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NYTimes People: Kawakubo, Rei
Rei Kawakubo News - The New York Times Log In Register Now Help Home Page Today's Paper Video Most Popular Times Topics Search All NYTimes.com Tuesday, November 13, 2012 Times Topics World U.S. N.Y. / Region Business Technology Science Health Sports Opinion Arts Style Travel Jobs Real...
GPS Technology Drives Global Treasure Hunt
with my five-year-old and nine-year-old as a family," said geocacher Rich Ness, assistant manager of Bloomington, Minnesota's REI outdoor-gear store (GPS coordinates: latitude 44° 51.6' N, longitude 93° 17.3' W). "I haven't come across a kid yet who didn't think it's the gre...
 Riemannian Geometry (PDF)
from above that (TM,M, pi) together with the maximal bundle atlas B̂ defined by B is a differen- tiable vector bundle. Definition 4.8. Let M be a differentiable manifold, then a section X : M → TM of the tangent bundle is called a vector field. The set of smooth vector fields X : M →...
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from above that (TM,M, pi) together with the maximal bundle atlas B̂ defined by B is a differen- tiable vector bundle. Definition 4.8. Let M be a differentiable manifold, then a section X : M → TM of the tangent bundle is called a vector field. The set of smooth vector fields X : M → TM is denoted by C∞(TM). Example 4.9. We have seen earlier that the 3-sphere S3 in H ∼= C2 carries a group structure · given by (z, w) · (α, β) = (zα− wβ̄, zβ + wᾱ). This makes (S3, ·) into a Lie group with neutral element e = (1
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=18
16 2. DIFFERENTIABLE MANIFOLDS Example 2.25. The result of Proposition 2.24 can be used to show that the following maps are all smooth. (i) φ1 : S 2 ⊂ R3 → <span class="highlight">S3</span> ⊂ R4, φ1 : (x, y, z) 7→ (x, y, z, 0), (ii) φ2 : S 3 ⊂ C2 → S2 ⊂ C×R, φ2 : (z1, z2) 7→ (2z1z̄2, |z1|2−|z2|2), (iii) φ3 : R1 → S1 ⊂ C, φ3 : t 7→ eit, (iv) φ4 : Rm+1 \ {0} → Sm, φ4 : x 7→ x/|x|, (v) φ5 : Rm+1 \ {0} → RPm, φ5 : x 7→ [x], (vi) φ6 : S m → RPm, φ6 : x 7→ [x]. In differential geometry we are especially interested in differentiable manifolds
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=19
→ p · q̄ and a real valued norm given by |p|2 = p · p̄. Then the 3-dimensional unit sphere <span class="highlight">S3</span> in H ∼= R4 with the restricted multiplication forms a compact Lie subgroup (<span class="highlight">S3</span>, ·) of (H∗, ·). They are both non-abelian. We shall now introduce some of the classical real and complex matrix Lie groups. As a reference on this topic we recommend the wonderful book: A. W. Knapp, Lie Groups Beyond an Introduction, Birkhäuser (2002). Example 2.31. Let Nil3 be the subset of R3×3 given by Nil3 = { 1 x z0 1 y 0 0 1
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=22
R3 and the Riemann sphere Ĉ are diffeomorphic. Exercise 2.8. Find a proof of Proposition 2.24. Exercise 2.9. Let the spheres S1, <span class="highlight">S3</span> and the Lie groups SO(n), O(n), SU(n), U(n) be equipped with their standard differentiable structures introduced above. Use Proposition 2.24 to prove the fol- lowing diffeomorphisms S1 ∼= SO(2), <span class="highlight">S3</span> ∼= SU(2), SO(n)×O(1) ∼= O(n), SU(n)×U(1) ∼= U(n). Exercise 2.10. Find a proof of Corollary 2.28. Exercise 2.11. Let (G, ∗) and (H, ·) be two Lie groups. Prove that the product
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=32
