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Riemannian Geometry (PDF)
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...
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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 S3 and S2 be the unit spheres in C2 and C× R ∼= R3, respectively. The Hopf map φ : S3 → S2 is given by φ : (x, y) 7→ (2xȳ, |x|2 − |y|2). For p ∈ S3 the Hopf circle Cp through p is given by Cp = {eiθ(x, y)| θ ∈ R
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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=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=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
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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.
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...
Rare Map Collection - Colonial America
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Mainichi Daily News (English Version)
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MSN - Mainichi Interactive
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