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Question 1. Using the transformation law patterns for upper ...

Question 1. Using the transformation law patterns for upper and lower indices, predict a transfor- mation law giving Laß in terms of Lij in the presence of a coordinate transformation f : U U, and then show that your guess is correct. (This shows that the Lij functions do indeed form tensor...

Question 1. This question concerns surfaces of revolution....

Question 1. This question concerns surfaces of revolution. (See Homeworks 5 and 6 for the definition and some earlier computations) (i) Compute the coefficients L's for a surface of revolution, and write your answer in matrix format. (ii) Compute the mean curvature H and the Gaussian curvatur...

1. [4 pts] Recall that for plonar curves, convexity is equiv...

1. [4 pts] Recall that for plonar curves, convexity is equivalent to planar curvature having con- stant. sign... but. only if the curve is simple and closed. Explain what aspect. of the prool uses the assumption that the curve is simple. 2. Recall the following simple surface function for the tor...

Question 1. Show that a unit speed meridian on a surface of ...

Question 1. Show that a unit speed meridian on a surface of revolution (see Homework 4) is always a geodesic (HINT: Use Prop 4.17 in the notes). Use similar techniques to determine when a circle of latitude is a geodesic. Phrase your answer in terms of the tangent vector X 1 along the circle of...

1. Show that R n and the open balls in R n (with the subs...

1. Show that R n and the open balls in R n (with the subspace topology) are homeomorphic. 2. Show that the “punctured sphere” S 2 − {p}, where S 2 = {x ∈ R 3 kxk = 1} and p ∈ S 2 equipped with the subspace topology is homeomorphic to the plane R 2 . Hin...

1. A diffeomorphism f : E' between surfaces is a conformal...

1. A diffeomorphism f : E' between surfaces is a conformal map, if there is a non- vanishing function A so that p(X) = (.1) for all P € and X E TpL The function l is called the conformal factor. a. Let Q : u V be an orientation preserving diffeomorphism of open subsets of R² and ...

13. Let V be the canonical connection on R3 given by the dir...

13. Let V be the canonical connection on R3 given by the directional derivative. Define VxY = Vxy + 1XxY, where X denotes the cross-product. Show that V is a connection and find its torsion and curvature. 15. Suppose that O is a 1-form on R3 such that O ^ da III 0. Let X, Y are vector fields, ...

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