# Report from the lectures

January 16.: I defined and gave examples of smooth manifolds

January 17.: I gave some more examples of smooth manifold, and defined smooth maps between manifolds. I defined smooth manifolds with boundaries (Spivak p. 32). I then introduced some notation for derivatives of mappings between smoth manifolds and defined the rank of a smooth map f: M-> N at point p IN M (Spivak p. 35- 40)

January 23. : A stated and explained the content of Sards Theorem. I also stated the rank Theorem (Theorem 9. Spivak p. 42) and Theorem10. (p. 44). I gave no proofs of these theorems. Then I talked about immersions and immersed submanifolds(Spivak p. 46-48).

January 24. I proceeded talking about immersed submanifolds, and about submanifolds ending by stating and proving Proposition 12 (p. 49). I then jumped back to p. 33 and explained the various constructions of special smooth functions given on that page, ending by stating and proving Lemma 2. Finally I explained what we mean by a partition of unity subordinated an open covering of a manifold. I will proceed with this proving the existence of such a partition of unity next Monday.

January 30: I proved the existence of a partition of unity subordinate an open covering of a smooth manifold.

January 31: I proved that any compact manifold can be embedded into an Euclidean space. As an example, I looked at problem 1 from the old exam MA 252 1997. Then I started lecturing from chapter 3 in Spivak defining the tangent space of a smooth manifold M at a point p in M. I will proceed with examine this tangent space on Monday (6/2).

February 6. : I defined the tangent space M_p of a smooth manifold M at a point p in M as the linear space of local derivations at p (of smooth functions on M). I also defined the derivative of a smooth mapping f: M-> N at a point p as a linear map between the corresponding tangent spaces M_p and N_f(p). I then proved the chain rule for the derivative of a composed mapping. Then I showed that the dimension of M_p was equal dimM, and I also showed that given a chart (x,U) around p, a basis of M_p is given by the partial derivatives {∂/∂x_1,..,∂/x_n }. Finally I gave an alternative definition of the tangent space M_p, namely the space of equivalence classes [c] of smooth curves c: (-ß,ß)-> M with c(0)=p, and equivalence relation given by c~d iff (xoc)'(0)=(xod)'(0) (where x is a chart around p).

February 7. : I defined the tangent bundle of a manifold M and showed that this is a differentiable manifold of dimension 2dimM. I explained how a map between two manifolds induces a (bundle)map between the corresponding tangent bundles. I then defined what we mean by a differentiable vector bundle in general. Finally I solved problem 1 b) and c) from MA 252 1999 and problem 2 a) and b) from 2002.

February 13. : I explained what we mean by vector field on a manifold. Assume f is a diffeomorphism from one manifold M to another manifold N, and X is a vector field on M, I explained that we can push X forward by f and get a vector field on N. I also stated that on S^n, there exists a nowhere vanishing vector field if and only if n is odd (I constructed such a vector field when n is odd). I also explained that any vector field on a manifold M, induces a global derivation on the space of smooth function on M and conversely each such derivation is induced by a suitable vector field. Then I recalled the definition of a smooth vector bundle, explained what we mean by bundle maps and equivalences among such bundles, and define what we mean by a trivial bundle. I gave some examples. Finally I defined what we mean by an orientable vector bundle and an orientable manifold. I proved that S^n is orientable.

February 14. ; I recalled the definition of an orientable manifold, proved that the torus T^2 is orientable, and explained why the Möbius strip and the projective space P^2 is not orientable. Then I proved that the antipodal map on S^n is orientation-preserving if and only if n is odd, and from this we consequently will get that P^n is orientable if and only if n is odd. Then I solved Problem 12 and b) c) of Problem 32 from chapter 2.

February 20.: Firstly, I talked in general about the dual space of a (finite-dimensional) vector space. Then I defined the cotangent bundle over a smooth manifold, and talked about its basic properties. I defined the differential df of a function f, and gave the formulae for df in local coordinates. I also explained, that a vector field and a section in the cotangent bundle (a 1-form) induce a function on the manifold. Then I started to talk about multilinear algebra in general has an introduction to the definition of the bundle of covariant k-tensors for a manifold. Torrow I will proceed with this and then, after the do part a) of Problem 32 chapter 2.

February 21. I did part a) of Problem 32 chap 2. Then I proceeded talking about multilinear algebra and I defined the bundle of covariant k-tensors. I then explained how a covariant tensor field A of order k, gives raise to a multilinear map script A:script V x....scriptV->script F linear over script F , where script V is the vector space of smooth vector fields on script F the space of smooth functions on the given manifold. I the formulated Theorem 2. p. 118.

