February 2005

JMM

Matematik 5GE

Assignment 1

Problem 1 Define CP^{1} to the set of complex 1-dimensional complex subspaces of C^{2}. Show that CP^{1} is a smoth manifold: Find local charts and show that the transition functions are smooth. Have we seen this particular manifold before?

Problem 2

Show that the map π: T M → M is smooth: Find an expression for π in local coordinates. Problem 3

• Show that there is a global trivialization T R^{n} ⟶^{≃} R^{n} × R^{n}. (Remember the discussion about geometric tangen vectors in R^{n}.)

• Show that there is a global trivialization T S^{1} ⟶^{≃} S^{1} × R. (Think about why we could find a trivialization for R^{n}.)

• Do you think that there is a global trivialization T S^{2} ⟶^{≃} S^{2} ×R^{2}? (No proof required.)

Manifolds M with global trivializations T M ⟶^{≃} M × R^{n} are said to be parallizable.

Problem 4 Let F: M → N be a smooth map.

(1) Define a map F_{∗}: T M → T N such that the diagram

\xymatrix{T M \ar[d]_{π} \ar[r]^{F_{∗}} & T N \ar[d]^{π} \\ M \ar[r]_{F} & N}

commutes.

(2) Show that F_{∗} is smooth: Find the expression for F_{∗} in local coordinates.

(Due: Tuesday March 1st )