MATH 568-SPRING 2012-COURSE LOG
Th 1/12 Lorentzian vector spaces
handout
1: Lorentzian vector spaces
Tu 1/17 Hyperquadrics in Minkowski space/ geodesics
handout
2: totally geodesic/totally umbilic submanifolds
Th 1/19 Totally geodesic/totally umbilic submanifolds
Review: Curvature
tensor, jacobi equation, sectional curvature (Riemannian case)
M 1/23 and W 1/25: sectional curvature of Lorentzian
manifolds/ constant curvature and hyperquadrics
handout
3: sectional curvature of Lorentzian manifolds
(includes 3 HW
problems, due Monday, Feb.6)
M 1/30 Ricci curvature, splitting of the curvature tensor
handout
4: Ricci curvature, constraints and evolution
(includes 4
exercises, due Monday, Feb 13)
W 2/1 Constraint and evolution equations; 1st variation of arc
length (ref: O'Neill, chapter 10)
M 2/6 Second variation of arc length
W 2/8 The index form: variational meaning, index theorem
(statement)
M 2/13 Bonnet-Myers theorem, Rauch comparison thm
W 2/15 Rauch comparison (cont'd), applications
M 2/20 volume growth
handout
5: volume estimates
W 2/22 volume growth for geodesic spheres
M 2/27 Volume growth: theorems based on Ricci curvature;
cut locus
W 2/29 lecture postponed
M 3/5 first variation for hypersurfaces
handout
7: first and second variation for hypersurfaces
(version of 3/7)
W 3/7 second variation for hypersurfaces/ focal points
([ON, pp.280-288][dC 10.4]
problem
set
M 3/12 energy-momentum tensor for a relativistic ideal
fluid/ Robertson-Walker spacetimes/ skew-product metrics
(O'Neill: ch. 10 and
ch. 7)
W 3/14 skew-product metrics
M 3/19, W 3/21: SPRING BREAK
M 3/26 lecture postponed
W 3/28 RW cosmologies, Friedmann models/ focal points (start)
handout
8: skew-product metrics, RW cosmological models
(includes six problems)
M 4/2 causality in Lorentzian manifolds
W 4/4 causality (cont)
M 4/9 Hawking singularity theorem
W 4/11 Schwarzschild solution
M 4/16 Schwarzschild
W 4/18 Schwarzschild
handout
9: Schwarzschild solutions
(preliminary version, 9 problems)
Last HW set: problems 3,4,5,8(iii),9 (from handout 9)
M 4/23 trapped surfaces and singularities
handout10:
trapped surfaces and timelike incompleteness
W 4/25 trapped surfaces and singularities
M 4/30 postponed lecture
W 5/2 postponed lecture