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 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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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
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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
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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
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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
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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.
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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
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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
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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
Colorado Schools Unit: Science of the People
29Goals 2000 Partnership for Educating Colorado Students Lesson 7: Dr. Bernardo Houssay What wil students be learning? STANDARD(S) Students know and understand the characteristics and structure of living things, the process of life, and how living things interact with each other and their environmen...
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29Goals 2000 Partnership for Educating Colorado Students Lesson 7: Dr. Bernardo Houssay What wil students be learning? STANDARD(S) Students know and understand the characteristics and structure of living things, the process of life, and how living things interact with each other and their environment. (S3) Students apply thinking skills to their reading, writing, speaking, listening, and viewing. (RW4) BENCHMARK(S) Studetns know and understand how the human body functions in health and disease and factors
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http://www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=3
www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=3
structure of living things, the process of life, and how living things interact with each other and their environment. (<span class="highlight">S3</span>) Students know and understand the processes and interactions of Earth’s systems and the structure and dynamics of Earth and other objects in space. (S4) Students know and understand interrelationships among science, technology, and human activity in the past, present, and future and how they can affect the world. (S5) Students understand that science involves a particular way of knowing and
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http://www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=27
www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=27
25Goals 2000 Partnership for Educating Colorado Students Lesson 6: Ynez Mexia What will students be learning? STANDARD(S) Students understand the process of scientific investigation and design, conduct, communicate about, and evaluate such investigations. (S1) Students know and understand the characteristics and structure of living things, the process of life, and how living things interact with each other and their environment. (<span class="highlight">S3</span>) Students apply thinking skills to their reading, writing, speaking
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http://www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=31
www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=31
29Goals 2000 Partnership for Educating Colorado Students Lesson 7: Dr. Bernardo Houssay What wil students be learning? STANDARD(S) Students know and understand the characteristics and structure of living things, the process of life, and how living things interact with each other and their environment. (<span class="highlight">S3</span>) Students apply thinking skills to their reading, writing, speaking, listening, and viewing. (RW4) BENCHMARK(S) Studetns know and understand how the human body functions in health and disease and factors
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http://www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=39
www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=39
37Goals 2000 Partnership for Educating Colorado Students Lesson 9: Dr. Eloy Rodriquez What will students be learning? OUTCOME(S) Students understand the process of scientific investigation and design, conduct, communicate about, and evaluate such investigations. (S1) Students know and understand the characteristics and structure of living things, the process of life, and how living things interact with each other and their environment. (<span class="highlight">S3</span>) Students understand that science involves a particular way of
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...
PAL Bibliography: Bret Harte
Californian, 1864 67. San Francisco, J. Howell, 1927. PS1821 .H6 San Francisco (from the sea) by Bret Harte. San Francisco, Calif. : The Kennedy Ten Bosch Co., 1927. Case PS1829 .S3 Tales of the gold rush, by Bret Harte, illus. by Fletcher Martin, with an introd. by Oscar Lewis.NY: Herita...
PAL Bibliography: Archibald Macleish
& other poems. Boston: Houghton Mifflin, 1968. PS3525 A27 W5 A continuing journey. Boston: Houghton Mifflin, 1968 1967. PS3525.A27 C69 Scratch. Suggested by Stephen Vincent Benet's short story "The devil and Daniel Webster." Boston: Houghton Mifflin, 1971. PS3525 A27 S3 Th...
PAL Bibliography: Josephine Miles
Classic essays in English. Boston, Little, Brown, 1965. PR1363 .M53 Saving the bay. San Francisco: Open Space, 1967. PS3525.I4835 S3 Kinds of affection. Middletown, Conn.: Wesleyan UP, 1967. PS3525 .I4835 K5 Fields of learning. Berkeley: Oyez, 1968. PS3525.I4835 F5 To all appearances;...
PAL Bibliography: James Wright
1977; This Journey, 1982; Above the River: The Complete Poems, 1992. Saint Judas. Middletown, Conn: Wesleyan UP, 1959. PS3545.R58 S3 The branch will not break; poems. Middletown, Conn: Wesleyan UP, 1963. PS3545 .R58 B7 Shall we gather at the river; poems by James Wright. Middletown, Conn...
CME Flip Book
produced a G5 magnetic storm — the most severe possible and an S3 radiation storm in space. The high energy solar particles expelled by the flare produce streaks and spots on the flip-book images as the crash into Print the following 3 pages. It works best if you can use stiff paper bu...
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produced a G5 magnetic storm — the most severe possible and an S3 radiation storm in space. The high energy solar particles expelled by the flare produce streaks and spots on the flip-book images as the crash into Print the following 3 pages. It works best if you can use stiff paper but standard printer paper is fine. Cut out each of the pages for the flip book along the solid line. All of the pages will be slightly different lengths. This makes it easier to flip through the book when it is finished. Arrange
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http://www.windows.ucar.edu/teacher_resources/flipbooks/CME.pdf#page=1
www.windows.ucar.edu/teacher_resources/flipbooks/CME.pdf#page=1
produced a G5 magnetic storm — the most severe possible and an <span class="highlight">S3</span> radiation storm in space. The high energy solar particles expelled by the flare produce streaks and spots on the flip-book images as the crash into Print the following 3 pages. It works best if you can use stiff paper but standard printer paper is fine. Cut out each of the pages for the flip book along the solid line. All of the pages will be slightly different lengths. This makes it easier to flip through the book when it is finished. Arrange
Another CME Flip Book
storm — the most severe possible and an S3 radiation storm in space. The high energy solar particles expelled by the flare produce streaks and spots on the flip-book images as the crash into Print the following 3 pages. It works best if you can use stiff paper but standard printer pape...
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storm — the most severe possible and an S3 radiation storm in space. The high energy solar particles expelled by the flare produce streaks and spots on the flip-book images as the crash into Print the following 3 pages. It works best if you can use stiff paper but standard printer paper is fine. Cut out each of the pages for the flip book along the solid line. All of the pages will be slightly different lengths. This makes it easier to flip through the book when it is finished. Arrange them in order
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http://www.windows.ucar.edu/teacher_resources/flipbooks/CME_flip.pdf#page=1
www.windows.ucar.edu/teacher_resources/flipbooks/CME_flip.pdf#page=1
storm — the most severe possible and an <span class="highlight">S3</span> radiation storm in space. The high energy solar particles expelled by the flare produce streaks and spots on the flip-book images as the crash into Print the following 3 pages. It works best if you can use stiff paper but standard printer paper is fine. Cut out each of the pages for the flip book along the solid line. All of the pages will be slightly different lengths. This makes it easier to flip through the book when it is finished. Arrange them in order
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