18.966 – Homework 2 – due Tuesday March 20, 2007. 1. Show that the sphere S 6 carries a natural almost-complex structure, induced by a vector cross-product on R 7 . Hint: view R 7 as the space of imaginary octonions. Octonions are the non-commutative, non-associative normed division algebra structure on R 8 = H ∀ e H with product given by the formula ( a + be )( a + b e ) = ( aa − b b ) + ( b a + ba ) e, a, b, a , b → H ( a is the conjugate of a , i.e. x + yi + zj + tk = x − yi − zj − tk ). (You may use the fact that ⊗ ( a + be )( a + b e ) ⊗ = ⊗ a + be ⊗⊗ a + b e ⊗ , where ⊗·⊗ is the usual Euclidean norm on R 8 .) 2. Let ( V, ) be a symplectic vector space of dimension 2 n , and let J : V ± V , J 2 = − Id be a complex structure on V . a) Prove that, if J is-compatible and L is a Lagrangian subspace of ( V, ), then JL is also Lagrangian and JL = L ± , where L ± is the orthogonal to L with respect to the positive inner product g ( u, v ) = ( u, Jv ). b)
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