MATH 462--DIFFERENTIAL GEOMETRY--SPRING 2020--A. FREIRE

Syllabus

1/9  Differential forms in R2 and R3: examples, wedge product (start)

1/14 Differential forms: wedge product, pullback, exterior differential
HW 1: 4,5,6 (Ch.1) Due 1/21

1/16 Hodge star and examples (problems 10-14, 16)

1/21 Hodge * and inner products/ Divergence in an orthogonal coord. syst./
Poincare lemma  for 1-forms on star-shaped domains.

1/23 Integration of one-forms along curves/closed forms along continuous curves/
exactness in terms of line integrals/ example of closed, non-exact one-form.
(dC, ch. 2) HW 2: 2, 4, 5, 8 (Ch.2)--due 2/4 or 2/6
HW 2 solutions

1/28 Homotopy invariance of line integrals/ simply-connected domains, Poincare lemma

1/30 Poincare lemma for p-forms on contractible domains/comments on homework problems

2/4 turning number of a closed curve; properties/index of a vector field on a disk (start)

2/6 topological proof of FTA/ Brouwer fixed point on the 2-disk/ Linearization lemma for vector fields

2/11 Borsuk-Ulam theorem/ Index of linear vector fields/ Kronecker's formula and Hopf's coorollary
Winding Number and applications

2/13 Definition of manifold. Topological properties (locally euclidean, Hausdorff, 2nd countable).
Example: stereographic coordinates on the 2-sphere

2/18 Examples: RP2, T2, Klein bottle/ diffble functions and maps/

2/20 Manifold structure in the quotient space of an idempotent diffeomorphism/surfaces in R^n/tangent space at a point

2/25 tangent bundle, differential of a map, vector fields, differential forms (on manifolds).

2/27 exterior differential on manifolds/ proof there are no zero-free vector fields on even-dimensional spheres
HW3 (due 3/5): Ch3, problems 2, 3, 7
Vector fields on even-dimensional spheres have zeroes
(J. Milnor's analytic proof, Am. Math Monthly 1978)
Note on the proof

3/3  orientable manifolds. Example: Mobius strip, antipodal maps of spheres
Note on the Mobius strip (includes 2 problems)

3/5 orientable manifolds: projective spaces. Integral of n-forms on compact orientable manifolds

3/10 partitions of unity (read) /Manifolds with boundary and Stokes' theorem

3/12 Stokes' theorem (end)/ Take-home midterm given (due Tuesday 3/24)
HW 3 solutions

3/17, 3/19 (Spring Break)-no classes

3/24 moving frames, Cartan's equation, Cartan's Lemma (Zoom)
3-24 notes (scan)

3/26 First and second fundamental forms/examples: surfaces of revolution, graphs (Zoom)
3-26 notes (scan)
HW4 (due 4-7): p.96: 1, 2, 4
Midterm and HW4 solutions

3/31: Local theory of plane curves (zoom)
3-31 notes (scan, 7 pages)
Plane curves: local and global results

4/2: global theory of plane curves: Whitney's theorem, turning tangents theorem
4-2 notes (scan, 4 pages)
Turning tangents theorem (proof, 4 pages)

4/7: intrinsic geometry: area form, Gauss curvature, covariant derivative
4-7 notes (scan, 5 pages)

4/9 No lecture day (University-wide)

4/14 covariant derivative: properties, parallel transport/ geodesic curvature of curves
4-14 notes (scan, 8 pages)    pages 3 and 5
Review problems, p. 74: 13, 15, 16

4/16 Gauss-Bonnet Theorem
4-16 notes (scan, 5 pages)
HW5 (due 4/23)  solutions   parallel transport on the sphere

4/21 local Gauss-Bonnet with boundaries and corners
4-21 notes (scan, 6 pages)

4/23 (last day)  Problems: applications of Gauss-Bonnet: euler charactetistic for planar domains and surfaces
(from triangulations or vector fields), critical points of Morse functions (6.2 in text)
4-23 notes (scan, 5 pages)

5/1, 10:15--12:15: FINAL EXAM (comprenhensive)
final exam  final-diagrams solutions