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uestion 2. (1) Metric of surface of revolution. Consider a smooth surface of revolution which has a regular surface patch σ: U → R³, for some open UC R², o 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 metri g of S with respect to o. In particular, show that if f'(u)² + h'(u)² = 1 fo all U₂ then g = du² + f(u)² dv².

uestion 2. (1) Metric of surface of revolution. Consider a smooth surface of revolution which has a regular surface patch σ: U → R³, for some open UC R², o 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 metri g of S with respect to o. In particular, show that if f'(u)² + h'(u)² = 1 fo all U₂ then g = du² + f(u)² dv².

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 2. (1) Metric of surface of revolution. Consider a smooth surface of revolution S which has a regular surface patch σ: U → R³, for some open UC R², 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)² + h'(u)² = 1 for all u, then g = = du² + f(u)² dv². Bonus: Suppose now that another smooth surface S has an atlas consisting of a single regular surface patch õ(u, v), defined on an open set Ũ, and its Riemannian metric 9 is given by 9 = du² + p(u)² dv², for a smooth function p(u) such that p'(u)| < 1. Using the previous part, or otherwise, show that S is locally isometric to a surface of revolution.
Transcribed Image Text:Question 2. (1) Metric of surface of revolution. Consider a smooth surface of revolution S which has a regular surface patch σ: U → R³, for some open UC R², 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)² + h'(u)² = 1 for all u, then g = = du² + f(u)² dv². Bonus: Suppose now that another smooth surface S has an atlas consisting of a single regular surface patch õ(u, v), defined on an open set Ũ, and its Riemannian metric 9 is given by 9 = du² + p(u)² dv², for a smooth function p(u) such that p'(u)| < 1. Using the previous part, or otherwise, show that S is locally isometric to a surface of revolution.
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