February 27. I proved Theorem 2. p. 118. I then jumped to chapter 5 and I lectured from the beginning of the chapter and ended by stating and proving Theorem 2 on p. 141. Tomorrow I will proceed with this stuff and then probably after the break look at problem 5 and 7 from chapter 3.

February 28. I looked at problem 7 chapter 3. Then I lectured from Spivak Chapter 5 covering the stuff up to an including Theorem 6 p. 147. Finally I sketched a solution of problem 5 (i) from chapter 3.

March 5. I solved problem 3, MA 252 2001. I then formulated and proved Theorem 7 p. 148. I defined the Lie derivative of vector fields and covariant vector fields with respect to a given vector field on a manifold. Finally, I stated Proposition 8 p. 151 and proved statement (3) of this proposition. Tuesday March 6. I will start looking at problem 16 from chapter 3, and then proceed lecturing from chapter 5.

March 6. I solved problem 16 a) b) c ) and d) chapter 3, and then proceeded lecturing from chapter 5, computing L_X w for a covariant vector field w. Monday March 12 I will start by computing the Lie derivative L_X Y (where X and Y are vector fields) (p. 153).

March 12. I computed the Lie derivative L_X Y in local coordinates, then I proceeded lecturing from chapter 5. I dropped the (coordinate-free) proof of Theorem 10, but apart from this I followed the text book ending by proving Lemma 13. p. 157. Tomorrow I will prove Theorem 14 (p. 158). Then I will jump to chapter 7.

March 13. I stated and proved Theorem 14 (p.158). Then I started lecturing from chapter 7. I have covered the stuff including Theorem 3. p. 205.

March 19. I solved problem 19 from chapter 3. Then I proceeded lecturing from chapter 7 ending by proving Corollary 6. p. 207.

March 20. I solved problem from the old exam Ma 252 2000. Then I proceeded lecturing from chapter 7. Defining the bundle of alternating k-tensors. I ended by proving Corollary 8 on p. 209.

March 26 and 27. I covered the stuff from p. 209 (Theorem 9) to p. 217 (Proposition 16). I also solved the problems 1d) 2 and 4 from Chapter 4 in Spivak.

April 10. I defined de Rham cohomology and showed that every closed differential form on R^2 is exact. I then showed that the form \omega on p. 220 is closed but not exact. Then I defined what i mean for a manifold to be contractible and showed that every closed form on a contractible manifold is exact, but I deferred the proof of Theorem 17. p. 224.

April 16. I lectured the proof of Theorem 17. p. 224. I then jumped to chapter 8 p. 277 and lectured Theorem 13 p. 277. I then started on chapter 8 from the beginning defining the integral of a k-form on a singular k-cube and on a k-chain.

April 17. I defined the boundary of a singular k-cube and k-chain. Showed Proposition 3 on p. 251 anf finally proved Stokes Theorem for the integral of forms over chains. I showed that the Fundamental Theorem of calculus and Greens Theorem are special consequences of this more general Theorem.

April 23. I defined the integral of an n-form with compact support over an orientable, n-manifold. I proved the general version of Stokes Theorem.

April 24. I used Stokes Theorem to prove that if M is compact and orientable the are n-forms which are not exact, and I constructed an explicit n-form on S^n of this type. I proved the formulae for integration in polar coordinates.(Spivak Corollary 8 p. 266)

April 30. I started lecturing from the notes by Bjørn Jahren about de Rham chomology. I ended by computing the de Rham cohomology of the real projective spaces. I will proceed lecturing from these notes on Monday (May 7).

May 7. I calculated the the de Rham cohomology of the complex projective spaces. I defined the de Rham cohomology with compact support. I proved Lemma 3.2 and calculated the de Rham cohomology with compact support for Euclidean spaces.

May 8. I defined the Mayer Vitoris sequence for de Rham cohomology with compact support. I explained the cup product and the ring structure of the de Rham cohomology (Lemma on p. 8) and stated Poincare duality for connected orientable manifolds and calculated the de Rham cohomology (with or without compact support) in the top dimension for orientable connected manifolds. Monday May 14. I will prove the Poincare duality Theorem.

May 14. I proved the Poincare duality Theorem

May 15. I finished the notes by Bjørn Jahren calculating the de Rham Cohomology with compact support of MXR, explaining why the de Rham cohomology of compact manifolds always are finite dimensional. Then I showed that the de Rham cohomology (with or without compact support) of non-orientable manifolds in the top dimension always is equal 0. I ended the lecture solving problem 3 of the old exam from 1997 and problem 2 from 1999.