1. Differentiable Manifolds 15 Examples and Exercises 1.20. Discuss the following submanifolds of E n in particular make drawings of them: The unit sphere S71'1 = { x G l n : (x, x) = 1} C W1. 2 The ellipsoid {x G Rn : f(x) := Y^i=i ^ ~ 1} a * ^ ^' w ^ ^ principal axis ai,.. . , an. The hyperboloid {x e W1 : f(x) := E l U ^ S = !} ^ = ±]L a * ^ ° w i t h principal axis a and index — J ] ^ . The saddle {x G M3 : xs = X1X2}. The foras: the rotation surface generated by rotation of (y — R)2 + z2 — r2^ 0 r R, with center the z-axis, i.e., {(x,y,z):(y/^T^-R)2 + z2=r2}. 1.21. A compact surface of genus g. Let f(x) := x(x — l)2(x — 2)2 ...{x - (g - l))2(x - g). For small r 0 the set {(x,y,z) : (y2 + f(x))2 + z2 = r2} describes a surface of genus g (topologically a sphere with g handles) in IR3. Visualize this: 1.22. The Moebius strip. It is not the set of zeros of a regular function on an open neighborhood of Rn. Why not? But it may be represented by the following parameterization: ( cos ip(R + r cos((p/2)) sin ip(R + r cos((p/2)) rsm((p/2) (r,y)€ (-1,1) x [0,2*0, where R is quite big. 1.23. Describe an atlas for the real projective plane which consists of three charts (homogeneous coordinates) and compute the chart changings. Then describe an atlas for the n-dimensional real projective space Pn(M) and compute the chart changes.
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2008 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.