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# One way to see higher dimensional surface

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PFI seminar on 2011/03/03

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### One way to see higher dimensional surface

1. 1. PFI2011 3 3
2. 2. • • GLn (R), GLn (C) • GLn (R), O(n) {( ) } a, b • SL2 (R) = |ab − cd = 1 c, d • R4 ab − cd = 1
3. 3. • Mn (R) : n• Mn (R) : n• GLn (R) : = {g ∈ Mn (R) | detg ̸= 0}• SLn (R) = {g ∈ Mn (R) | detg = 1}• SLn (C) = {g ∈ Mn (C) | detg = 1}• O(n) : = {g ∈ Mn (R) | gt g = In }• SO(n) : = {g ∈ Mn (R) | gt g = In , detg = 1}• U(n) : = {g ∈ Mn (C) | g∗ g = In }• SU(n) : = {g ∈ Mn (C) | g∗ g = In , detg = 1}
4. 4. 2• Mn (R) Rn 2• Mn (C) Cn 2 2 • → Rn , Cn•
5. 5. { } ∑ n+1 n• S = 1 n+1 (x , . . . , x )∈R n+1 | i 2 (x ) = 1 i=1
6. 6. 1 {( )} cos θ sin θ• SO(2) = = S1 − sin θ cos θ { } ∑ n+1• O(n + 1)/O(n) = S = n 1 n+1 (x , . . . x )| i 2 (x ) = 1 i=1   0  . • O(n) ∋ A →   A . .  ∈ O(n + 1) O(n)  0 0 ··· 0 1 O(n + 1)• O(3, R) R3 /O(2, R) {(0, 0, z)} = S1 × R •• SL2 (R) = S1 × R2
7. 7. 2(1/2)• H = {a + bi + cj + dk|a, b, c, d ∈ R} • i = j = k2 = −1, ij = k, jk = i, ki = j, ji = −k, 2 2 kj = −i, ik = −j • (e.g. ij ̸= ji)• z = a + bi + cj + dk ∈ H • ¯ = a − bi − cj − dk z • |z|2 = z¯ z• S3 = {(a, b, c, d) ∈ R4 | z = a + bi + cj + dk |z| = 1}
8. 8. 2(2/2)• ( ) a + bi c + di • z = a + bi + cj + dk ↔ −c + di −a + bi• → S3 H SU(2) Figure:
9. 9. • S3 4 → Figure: Dimensions
10. 10. 1• f : R2 → R f(x, y) = x2 − y3 • f −1 (0) : x = 0• {(x, y) ∈ R2 |x2/3 + y2/3 = 1}• http://www-history.mcs.st-and.ac.uk/Curves/Curves.html
11. 11. f : Rr → R a ∈ Rr ∂f (a)(1 ≤ i ≤ r) 0∂xi f −1 (f(a))
12. 12. 1 • f : R2 → R f(x, y) = x2 − y2 • ∂x f = 2x, ∂y f = −2y • ∂x f = ∂y f = 0 (x, y) = (0, 0) • f −1 (0) : 2 • f −1 (c) (c ̸= 0) :
13. 13. 2 • det : Mn (R) → R ; g → detg • SLn (R) = det−1 (1) SLn (R)
14. 14. 2• • (Rn or Cn ) R R• S1 = {(x, y) ∈ R2 | x2 + y2 = 1},• D = {(x, y) ∈ R2 | y ≥ 0}
15. 15. ∑ n• f : Mn (R) → R ; g → Tr(t gg) f(g) = 2 gij i,j=1 • g ∈ O(n) −1 f(g) = n √ O(n) ⊂ f (n) O(n) Mn (R) n
16. 16. • O(p, q) = {g ∈ Mn (R) | t gIp,q g = Ip,q }(p + q = n) q>0   1  ..   .     1  • Ip,q =     −1   ..   .  {( ) −1 } cosh θ sinh θ • O(1, 1) = ∈ R |0 ≤ θ < 2π 4 sinh θ cosh θ • O(3, 1)
17. 17. 3• Figure:
18. 18. 1 • On (R) −1 1 (= SO(n)) • π0 (On (R)) = Z2 • dimZ π0 (X) = #{X/ ∼} • x∼y⇔x y
19. 19. (2/2) • g g′ On (R) g g′ c c • c : [−1, 1] → On (R) ; c(−1) = g, c(1) = g′ • det ◦ c • (det ◦ c)(−1) = det(g) = −1 • (det ◦ c)(1) = det(g′ ) = 1 • (det ◦ c)(a) = 0 a ∈ (−1, 1) • c(a) ̸∈ On (R) c
20. 20. 2 Figure: • π0 ( ) = Z2
21. 21. 4•
22. 22. • π1 (X, x0 ) = {f : [0, 1] → X|f(0) = f(1) = x0 }/ ∼ • f∼g⇔f g•• http://www.slideshare.net/KentaOono/pf-iseminar• π1 (X, x0 ) = 0 ⇒ 1
23. 23. • X 1 1 I→X p = I(0) q = I(1) ···• www.math.sci.hokudai.ac.jp/ ˜ ishikawa/isoukika/isoukika5.pdf
24. 24. • πn (X, x0 ) = {f : [0, 1]n → X|f(a) = x0 ∀a ∈ ∂([0, 1]n )}/ ∼• f∼g↔f g
25. 25. •• :π1 (SOn (R)) = Z/2Z = {0, 1} (if n ≥ 3)
26. 26. (1/2)• su(n) = = {X ∈ Mn (C)|X + ∗ X = 0}• su(2) 3 R3• g ∈ SU(2), X ∈ su(2) Adg (X) = gXg−1 gXg−1 ∈ su(2)• Adg su(2)• Adg ∈ SO(3) • SU(2) → SO(3) ; g → Adg• 2 1 X = Adg g X 2
27. 27. (2/2)• Spin(n) → SO(n) :2
28. 28. • (dimR Mn (R) = n2 )• • (Sn ) • X → πn (X), Hn (X) • • C∞ (M) = {f : M → R|f } • f:M→R • • f : [−1, 1] → M = • f : S1 → M, =