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.

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