Fall 2010, MWF 1-2, Science Center 411
My name: Andrew Cotton-Clay (please call me Andy)
Office: Science Center 527
Office Hours: Tu 4:30-5:30 and W 3-4.
E-mail: acotton at math
TF: Si Li, sili at math
Si's office hours: Mondays, 4-5 in Science Center 112
Syllabus: Math 230a Syllabus
Course iSite: Math 230a iSite (obtain textbook here!)
Due Sep 8
Due Sep 15
Due Sep 22
Due Sep 29 (do any 3 problems; corrected version (gh replaced with hg and problem 4 fixed))
Due Oct 6
Due Oct 13
Due Oct 20
Due Oct 27
Due Nov 3
Due Nov 10
Due Nov 17
Due Nov 24
Due Dec 1
Wed 9/1: Smooth structures, inverse and implicit function theorems.
[T,Ch.1]
Fri 9/3: Examples: S^n, real/complex projective spaces and grassmannians,
Lie groups (GL_n, SL_n, O_n, SO_n, U_n, SU_n). [T, Ch.2]
Wed 9/8: Vector bundles, pullback, isomorphism, sections. Cocycle
definition. Examples: trivial, Mobius, bundles over S^n (and isomorphism
classes as homotopy classes of maps from S^(n-1) to GL_n). Beginning of
tangent bundle definition. [T, 3a-c, 3g-h, 5a]
Fri 9/10: Tangent bundle definition. Tensors. TM from equivalence classes
of curves. T^*M from functions to 1st order. Examples (immersions,
submersions, S^n, O_n). Duality of TM, T^*M. Push forward and pull back of
tensors. Associated bundles from representations of GL_n. [T, 3d-f,
4c-g, 5c]
Mon 9/13: Definition of d (and d commutes with pullback). De Rham
cohomology, functoriality, and homotopy invariance. Statement of vector
field ODE theorem. Lie derivatives, contraction, Cartan's formula, Lie
bracket. [T, 8c, 12a-c]
Wed 9/15: Proof of ODE theorem. Examples on R. Mayer-Vietoris and
H^*(S^1). Nonvanishing vector fields are locally d/dx. [T, 8A] for ODE
theorem (for de Rham cohomology and Mayer-Vietoris, see e.g. Bott and Tu
"Differential forms in algebraic topology")
Fri 9/17: Frobenius integrability theorem. Formula for d in terms of
contraction, vector fields, and Lie bracket (statement only).
Integrability theorem in terms of forms. Nonexample of standard contact
structure on R^3. [T, 13c, 13A2] for vector field version of
integrability theorem
Mon 9/20: Lie algebras, left-invariant vector fields, formulas for GL_n,
exponential of a matrix, exponential map and smoothness, adjoint
representation, Lie subalgebras give immersed Lie subgroups. [T, 5d-f]
for some (see e.g. Lee "Introduction to smooth manifolds" for the rest).
Wed 9/22: Metrics and pseudo-metrics. Acceleration of a parametrized curve
along a submanifold of R^n. Covariant derivatives: definition, affine over End
E-valued 1-forms. [T, 7acd, 11a-c]
Fri 9/24: Hessians. Changes of trivializations and coordinates for connections.
Restriction, pullback, and projection to subbundles for connections. [T, 11bch]
Mon 9/27: Parallel transport. Holonomy. Horizontal lifts. Conditions for local triviality
in terms of these. [T, 11f, 15e] (11f from the perspective of principal bundles)
Wed 9/29: Horizontal-like vector fields. Curvature. Local triviality and curvature.
Covariant derivative on higher E-valued forms and alternative definition of
curvature. Curvature and holonomy. [T, 12defh, 13h] (some of this from the
perspective of principal bundles)
Fri 10/1: Connections on associated bundles. Equivalent definitions of metric connections.
Induced connections on submanifolds of R^n are metric. Torsion of connections on TM.
Torsion vanishes for induced connection on submanifolds of R^n. [T, 15abf]
Mon 10/4: Torsion on T^*M and equality with torsion on TM. Torsion-free implies d_nabla
skew-symmetrized agrees with d on higher forms. Existence and uniqueness of Levi-Civita
connection and expression in terms of the metric. First and second fundamental forms for
submanifolds of R^n and shape operator for hypersurfaces. [T, 15b-f]
Wed 10/6: Geodesics. Variation of length and energy. Geodesic equation. Exponential map.
Gauss lemma and normal coordinates. Short geodesics are length minimizing. [T, 8abc-9abcd]
Fri 10/8: Metric topology is manifold topology. Geodesically complete manifolds and Hopf-Rinow
theorem. Examples of R^n, S^n, H^n, and hypersurfaces. [T, 8d, 9d]
Wed 10/13: Symmetric spaces. Biinvariant metrics for compact Lie groups.
Lie exponential is (pseudo-)Riemannian exponential for biinvariant
(pseudo-)metric. Ad-invariant (pseudo-)metric on Lie algebra gives
biinvariant (pseudo-)metric. Example of -tr(ab) on O(n). Picture of
SL_2(R) and non-surjectivity of exponential. [T,8d] for a different
perspective.
Fri 10/15: Properties of the curvature tensor of the Levi-Civita
connection (skew in two pairs, cyclic sum identity, exchange of two pairs).
Sectional curvature definition; determines full tensor. Constant
curvature tensor. Computation of curvature for S^n and H^n. [T, 15g,
16a]
Mon 10/18: Various forms and proofs of the Bianchi identity. For curvature
of a connection on a vector bundle: formal proof via exterior covariant
differentiation; proof in a trivialization; proof of vector field version
via cyclic sums. For Riemannian curvature: using torsion-free from vector
field version; in coordinates. Application: constant curvature at each
point in dimension at least three implies constant curvature. [T, 14a,
15g].
