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 Riemannian Geometry (PDF)
→ p · q̄ and a real valued norm given by |p|2 = p · p̄. Then the 3-dimensional unit sphere S3 in H ∼= R4 with the restricted multiplication forms a compact Lie subgroup (S3, ·) of (H∗, ·). They are both non-abelian. We shall now int...
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→ p · q̄ and a real valued norm given by |p|2 = p · p̄. Then the 3-dimensional unit sphere S3 in H ∼= R4 with the restricted multiplication forms a compact Lie subgroup (S3, ·) 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=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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