1 (1) Metric of surface of revolution. Consider a smooth surface of revolution S which has a regular surface patch o: UR

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1 1 Metric Of Surface Of Revolution Consider A Smooth Surface Of Revolution S Which Has A Regular Surface Patch O Ur 1 (37.6 KiB) Viewed 23 times

1 (1) Metric of surface of revolution. Consider a smooth surface of revolution S which has a regular surface patch o: UR3, for some open U CRP, of the form o(u, v) = (f(u) cosv, f(u) sin v, h(u)). where f(u) and h(u) are smooth functions. Compute the Riemannian metric g of S with respect to o. In particular, show that if f'(u)? + H'(u) = 1 for all u, then g = du? + f(u) dv2. (2) Bonus: Suppose now that another smooth surface Š has an atlas consisting of a single regular surface patch õ(u, v), defined on an open set Ū, and its Riemannian metric ğ is given by g = du? + p(u)?dva, for a smooth function plu) such that pu) < 1. Using the previous part, or otherwise, show that S is locally isometric to a surface of revolution.