embedding if and only if k = ±1. Example 3.23. Let q ∈ <span class="highlight">S3</span> be a quaternion of unit length and φq : S 1 → <span class="highlight">S3</span> be the map defined by φq : z 7→ qz. For w ∈ S1 let γw : R → S1 be the curve given by γw(t) = weit. Then γw(0) = w, γ̇w(0) = iw and φq(γw(t)) = qwe it. By differentiating we yield dφq(γ̇w(0)) = d dt (φq(γw(t)))|t=0 = d dt (qweit)|t=0 = qiw. Then |dφq(γ̇w(0))| = |qwi| = |q||w| = 1 6= 0 implies that the differen- tial dφq is injective. It is easily checked that the immersion φq is an embedding. In the next
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=35
pi : Rn → Rm given by pi : (x1, . . . , xn) 7→ (x1, . . . , xm). Its differential dpix at a point x is surjective since dpix(v1, . . . , vn) = (v1, . . . , vm). This means that the projection is a submersion. An important sub- mersion between spheres is given by the following. Example 3.30. Let <span class="highlight">S3</span> and S2 be the unit spheres in C2 and C× R ∼= R3, respectively. The Hopf map φ : <span class="highlight">S3</span> → S2 is given by φ : (x, y) 7→ (2xȳ, |x|2 − |y|2). For p ∈ <span class="highlight">S3</span> the Hopf circle Cp through p is given by Cp = {eiθ(x, y)| θ ∈ R
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=36
ψk : z 7→ zk. For which k ∈ N0 are φk, ψk immersions, submersions or embeddings. Exercise 3.7. Prove that the map φ : Rm → Cm given by φ : (x1, . . . , xm) 7→ (eix1 , . . . , eixm) is a parametrization of the m-dimensional torus Tm in Cm. Exercise 3.8. Find a proof for Theorem 3.26. Exercise 3.9. Prove that the Hopf-map φ : <span class="highlight">S3</span> → S2 with φ : (x, y) 7→ (2xȳ, |x|2 − |y|2) is a submersion.
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=41
from above that (TM,M, pi) together with the maximal bundle atlas B̂ defined by B is a differen- tiable vector bundle. Definition 4.8. Let M be a differentiable manifold, then a section X : M → TM of the tangent bundle is called a vector field. The set of smooth vector fields X : M → TM is denoted by C∞(TM). Example 4.9. We have seen earlier that the 3-sphere <span class="highlight">S3</span> in H ∼= C2 carries a group structure · given by (z, w) · (α, β) = (zα− wβ̄, zβ + wᾱ). This makes (<span class="highlight">S3</span>, ·) into a Lie group with neutral element e = (1
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=83
(− � , � )→ O(n) is a geodesic if and only if γt · γ̈ = γ̈t · γ. Exercise 7.3. Find a proof for Proposition 7.23. Exercise 7.4. Find a proof for Corollary 7.24. Exercise 7.5. For the real parameter θ ∈ (0, pi/2) define the 2- dimensional torus T 2θ by T 2θ = {(cos θeiα, sin θeiβ) ∈ <span class="highlight">S3</span>| α, β ∈ R}. Determine for which θ ∈ (0, pi/2) the torus T 2θ is a minimal submanifold of the 3-dimensional sphere <span class="highlight">S3</span> = {(z1, z2) ∈ C2| |z1|2 + |z2|2 = 1}. Exercise 7.6. Find a proof for Corollary 7.27. Exercise 7.7. Determine the totally
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=95
zkw̄k and let Tm = {z ∈ Cm| |z1| = ... = |zm| = 1} be the m-dimensional torus in Cm with the induced metric. Find an isometric immersion φ : Rm → Tm, determine all geodesics on Tm and prove that the torus is flat. Exercise 8.6. Find a proof for Proposition 8.17. Exercise 8.7. Let the Lie group <span class="highlight">S3</span> ∼= SU(2) be equipped with the metric g(Z,W ) = 1 2 Re(trace(Z̄tW )). (i) Find an orthonormal basis for TeSU(2). (ii) Prove that (SU(2), g) has constant sectional curvature +1. Exercise 8.8. Let Sm be the unit sphere in
Arctic Animals 4-8
Be au fo rt Se a Am u r R iv er Se a of O kh ot sk Be rin g Se a G ul f o f A la sk a Ba ffi n Ba y Ch uk ch i S ea Yu k o n R i v e r CA N A D A A LA S K A Ar c ti c C ir c le H ud so n Ba y Ri ve r Sa gu en ay N or th P o le RU S S IA G R E E N LA N D Ar c tic O ce an NO RW AY SW ED EN FIN LA ND I...
