006 Spring Semester Math 505

Math 505 Differentiable Manifolds

Syllabus

1. Smooth manifolds.

(1) Topological m anifolds, local coordinates

(2) manifolds in ℝ^{n} as an example, implicit description

(3) smooth structure, analytical and algebraic structures (without details)

(4) examples: coordinates on the sphere, more sophisticated exaple: quotients (example: projective spaces, real or complex)

(5) problems and exercises

(6) manifolds with boundary, boundary as a submanifold, problem with the product (7) orientation as the agreement of the charts; example: direction on a curve, rotation

in a surface

2. Smooth maps.

(1) smooth functions and maps

(2) examples: functions on the quotient spaces

(3) bump functions and partitions of unity (without details)

3. Tangent bundle.

(1) tangent vectors and vector fields in a chart

(2) tangent vector to a curve

(3) push-forwards of vectors: for any maps and for coordinate change

(4) dynamical (geometric) interpretation of a vector as an equivalence class of curves (5) tangent space at a point

(6) the case of a manifold with boundary

(7) the differential at a point for a map in a chart and its invariance

(8) tangent bundle as a manifold

(9) the differential between the tangent bundles

(10) frames, frame fields, parallel transition in a chart

(11) interpretation of orientability via frames

4. Cotangent space.

(1) covectors and covector fields (1-forms) in a chart

(2) differential of functions

(3) cotangent spaces at points and the pull-back at a point

(4) the cotangent bundle as a manifold and the pull-back of 1-forms

(5) line integral, conservative covector fields 5. Submanifolds.

(1) immersions and embeddings, submanifolds

(2) the regular value theorem in ℝ^{n} and for manifolds

(3) the normal vector fields

(4) orientation of hypersurfaces

(5) restriction of 1-forms to submanifolds and the vector fields

6. Tensors.

(1) algebra of tensors in a vector space

(2) tensors and tensor fields on manifolds (in a chart and transformations)

(3) symmetric tensors and Riemannian metrics

7. Differential forms.

(1) algebra of alternating tensors

(2) the wedge product (in a vector space)

(3) differential forms on manifolds

(4) the exterior derivative

(5) examples: symplectic forms, the volume form

8. Integration.

(1) integration of differential forms: along an oriented submanifold or a cycle (example: manifolds with corners)

(2) Stokes’ theorem

(3) Integration on Riemannian manifolds

(4) De Rham cohomology groups

(5) De Rham theorem

9. Integral curves and flows.

(1) integral curves

(2) flows the fundamental theorem on flows (the ODE theorem)

(3) vector field as a differential operator

10. Lie derivatives.

(1) the Lie derivative of functions

(2) Lie bracket

(3) commuting vector fields

(4) Lie derivative of tensor fields

(5) contraction of differential forms along vector fields

(6) the magic Kartan’s formula

(7) Poisson bracket

11. Integral manifolds and foliations. (1) tangent distributions

(2) integral manifolds and involutivity

(3) the Frobenius theorem

(4) example: contact manifolds and Legendrian curves