Messages - Page 2

I talked about bundles of covariant tensors, using Riemannian metrics as an example.

Exercises for Monday March 11th: Chapter 3, problems 19, 23, 24, 25, 26, 27.

I talked about differentials as cotangent fields, and started talking about multilinear maps and tensor products.

Exercises for Monday March 4th: Chapter 3, problems 9, 11, 12, 15 and 16.

I finished chapter 3 with a discussion of orientations, gave an interlude about Lagrangian and Hamiltonian mechanics, and started chapter 4 with the cotangent bundle.

Exercises for Monday February 25th: Chapter 2, problems 33, 34; Chapter 3, problems 3, 4, 6, 7; MA 252 exam 1997, problem 1.

Please work on the problems in advance, and come to the class with an idea of which problems you would like to see discussed. We will only have time to go through a few of the problems.

Hints: For problem 33, to show that 1 is a regular value of the determinant map from M(n, R) to R, it suffices to consider the curve c(t) = tA through any matrix A of determinant 1, for t near 1. This replaces the explicit formula for the derivative of the determinant. For problem 3, note that the map GL(n, R) --> GL(n, R) taking an invertible matrix A to its inverse is continuous. For problem 4 either use that in a vector bundle E --> B the space B has the quotient topology from E, or use the zero section to think of B as a subspace of E. For problem 6 make use of problem 34 from chapter 2. For problem 7...

I gave alternative definitions of the tangent bundle in terms of equivalence classes of curves, and in terms of point derivations.

I constructed the tangent bundle of a smooth manifold.

I continued chapter 3, on vector bundles and bundle maps.

I started chapter 3 on the tangent bundle of a differentiable manifold. Other textbook sources for this material include Per Holm's lecture notes at http://folk.uio.no/pholm/mangfold.html and Victor Guillemin and Alan Pollack's book "Differential Topology".

There will be no class on Monday February 18th or Wednesday February 20th.

Exercises for Monday February 11th: Chapter 2, problems 13, 21, 33, 34; MA 252 exam 2002, problem 1.

Exam problems from 1997 to 2002 can be found under "Oppgaver" in the left hand column on the course and semester pages. Translated to English, problem 1 from 2002 reads: Let N and P be smooth manifolds of dimension n and p, respectively, and let f : N --> P be a smooth map. Let psi = (psi^1, ..., psi^p) be a chart in P centered at a point f(a) (meaning that psi(f(a)) = 0), and define Q (a subset of P) by the equations psi^1 = 0, ..., psi^l = 0. (a) Explain why Q is a submanifold of P. What is its dimension? (b) Prove that if the map psi' o f with components psi^1 o f, ...., psi^l o f has rank l in the point a, then f^{-1}Q is a submanifold of N in a neighborhood of a. What is its dimension?

Tillitsvalgt student (student representative) er Nikolai Bj?rnest?l Hansen <nikolabh "at" math.uio.no>.

I talked about constructing C^\infty functions and embedding manifolds in Euclidean space (Spivak, pp. 33-34, 52-53).

Exercises for Monday February 4th: Chapter 2, problems 10, 14a, 15, 16, 24, 26, 28.

I talked about immersions, embeddings, the shrinking lemma and partitions of unity (Spivak, pp. 46-51).

I talked about standard forms for differentiable functions near a point (Spivak, pp. 42-45).

I talked about the classification of manifolds up to diffeomorphism, partial derivatives, the chain rule, the rank of a function at a point, critical and regular points and values, subsets of measure zero and Sard's theorem (Spivak, pp. 30, 35-42).

Exercises for Monday January 28th: Chapter 2, problems 2, 5, 6, 7a, 8, 9, 12.

A preliminary syllabus has been added to the spring 2013 front page.

I talked about differentiable maps and diffeomorphisms (Spivak, p. 31).

I started with chapter 1, including the definition of a topological manifold and examples like spheres, tori, orientable surfaces and the projective plane.

I finished chapter 1, including non-orientable surfaces, topological immersions, manifolds with boundary and sigma-compactness. Thereafter I started chapter 2 on differentiable structures, discussing C^\infty-related charts, atlases and the definition of a differentiable manifold in terms of a maximal atlas (Spivak, pp. 27-29).

Exercises for Monday January 21st: Chapter 1, problems 3ab, 4, 5, 6, 7, 8, 13, 16.

Here problems 3a, 5 and 6 are about clarifying points made in the text. Problems 3b, 4 and 7 should not be too hard. For problem 8, assume n>1. Problems 13 and 16 can be answered somewhat informally

The two first lectures are Monday January 14th and Wednesday January 16th. We start with chapter 1 on topological manifolds.