Geometry And Topology

Analytical geometry of three dimensions by William H. McCrea

By William H. McCrea

Written by means of a individual mathematician and educator, this short yet rigorous textual content is aimed toward complex undergraduates and graduate scholars. It covers the coordinate procedure, planes and contours, spheres, homogeneous coordinates, basic equations of the second one measure, quadric in Cartesian coordinates, and intersection of quadrics. 1947 variation.

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Consider for instance (R2n1 , σ1 ) and (R2n2 , σ2 ), the standard symplectic spaces of dimension 2n1 and 2n2 ; let Sp(n1 ) and Sp(n2 ) be the respective symplectic groups. The direct sum Sp(n1 ) ⊕ Sp(n2 ) is the group of automorphisms of 2n1 ⊕ R2n2 , σ1 ⊕ σ2 ) (R2n z , σ) = (R defined, for z1 ∈ R2n1 and z2 ∈ R2n2 , by (s1 ⊕ s2 )(z1 ⊕ z2 ) = s1 z1 ⊕ s2 z2 . It is evidently a subgroup of Sp(n): Sp(n1 ) ⊕ Sp(n2 ) ⊂ Sp(n) which can be expressed in terms of block-matrices as follows: let S1 = A1 C1 B1 D1 and S2 = A2 C2 B2 D2 be elements of Sp(n1 ) and Sp(n2 ), respectively.

Let us make a precise construction of the isomorphism π1 [Sp(n)] ∼ = π1 [U(n, C)]. 26. The mapping ∆ : Sp(n) −→ S 1 defined by ∆(S) = det u where u is the image in U(n, C) of U = S(S T S)−1/2 ∈ U(n) induces an isomorphism ∆∗ : π1 [Sp(n)] ∼ = π1 [U(n, C)] and hence an isomorphism π1 [Sp(n)] ∼ = π1 [S 1 ] ≡ (Z, +). Proof. 24 above and its proof, any loop t −→ S(t) = R(t)eX(t) in Sp(n) is homotopic to the loop t −→ R(t) in U(n). Now S T (t)S(t) = e2X(t) (because X(t) is in sp(n) ∩ Sym(2n, R)) and hence R(t) = S(t)(S T (t)S(t))−1/2 .

The unitary group U(n, C) acts in a natural way on (R2n z , σ) (cf. 6) and that action preserves the symplectic structure. 12. 11) of Sp(n). Proof. 7) the inverse of U = µ(u), u ∈ U(n, C), is U −1 = AT −B T BT = UT , AT hence U ∈ O(2n, R) which proves the inclusion U(n) ⊂ Sp(n) ∩ O(2n, R). Suppose conversely that U ∈ Sp(n) ∩ O(2n, R). Then JU = (U T )−1 J = U J which implies that U ∈ U(n) so that Sp(n) ∩ O(2n, R) ⊂ U(n). 14) 34 Chapter 2. The Symplectic Group of course these conditions are just the same thing as the conditions (A + iB)∗ (A + iB) = (A + iB)(A + iB)∗ = I for the matrix A + iB to be unitary.

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