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Exercice Consider the circle S¹, defined by S¹ = {(x, y) = R² | 2² + y² = 1}. Let U₁ = {(r,y) € S¹ r>0}, and 9₁: U₁(-), with 1(x, y) = arctan, U₂= {(1,y) € S¹ | <0}, Us=(r,y) Sty>0}, U₁= {(r,y) € S¹r>0}. 1. Construct functions 92, 93, and 4, such that {(U₁.91), (U2, 42), (U3, 43), (U₁, Py)} is a smooth atlas. 2. Show theoretically that the torus T2 = S¹ x S¹ is a smooth Manifold. 3. Use the Atlas constructed in question 1 to construct a smooth atlas on T². 4. Let 0

Exercice Consider the circle S¹, defined by S¹ = {(x, y) = R² | 2² + y² = 1}. Let U₁ = {(r,y) € S¹ r>0}, and 9₁: U₁(-), with 1(x, y) = arctan, U₂= {(1,y) € S¹ | <0}, Us=(r,y) Sty>0}, U₁= {(r,y) € S¹r>0}. 1. Construct functions 92, 93, and 4, such that {(U₁.91), (U2, 42), (U3, 43), (U₁, Py)} is a smooth atlas. 2. Show theoretically that the torus T2 = S¹ x S¹ is a smooth Manifold. 3. Use the Atlas constructed in question 1 to construct a smooth atlas on T². 4. Let 0

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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differential geometry smooth atlas part 1 2 3
Exercice Consider the circle S¹, defined by S¹ = {(x, y) = R² | x² + y² = 1}. Let U₁ = {(r,y) S¹ r>0}, and 91: U₁(-), with 1(r, y) = arctan, U₂= {(r,y) S¹ <0}, Us={(r,y) Sy>0}, U₁= {(r,y) € S¹|r>0}. 1. Construct functions 92, 93, and 94, such that {(U₁, 1), (U2, 42), (U3, 43), (U4, Py)} is a smooth atlas. 2. Show theoretically that the torus T2 = S¹ x S¹ is a smooth Manifold. 3. Use the Atlas constructed in question 1 to construct a smooth atlas on T². 4. Let 0 <r <1 and let T be the smooth manifold of R³ defined by the equation: (² + y² +r²-²-1)² - 4(x² + y²) (r² - ²) = 0. Show that (x, y, z) € T if and only if (√² + y² - 1)² +2²=². 5. Use part 4 and imitate part 1 to construct a smooth atlas on T. (Show that any point in T can be written as ((1+rcosv)cosu, (1+rcosv) sinu, rsinv) with (u, v) in an appropriate domain.) 6. Let F: T² →→→T, defined in the following way: for a point PE T2 choose a coordinate chart in the atlas constructed in question 3, and let (u, v) be local coordinates, put F(P) = ((1+rcose)cosu, (1+rcosv)sinu, rsine). Give convincing proofs that F is well defined and that F is a smooth map. 7. Is F a diffeomorphism? =(1) € T², give an example of a derivation X at P. 9. Give an interpretation of X, in terms of a curve (just explain geometrically what does this mean). 10. What is the image of X by F.?
Transcribed Image Text:Exercice Consider the circle S¹, defined by S¹ = {(x, y) = R² | x² + y² = 1}. Let U₁ = {(r,y) S¹ r>0}, and 91: U₁(-), with 1(r, y) = arctan, U₂= {(r,y) S¹ <0}, Us={(r,y) Sy>0}, U₁= {(r,y) € S¹|r>0}. 1. Construct functions 92, 93, and 94, such that {(U₁, 1), (U2, 42), (U3, 43), (U4, Py)} is a smooth atlas. 2. Show theoretically that the torus T2 = S¹ x S¹ is a smooth Manifold. 3. Use the Atlas constructed in question 1 to construct a smooth atlas on T². 4. Let 0 <r <1 and let T be the smooth manifold of R³ defined by the equation: (² + y² +r²-²-1)² - 4(x² + y²) (r² - ²) = 0. Show that (x, y, z) € T if and only if (√² + y² - 1)² +2²=². 5. Use part 4 and imitate part 1 to construct a smooth atlas on T. (Show that any point in T can be written as ((1+rcosv)cosu, (1+rcosv) sinu, rsinv) with (u, v) in an appropriate domain.) 6. Let F: T² →→→T, defined in the following way: for a point PE T2 choose a coordinate chart in the atlas constructed in question 3, and let (u, v) be local coordinates, put F(P) = ((1+rcose)cosu, (1+rcosv)sinu, rsine). Give convincing proofs that F is well defined and that F is a smooth map. 7. Is F a diffeomorphism? =(1) € T², give an example of a derivation X at P. 9. Give an interpretation of X, in terms of a curve (just explain geometrically what does this mean). 10. What is the image of X by F.?
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