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| (1) | Read chapter 4 of Do Carmo: “Curvature”. |
| (2) | Prove that  has constant sectional curvature 1. Do this from scratch by considering orthonormal vectors  in  , and computing
 and  to vector fields on  by extending them to constant vector fields on  and projecting onto  . Comment: in chapter 6, we’ll learn an easier way to solve this problem. |
| (3) | Consider the surface of revolution described in problem 1 of DoCarmo’s chapter 3. Assume that the curve being rotated about the z-axis is parameterized by arclength; that is, assume that  . Prove that the sectional curvature at any point  of this surface is  . Hint: you have formulas for the metric components  . Use these to find formulas for the Christoffel symbols  and the curvature tensor component  . |
| (4) | Using the previous problem, prove that no complete surface of revolution in can have constant curvature  . Comment: this is a special case of Hilbert’s theorem, which says that a complete surface with constant negative curvature cannot be isometrically immersed in . |
| (5) | Describe the curvature tensor of the product of two Riemannian manifolds,  (with the product metric), in terms of the curvature tensors of  and  . If and both have nonnegative sectional curvature, does their product have nonnegative sectional curvature? If and both have positive sectional curvature, does their product have positive sectional curvature?
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