|
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
18
0
http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=18
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
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
32
0
http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=32
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
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
36
0
http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=36
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.
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
95
0
http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=95
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
Glossary Terms: Safety Phrases
sufficient to adequately communicate the safety precautions for a particular material. Single Safety Phrases S1Keep locked up. S2Keep out of the reach of children. S3Keep in a cool place. S4Keep away from living quarters. S5Keep contents under ... (appropriate liquid to be specified by the...
Logic and Circuits
are TWO complete circles. S1 and S3 will light the bulb. S2 and S3 will light the bulb. Logically this circuit is You can arrange any patterns you wish your students to investigate. You can assign a logical sentence and ask the students to arrange the circuits, or you can show a...
PAL Bibliography: Elizabeth Stoddard
E-Mail Source: Legacy Primary Works The Morgensons, 1862 (novel); Two Men, 1865 (novel); Temple House, 1867 (novel); Lolly Dinks' Doings, 1874 (children's tales); Poems, 1895. Wrote a bi-monthly column for Daily Alta California, a San Francisco newspaper, 1854-1858. Temple House; a novel. Ph...
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...
Photoelectric Effect
80px; left: 50px; top: 59px; background-color: #FFF; opacity: 0.6; filter: alpha(opacity = 60);"><table style="position: absolute; width: 200px; height: 80px; left: 50px; top: 59px;"><tr><td style="text-align: center; color: #000; font-size: 24px; font-family: A...
Volume of a cube
the orange dot to resize the cube. The volume is calculated as you drag. How to find the volume of a cube Recall that a cube has all edges the same length (See Cube definition). The volume of a cube is found by multiplying the length of any edge by itself twice. So if the length of an edge is 4,...
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...
1
0
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
23
0
http://www.seaworld.org/just-for-teachers/guides/pdf/arctic-animals-4-8.pdf#page=23
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
IT Conversations
({average:3,individual:0,count:1}) details... Alyssa Henry - Pivot (Amazon S3) Amazon has a virtual supercomputer you can rent by the hour, and it is cheap too. Alyssa Henry from Amazon discusses the company's Elastic MapReduce web service for businesses, researchers, data analysts, a...
Rare Map Collection - Colonial America
1692 S3Amerique Septentrionale divisee en ses principales parties. Neg 5080Sanson 1692365kb Map 1695 C6Mare del nord. Neg 5082Coronelli 1695 479kb Map 1696 S3Carte general de la Caroline. Neg 3387Sanson 1696 286kb Map 1696 S32Carte particuliere de la Caroline. Neg 5083Sanson...
|
