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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=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=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
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