Wed 10/20: Raising and lowering and contraction of indices. Ricci and scalar curvature.
Curvature in dimension two. Levi-Civita connection of Lie group with biinvariant metric.
[T, 16c, 16i]
Fri 10/22: Coordinate independent Koszul formula for Levi-Civita connection; recalculation of
Levi-Civita connection for biinvariant metric and another proof that integral curves are geodesics.
Curvature for biinvariant metrics. Calculation of curvature for SO(3). [T, 16i]
Mon 10/25: Second fundamental form and shape operators in general. Gauss's equation for curvature of
a submanifold. Codazzi-Mainardi equation (only for hypersurfaces). Umbilic points, and proof that
totally umbilic hypersurfaces in R^n are hyperplanes and spheres. Variations of geodesics and Jacobi
field equation. (See e.g. Kobayashi-Nomizu vol 2 chapter 7.)
Wed 10/27: Explicit variation of geodesics giving a given Jacobi field. Jacobi fields are a
reparametrization term plus an orthogonal term. Conjugate points and Jacobi fields. Proof that
Exp_p is a covering map for complete nonpositively curved manifolds. Geodesic deviation. [T, 16g-h]
Fri 10/29: Taylor expanding the metric in normal coordinates in terms of the curvature tensor. (See
e.g. Cheeger and Ebin "Comparison theorems in Riemannian geometry", chapter 1.)
Mon 11/1: Jacobi fields in constant curvature. Theorems of Bonnet and Myers. Rauch comparison
statement. Reformulation as a theorem about second order ODE's and beginning of proof. (See e.g.
Cheeger and Ebin, chapter 1.)
Wed 11/3: Rauch comparison theorem proof. Corollary comparing lengths of curves. Minimal implies no
conjugate points in interior. (See e.g. Cheeger and Ebin, chapter 1.)
Fri 11/5: Cartan-Ambrose-Hicks theorem. Constant curvature H implies universal cover is the standard
constant curvature H space. Introduction to General Relativity: Tidal deviation in classical
mechanics. Stress-energy tensor in special relativity (in the case of dust) and conservation of
energy and momentum is equivalent to divergence (in one variable) is zero. (See e.g. Cheeger and Ebin,
chapter 1, for the first part. See e.g. Wald "General Relativity", chapter 4 for the second.)
Mon 11/8: Contracted Bianchi identity. Einstein's equation and classical limit via tidal deviation
equals geodesic deviation. Principal bundles: definition, cocycle construction, associated vector
bundles. (See e.g. Wald, chapter 4, for the first part. [T, 10a-b,i] for the
second.)
Wed 11/10: Frame bundles and agreement of cocycle data for GL_n(R)-principal and R^n-vector bundles.
Reduction of structure group definition and example. Examples via representations of pi_1(M) in G.
G is an H principal bundle over G/H. Principal bundles are trivial if and only if they admit a section. [T, 10cdefh].
Fri 11/12: Example of universal cover. Reduction when G/H contractible. Connections on principal bundles:
Definition in terms of G-invariant horizontal distributions. Identification with lie algebra valued
one-forms on P which are Ad-related and standard on vertical vectors. Affine over (P x_Ad g)-valued
one-forms on M. [T, 11d]
Mon 11/14: Gauge group. Identification with sections of (P x_conj G) over M, and with maps to G in
abelian case. Example of maps to the center. In a trivialization as left-multiplication by a map to G.
Change of gauge formula for a connection. Example of R/Z and U(1) connections (and flat connections)
modulo gauge. [T, 11d]
Wed 11/16: Recap of change of gauge formula for a connection. Definition of exterior covariant
derivative and curvature for principal bundles. Curvature vanishes if and only if horizontal
distribution is integrable. [T, 12h]
Fri 11/18: Formula for curvature for principal bundles and change of gauge. Maurer-Cartan form
and identity. Bianchi identity. Weil homomorphism from Ad-invariant polynomials to de Rham
cohomology. [T, 12g, 14abc]
Mon 11/21: Polarization of polynomials and symmetric k-linear forms. Invariant polynomials on u(n).
Chern classes. Calculation of the first chern class for bundles over S^2 via clutching functions, with
examples of TCP^1 and the tautological line bundle. Naturality and Whitney sum formula. [T, 14deg]
Wed 11/23: First chern class for U(1) bundle is poincare dual to the zero set of a section.
Chern-Simons one- and three-forms and explicit check that d of them gives f(F_A) and f(F_A wedge F_A)
respectively. Second chern classes of SU(2) bundles over S^4 are the integers, via integral over S^3
of the Chern-Simons 3-form for a change of trivialization. [T. 14ghi]
Mon 11/29: Invariant polynomials for O(n), SO(2n-1), SO(2n). Pontrjagin classes and relation to Chern
classes. Euler class and top Chern class as example. Pfaffian: definition, examples, SO(2n) invariance
of, square is determinant. Statement of Chern-Gauss-Bonnet. (See e.g. Kobayashi and Nomizu, vol 2,
chapter 12.)
Wed 12/1: Euler characteristic of constant curvature H manifolds in terms of H, dimension, and
volume via computation of the pfaffian. Proof of Chern-Gauss-Bonnet for bundles with fiber rank equal
to the manifold dimension. Euler class is poincare dual to the zero set of a section for the arbitrary
rank case.