Math Math 267: Reading and Homework Assignments

Day 01: 12 January

Review of the Classical Inverse and Implicit Function Theorems, examples, and applications, particularly to the study of level sets of smooth functions. The orthogonal group O(n) regarded as a level set.

Exercises for Day 1

Day 02: 14 January

Proof that the Inverse Function Theorem implies the Implicit Function Theorem. Definition of embedded submanifolds in vector spaces, the definition of n-dimensional charted spaces and n-dimensional manifolds.

Exercises for Day 2

Here are some notes on the definitions of charted spaces and manifolds. (I'll say more about this in the lecture on Day 4.)

Day 03: 17 January

No class: (Martin Luther King, Jr. holiday observed)

Day 04: 19 January

This class was taught by Hugh Bray because Robert Bryant was trapped in Washington, DC by the snow fall. Hugh covered examples of manifolds, the definition of smooth mappings and diffeomorphisms, constructing manifolds as quotients (which yielded the n-torus and n-dimensional real projective spaces) and described connected sums and stated the classification theorem for surfaces.

Exercises for Day 4

Day 05: 21 January (2:50–4:05)

This lecture brought to you by the Chain Rule!

Products of manifolds, patching/extending smooth functions from neighborhoods to the entire manifold (via the bump function construction), and the definition of tangent vectors at a point of a manifold.

Exercises for Day 5

Day 06: 24 January

No class: (Special lectures this afternoon by Altschuler and Wu)

Day 07: 26 January (2:50–4:05)

Properties of tangent spaces, the Chain Rule, the Inverse and Implicit Function Theorems, regular values, submersions, immersions, submanifolds, embedded submanifolds. The tangent bundle as a manifold.

Exercises for Day 7

Day 08: 28 January (2:50–4:05)

Vector fields, flows, the Lie bracket, the Flow-Box Theorem, definitions of 1-forms and Riemannian metrics.

Exercises for Day 8

Day 09: 31 January

The Simultaneous Flow Box Theorem, more on the Lie brackets

Exercises for Day 9

Day 10: 02 February

Examples of Riemannian metrics, isometries, the associated metric distance function and the equality of metric an manifold topologies.

Exercises for Day 10

Day 11: 04 February

The length and energy functionals for curves in a Riemannian manifold, the computation of the Euler-Lagrange equations in local coordinates for the energy minimizers.

Exercises for Day 11

Day 12: 07 February

More on geodesics, the vector field E on TM and the exponential map.

Exercises for Day 12

Day 13: 09 February

Local minimizing properties of geodesics.

Exercises for Day 13

Day 14: 11 February

Completeness and metric completeness. The Hopf-Rinow Theorem

Exercises for Day 14

Day 15: 14 February

No class: (Instructor in Berkeley for Chern memorial)

Day 16: 16 February (2:50–4:05)

Affine connections: axiomatic treatment, locality, the notion of torsion.

Exercises for Day 16

Day 17: 18 February (2:50–4:05)

The Levi-Civita connection associated to a metric, its existence and uniqueness. The notion of covariant derivative along a curve, parallel vector fields along a curve. Properties of parallel transport.

Exercises for Day 17

Day 18: 21 February

The Riemann curvature tensor, properties and symmetries. Examples.

Exercises for Day 18

Day 19: 23 February

Vanishing of Riemann curvature implies flatness.

Exercises for Day 19

Day 20: 25 February

The surface case. Computation of the Gauss curvature in terms of an orthonormal frame field. Examples.

Exercises for Day 20 (Coming Soon)

Day 21: 28 February

Sectional curvature, the fact that the sectional curvature function determines the Riemann curvature tensor. Constant sectional curvature.

Exercises for Day 21

Day 22: 02 March

Coordinates adapted to a geodesic on a surface. Normal form in terms of the curvature. The local uniqueness of surfaces of constant Gauss curvature.

Exercises for Day 22

Day 23: 04 March

Second variation of energy and arc length for surfaces.

Exercises for Day 23 (Coming Soon!)

Day 24: 07 March (2:50–4:05)

Vector fields along a mapping, induced connection and curvature identities. Second variation of energy in general dimension. Ricci curvature. The Bonnet-Myers Theorem.

Exercises for Day 24

Day 25: 09 March (2:50–4:05)

Jacobi fields and the derivative of the exponential map. Covering spaces. The Cartan-Hadamard Theorem.

Exercises for Day 25 (Coming Soon!)

Day 26: 11 March

No class today because of the long classes on 7 and 9 March. Have a good break!

Spring Break: 12-20 March

(20 March 2005 is the Vernal Equinox as well as the Persian New Year)

Day 27: 21 March

Exterior forms on a vector space, the exterior product and exterior algebra.

Exercises for Day 27 (Coming Soon!)

Day 28: 23 March

Exterior differential forms on a manifold. Examples. Pullback of differential forms.

Exercises for Day 28 (Coming Soon!)

Day 29: 25 March

Forms and orientability. Integration.

Exercises for Day 29 (Coming Soon!)

Day 30: 28 March

Class was cancelled today due to illness of instructor. Note that the remaining classes this week are long classes!

Day 31: 30 March (2:50–4:05)

Partitions of unity. Domains with boundary. Induced orientation. The exterior derivative.

Exercises for Day 31 (Coming Soon!)

Day 32: 01 April (2:50–4:05)

Stokes' Theorem.

Exercises for Day 32 (Coming Soon!)

Day 33: 04 April

The covariance of the exterior derivative under smooth mappings. Applications: The Brouwer Fixed Point Theorem. Vanishing of vector fields on even-dimensional spheres.

Exercises for Day 33 (Coming Soon!)

Day 34: 06 April

The deRham complex, deRham Cohomology and induced mappings. The Homotopy Lemma and Poincaré's Lemma.

Exercises for Day 34 (Coming Soon!)

Day 35: 08 April

Applications of deRham cohomology.

Day 36: 11 April (2:50–4:05)

The long exact sequence in cohomology. The cohomology groups of the n-sphere. Finite dimensionality of the cohomology of compact manifolds.

Day 37: 13 April (2:50–4:05)

Lie Groups. Definitions, examples, basic results. Here are lecture notes and exercises for the next several days: Notes on Lie groups