Lecture 1 (Wednesday, January 20)
Complete manifolds, Hopf-Rinow theorem
Reference: [dC] Chapter 7, Section 1, 2
Lecture 2 (Monday, January 25)
Hadamard Theorem
Reference: [dC] Chapter 7, Section 3
Lecture 3 (Wednesday, January 27)
Determination of the metric by the curvature;
the hyperbolic space: the half space model and the disk model
Reference: [dC] Chapter 8, Section 1, 2, 3
Lecture 4 (Monday, February 1)
Geodesics in the half space model and the disk model
of the hyperbolic space
References: [dC] page 73--74, [GHL] page 185--186
Lecture 5 (Wednesday, February 3)
Riemannian symmetric spaces
Reference: [Bo] Chapter VII, Section 8, 9
Lecture 6 (Monday, February 8)
Space forms
Reference: [dC] Chapter 8, Section 4
Lecture 7 (Wednesday, February 10)
Gauss-Bonnet theorem, constant sectional curvature metrics on compact surfaces
References: do Carmo, Differential Geometry of Curves and Surfaces, Section 4-5; [GHL] 3.L.4, 3.L.5
Lecture 8 (Monday, February 15)
Conformal maps; Theorem of Liouville
Reference: [dC] Chapter 8, Section 5
Lecture 9 (Wednesday, February 17)
First variation of energy
Reference: [dC] Chapter 9, page 191--196
Lecture 10 (Monday, February 22)
Second variation of energy, Bonnet-Myers theorem, Weinstein's theorem
Reference: [dC] Chapter 9, page 197--206
Lecture 11 (Wednesday, February 24)
Synge's theorem; the index lemma, Rauch's comparison theorem
References: [dC] Chapter 9, page 206--207; Chapter 10, page 210--217
Lecture 12 (Monday, March 1)
Applications of the Rauch's comparison theorem; applications of
the index lemma
Reference: [dC] Chapter 10, page 218--226
Lecture 13 (Wednesday, March 3)
Some corollaries of J.D. Moore's theorem; focal points
Reference: [dC] Chapter 10, page 226--230
Lecture 14 (Monday, March 8)
Focal points and critical values of the exponential map;
index lemma for focal points; the Rauch's comparison theorem for focal points
Reference: [dC] Chapter 10, page 231--235
Lecture 15 (Monday, March 22)
Fibre bundles, principal fibre bundles, associated bundles
Reference: [KN] Chapter I, Section 5, page 50--56
Lecture 16 (Wednesday, March 24)
Connections and connection 1-forms on a principal bundle
Reference: [KN] Chapter II, Section 1
Lecture 17 (Monday, March 29)
Induced connections on associated vector bundles, horizontal lifts
References: [KN] Chapter II, Section 1 page 65, Section 3 page 68--70
Lecture 18 (Wednesday, March 31)
Parallel transport/displacement; holonomy groups; pseudotensorial forms and tensorial forms
References: [KN] Chapter II, Section 3 page 70--71, Section 4, Section 5 page 75--76
Lecture 19 (Monday, April 5)
G-equivariant vector bundles; exterior covariant derivatives; definition of the curvature
Lecture 20 (Wednesday, April 7)
The structure equation; Bianchi's identity; induced connection on an associated vector bundle
and its curvature
Reference: [KN] Chapter II, Section 5 page 77--79
Lecture 21 (Monday, April 12)
Chern classes, Chern characters
References: [KN] Chapter XII, Section 3, 4.
Lecture 22 (Wednesday, April 14)
Properties of Chern classes and Chern characters; spacetime
References: [KN] Chapter XII, Section 1, 3, 4.
Lecture 23 (Monday, April 19)
Intial data set, ADM mass/energy and momentum, statement of the positive energy theorem
Reference: [PT] Section 1
Lecture 24 (Wednesday, April 21)
Spinors
Reference: [PT] Section 2
Lecture 25 (Monday, April 26)
The hypersurface Dirac operator; the Weitzenbock formula
Reference: [PT] Section 3
Lecture 26 (Wednesday, April 28)
The integral form of the Weitzenbock formula; the positive energy theorem
Reference: [PT] Section 3, Section 4