Problem 8: Let M be a 2 dimensional differentiable manifold with coordinates u and v, that is endowed with a covariant m

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Problem 8 Let M Be A 2 Dimensional Differentiable Manifold With Coordinates U And V That Is Endowed With A Covariant M 1 (28.11 KiB) Viewed 23 times

Problem 8: Let M be a 2 dimensional differentiable manifold with coordinates u and v, that is endowed with a covariant metric tensor g described by the line element ds? = 4 du? + 2uvdudu – du? Suppose also, that H is a type(2,0) tensor field defined at every point on the manifold such that 2 H = 0 (a) Find (for all values of i and j, i = 1, 2 and j = 1,2), the components: i. Hij ii. Hu] (Express your final answers in matrix form) (b) Without performing any calculation, write down the components, H), in matrix form. Provide a brief justification for your answer.