IT'S DIFFERENTIAL GEOMETRY. PLEASE AT LEAST DO THE PROBLEM 2, I DON'T KNOW HOW TO DO IT. AND IT'S DUE TOMORROW. THANK YOU!

Problem 1. Consider M - { (21, 12, 23, TA, 25) ( R5; 2 + 2 -23 +21+2;}. a. Show that M - {(0, 0, 0, 0, 0)] is a submanifold of RS. b. Show that, for every vector u of the form (41, 0, 1, 0,0), (#1,0, 0, 1,0), (+1, 0, 0, 0, 1), (0, +1, 1, 0,0), (0, +1, 0, 1,0), or (0, #1, 0, 0, 1), there exists a differential curve a: (-s, =) -> R whose image is contained in M and such that a(0) = (0, 0, 0, 0,0) and a' (0) = v. c. Show that M is not a submanifold of IRS. Problem 2. Let f: C3 -+ C be defined by f(u, v, w) = u tv + w. This can also be seen as a map R -> R' by using the standard identifications C = R and C3 = RS. In particular, one can consider the tangent map Ty,v,w) f: Re -> R2 a. Interpreting the tangent map as a map T(u,,w)f : C3 -> C (which is the smarter way to do things), compute T(4,w.w) f(a, b, c) e C for every (a, b, c) e C3. Make sure that you justify the method that you are using. Possible hint: use the geometric interpretation of the tangent map, in terms of curves. b. Show that M = {(u, v, w) E C3; u? + 13 + w = 0. (u, v, w) / 0} is a submanifold of C3 - R. What is its dimension? c. Let 9: C3 - R be defined by g(u, v, w) = ju|2 + ju|2 + |w/2. Show that, for every (u, v, w) E M, there is a vector V e C' such that T(u,v.5) f(V) = 0 and T(u,v,w)g(V) * 0. Possible hint: consider the curve t - a(t) (21tu, eldtu, ebtw) E M. d. Consider the 5 dimensional sphere as S' = { (2, v, w) = C3; lul + 1012 + 1w/2 - 1} (Note that this is the same as the standard definition once we identify C3 to RS.) Show that N = M n S is a submanifold of C3 - R. What is its dimension? Hint: Consider f x 9: C -> C x RR