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Does there exist a regular simple closed curve on a compact surface which cuts the surface into two regions, each with total Gaussian curvature of 2π ? Select one: O a. Yes, and an example is given by a circle on a torus of revolution obtained by rotating this circle about an axis. O b. Yes, and an example is given by the equator on the unit sphere. O c. Yes, and an example is given by the unit circle defined by z = 0 on the hyperboloid of one sheet x² + y² − z² = 1. O d. O e. O f. O g. Oh. O i. O j. Ok. No, by the Jordan curve theorem. No, by Hopf's Umlaufsatz. No, by Fenchel's theorem. No, by the isoperimetric inequality. No, by Green's theorem. No, by the four vertex theorem. No, by Gauss' Theorema Egregium. No, by the Gauss-Bonnet theorem.

Does there exist a regular simple closed curve on a compact surface which cuts the surface into two regions, each with total Gaussian curvature of 2π ? Select one: O a. Yes, and an example is given by a circle on a torus of revolution obtained by rotating this circle about an axis. O b. Yes, and an example is given by the equator on the unit sphere. O c. Yes, and an example is given by the unit circle defined by z = 0 on the hyperboloid of one sheet x² + y² − z² = 1. O d. O e. O f. O g. Oh. O i. O j. Ok. No, by the Jordan curve theorem. No, by Hopf's Umlaufsatz. No, by Fenchel's theorem. No, by the isoperimetric inequality. No, by Green's theorem. No, by the four vertex theorem. No, by Gauss' Theorema Egregium. No, by the Gauss-Bonnet theorem.

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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Does there exist a regular simple closed curve on a compact surface which cuts the surface into two regions, each with total Gaussian curvature of 2π? Select one: a. Yes, and an example is given by a circle on a torus of revolution obtained by rotating this circle about an axis. O b. Yes, and an example is given by the equator on the unit sphere. O c. Yes, and an example is given by the unit circle defined by z = 0 on the hyperboloid of one sheet x² + y² = z² = 1. O d. No, by the Jordan curve theorem. No, by Hopf's Umlaufsatz. O f. No, by Fenchel's theorem. g. e. No, by the isoperimetric inequality. Oh. No, by Green's theorem. O i. No, by the four vertex theorem. Oj. No, by Gauss' Theorema Egregium. Ok. No, by the Gauss-Bonnet theorem.
Transcribed Image Text:Does there exist a regular simple closed curve on a compact surface which cuts the surface into two regions, each with total Gaussian curvature of 2π? Select one: a. Yes, and an example is given by a circle on a torus of revolution obtained by rotating this circle about an axis. O b. Yes, and an example is given by the equator on the unit sphere. O c. Yes, and an example is given by the unit circle defined by z = 0 on the hyperboloid of one sheet x² + y² = z² = 1. O d. No, by the Jordan curve theorem. No, by Hopf's Umlaufsatz. O f. No, by Fenchel's theorem. g. e. No, by the isoperimetric inequality. Oh. No, by Green's theorem. O i. No, by the four vertex theorem. Oj. No, by Gauss' Theorema Egregium. Ok. No, by the Gauss-Bonnet theorem.
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