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Be au fo rt Se a Am u r R iv er Se a of O kh ot sk Be rin g Se a G ul f o f A la sk a Ba ffi n Ba y Ch uk ch i S ea Yu k o n R i v e r CA N A D A A LA S K A Ar c ti c C ir c le H ud so n Ba y Ri ve r Sa gu en ay N or th P o le RU S S IA G R E E N LA N D Ar c tic O ce an NO RW AY SW ED EN FIN LA ND ICELA ND M ur m an sk Di ks on Ti ks i Pe ve k Ba rro w Pr ud ho e Ba y Tu kto ya ktu k Ch ur ch ill Go dh av n Re so lut e Re so lut eT hu le Al er t Re yk jav ik sc al e ~ 1 cm = 2 33 k m 23 3
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www.seaworld.org/just-for-teachers/guides/pdf/arctic-animals-4-8.pdf#page...
Be au fo rt Se a Am u r R iv er Se a of O kh ot sk Be rin g Se a G ul f o f A la sk a Ba ffi n Ba y Ch uk ch i S ea Yu k o n R i v e r CA N A D A A LA S K A Ar c ti c C ir c le H ud so n Ba y Ri ve r Sa gu en ay N or th P o le RU S S IA G R E E N LA N D Ar c tic O ce an NO RW AY SW ED EN FIN LA ND ICELA ND M ur m an sk Di ks on Ti ks i Pe ve k Ba rro w Pr ud ho e Ba y Tu kto ya ktu k Ch ur ch ill Go dh av n Re so lut e Re so lut eT hu le Al er t Re yk <span class="highlight">jav</span> ik sc al e ~ 1 cm = 2 33 k m 23 3
NYTimes People: Elbaz, Alber
Margiela, Martin, Simons, Raf, Elbaz, Alber, Dior, Christian, Mouret, Roland, Chalayan, Hussein, Jil Sander AG, Lanvin, Christian Dior SA Anna Dello Russo's Fashion Shower By ERIC WILSON Teaser videos are the new thing in viral marketing, and after H&M's latest campaign, you wi...
Fingerprints
spotted Kate's green REI headlamp, which the thief had miraculously overlooked. He was spellbound. "I'll have to dust this carefully for fingerprints back at the station," he said, turning it over in his hands. I asked him when we could pick up the headlamp. "Patom," he...
Venezuela Analitica
Mundo Economía Editorial Nuestros Columnistas Videos Galerías YouTube Foros Twitter Facebook Clima Efemérides Juegos ¿Objetivos educativos o políticos? Gustavo Roosen Presentada originalmente como referida sólo a la educación bás...
The Female Robinson Crusoe: Who Was Ada Blackjack?
not thrive on extreme outdoor situations and adventures. I do not enjoy the cold nor do I particularly enjoy "roughing it." I like my conveniences, my creature comforts, and I am much more likely to subscribe to In Style magazine than to the REI catalogue. Perhaps that is why, whene...
Reiki: Hype or Help?
Say Yes to the Dress: Bridesmaids Secret Princes Sister Wives Toddlers & Tiaras Virgin Diaries What Not to Wear More Shows | ON TV Daily TV Schedule Weekly TV Schedule | VIDEO Full Episodes Take Charge of Your Everyday All TLC Videos | SOCIAL 19 Kids &...
NYTimes People: Watanabe, Junya
a Japanese designer who broke out from under the shadow of the towering presence of Rei Kawakubo, the creative force behind Comme des Garçons. A former protégé of Ms. Kawakubo and a knitwear designer, Mr. Watanabe launched his own line in 1993. Within years, his clothes...
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