Exercice Consider the circle S¹, defined by S¹ = {(x, y) = R² | 2² + y²=1}. Let U₁ = {(r,y) € S¹ r>0}, and 91: U₁ → (-₁), with 1(x, y) = arctan, S¹ | 1 <0}, U₂= {(1,y) Us=(r,y) S¹y>0}, U₁= {(r,y) € S¹|r>0}. 1. Construct functions 92, 93, and 94, 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.
Exercice Consider the circle S¹, defined by S¹ = {(x, y) = R² | 2² + y²=1}. Let U₁ = {(r,y) € S¹ r>0}, and 91: U₁ → (-₁), with 1(x, y) = arctan, S¹ | 1 <0}, U₂= {(1,y) Us=(r,y) S¹y>0}, U₁= {(r,y) € S¹|r>0}. 1. Construct functions 92, 93, and 94, 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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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differebtial geometry smooth manifolds part 6 7 8 9 10
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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