|
Education
Web
 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 →...
1
0
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
19
0
http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=19
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
22
0
http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=22
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
35
0
http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=35
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
41
0
http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=41
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
83
0
http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=83
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
Poetry Out Loud: Kennedy, X J
Poetry Out Loud Menu About Rules FAQ State Contacts News My Favorite Poems State Partners Login National Recitation Contest Find Poems My Favorite Poems Watch Video Listen to Poetry Tips on
The Skinny on Nudism in the U.S.
The Skinny on Nudism in the U.S. National Geographic News, Reporting Your World Daily Thursday, October 28, 2010 MAIN ANIMAL NEWS ANCIENT WORLD ENVIRONMENT NEWS CULTURES NEWS SPACE/TECH NEWS WEIRD PHOTOS VIDEO The Skinny on Nudism in the U.S. Jennifer
NYTimes People: Tunick, Spencer
Spencer Tunick 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 Re...
SStudies in Motion
Grade Level 9-12 Please Note: The video program contains several scenes featuring black and white archival photographs of nude male and female models, including some photographs of Eakins himself. Subject Area(s) Studio Art Technology Language Arts (See Extension Ideas) Objectives &bul...
1
0
Grade Level 9-12 Please Note: The video program contains several scenes featuring black and white archival photographs of nude male and female models, including some photographs of Eakins himself. Subject Area(s) Studio Art Technology Language Arts (See Extension Ideas) Objectives • Students will use digital video to capture human motion • Students will demonstrate their understanding of human motion by producing a realistic representation • Students will compare illustrations made before and after the project
1
0
http://www.pbs.org/eakins/pdfs/lesson2.pdf#page=1
www.pbs.org/eakins/pdfs/lesson2.pdf#page=1
Grade Level 9-12 Please Note: The <span class="highlight">video</span> program contains several scenes featuring black and white archival photographs of <span class="highlight">nude</span> male and female models, including some photographs of Eakins himself. Subject Area(s) Studio Art Technology Language Arts (See Extension Ideas) Objectives • Students will use digital <span class="highlight">video</span> to capture human motion • Students will demonstrate their understanding of human motion by producing a realistic representation • Students will compare illustrations made before and after the project
5
0
http://www.pbs.org/eakins/pdfs/lesson2.pdf#page=5
www.pbs.org/eakins/pdfs/lesson2.pdf#page=5
Assessment Recommendations • Participation and attitude in all activities • Completion of assigned work • Ability to identify details in comparing and evaluating their own work Extension Ideas Art • Hand drawn flipbooks of human motion • Computer animations Language Arts Marcel Duchamp Captures Movement 1. Procure images (or visit the Philadelphia Museum of Art) of Marcel Duchamp’s <span class="highlight">Nude</span> Descending a Staircase 2. Have students examine and make observations about how Duchamp captures movement in this piece
On Being Different
How did Eakins use the new technology of the day in his painting? 3. Play video Optional-DVD Bonus Mini-documentary- The Centennial Exhibition of 1876- (5:35 minutes) Scenes 7-14 (16:45) Note: Scene 13 (30:58-35:18) includes black and white archival photographs of nude male and female...
1
0
How did Eakins use the new technology of the day in his painting? 3. Play video Optional-DVD Bonus Mini-documentary- The Centennial Exhibition of 1876- (5:35 minutes) Scenes 7-14 (16:45) Note: Scene 13 (30:58-35:18) includes black and white archival photographs of nude male and female models. 4. Facilitate the discussion with the following questions and answers: Questions for Discussion and Understanding What unusual subjects did Eakins paint when he returned to Philadelphia from Europe? (Answers
1
0
http://www.pbs.org/eakins/pdfs/lesson3.pdf#page=1
www.pbs.org/eakins/pdfs/lesson3.pdf#page=1
correlations between their own ideas of being different, their school environment, the determination to pursue goals and Thomas Eakins’ life. The lesson utilizes the entire documentary film Thomas Eakins: Scenes From Modern Life, and is shown in three separate sessions. Grade Level 9-12 Please Note: The <span class="highlight">video</span> program contains several scenes featuring black and white archival photographs of <span class="highlight">nude</span> male and female models, including some photographs of Eakins himself. These scenes are noted in the lesson plan. Subject Area
5
0
http://www.pbs.org/eakins/pdfs/lesson3.pdf#page=5
www.pbs.org/eakins/pdfs/lesson3.pdf#page=5
How did Eakins use the new technology of the day in his painting? 3. Play <span class="highlight">video</span> Optional-DVD Bonus Mini-documentary- The Centennial Exhibition of 1876- (5:35 minutes) Scenes 7-14 (16:45) Note: Scene 13 (30:58-35:18) includes black and white archival photographs of <span class="highlight">nude</span> male and female models. 4. Facilitate the discussion with the following questions and answers: Questions for Discussion and Understanding What unusual subjects did Eakins paint when he returned to Philadelphia from Europe? (Answers
7
0
http://www.pbs.org/eakins/pdfs/lesson3.pdf#page=7
www.pbs.org/eakins/pdfs/lesson3.pdf#page=7
his family and what was the result? - What did Eakins paint during the last years of his life? Why? 6. Play <span class="highlight">video</span> Scenes 15- 21 (15:38) Note: Scenes 15-16 (36:48-40:06) include black and white archival photographs of <span class="highlight">nude</span> male and female models.
Zhongwen.com
Learn Chinese Characters Learn Chinese Characters Online Chinese-English dictionary. Learn Chinese characters. Chinese characters, Chinese dictionary, learn Chinese, learn Chinese characters, Chinese-English dictionary, study Chi...
Safe Kids Home Page
The convention was ratified by all UN members except the United States, Somalia and South Sudan. European laws protect children’s privacy even from their own parents. U.S. parents have a great deal of legal authority over their children, but there is nothing in the First Amendment that says y...
NYTimes People: Cox, Renee
Renee Cox 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 Es...
What's the big deal about when I have sex?
stages of development, and birth. Each class is engaged through lively discussion, the use of Life Magazine photos, three-dimensional replicas of babies in various embryonic and fetal stages, and an ultrasound video showing the activity inside the womb from a four-week embryo's heart...
|
