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hw4-difftop

hw4-difftop - Math 343 Homework 4 Due Wednesday 1 To be...

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Math 343 Homework 4 Due Wednesday February 24, 2010. 1 To be handed in for grading 1. Suppose that Z is an l -dimensional submanifold of X and that z Z . Show that there exists a local coordinate system { x 1 , x 2 , . . . x k } defined in a neighborhod U of z in X such that Z U is defined by the equations x l +1 = 0 , . . . , x k = 0. State any results you use and show clearly why they work. 2. If f : M N and g : N P are immersions, show that g f : M P is an immersion as well. 3. Let f : R 3 R be given by f ( x, y, z ) = x 2 + y 2 - z 2 . (a) Check that 0 is the only critical value of f . (b) Prove that if a and b are either both positive or both negative, then f - 1 ( a ) and f - 1 ( b ) are diffeomorphic. (c) Draw pictures to explain what happens to the topology of f - 1 ( c ) as c moves through the criticial value. 4. Let M = all 2 × 2 matrices A = x y z w with real coefficients and det A = 1 and AA T diagonal (that is, off-diagonal entries all 0). (a) Describe M as the solution set of 2 equations in 4 variables. (b) Show M is a manifold. State any results you use and show clearly why they can be used. 5. For which values of a does the hyperboloid defined by x 2 + y 2 - z 2 = 1 intersect the sphere x 2 + y 2 + z 2 = a transversally in R 3 ? What does the intersection look like for different values of a ? 1

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2 Extra Practice 1. (a) If f and g are immersions, show that f × g is. (b) If f is an immersion, show that its restriction to any submanifold on its domain is an immersion.
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hw4-difftop - Math 343 Homework 4 Due Wednesday 1 To be...

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