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(a) Letr> 0 be given. Consider the surface Cr= {(x, y, z) | x² + y² = r²}. Show that C, is a regular surface. Then show that all the geodesics on C, take on the form of either straight lines, circles, or helices. (b) What does it mean for two regular surfaces to be locally isometric? (c) Show that C, is locally isometric to the plane. (d) If two regular surfaces are locally isometric, do they necessarily have the same mean curvature? If your answer is yes, then prove it, and if your answer is no, then provide a counterexample.

(a) Letr> 0 be given. Consider the surface Cr= {(x, y, z) | x² + y² = r²}. Show that C, is a regular surface. Then show that all the geodesics on C, take on the form of either straight lines, circles, or helices. (b) What does it mean for two regular surfaces to be locally isometric? (c) Show that C, is locally isometric to the plane. (d) If two regular surfaces are locally isometric, do they necessarily have the same mean curvature? If your answer is yes, then prove it, and if your answer is no, then provide a counterexample.

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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Q4 (a) Letr> 0 be given. Consider the surface + M Cr= {(x, y, z) x² + y² = r²}. Show that C, is a regular surface. Then show that all the geodesics on C, take on the form of either straight lines, circles, or helices. (b) What does it mean for two regular surfaces to be locally isometric? (c) Show that C, is locally isometric to the plane. (d) If two regular surfaces are locally isometric, do they necessarily have the same mean curvature? If your answer is yes, then prove it, and if your answer is no, then provide a counterexample.
Transcribed Image Text:Q4 (a) Letr> 0 be given. Consider the surface + M Cr= {(x, y, z) x² + y² = r²}. Show that C, is a regular surface. Then show that all the geodesics on C, take on the form of either straight lines, circles, or helices. (b) What does it mean for two regular surfaces to be locally isometric? (c) Show that C, is locally isometric to the plane. (d) If two regular surfaces are locally isometric, do they necessarily have the same mean curvature? If your answer is yes, then prove it, and if your answer is no, then provide a counterexample.
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