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By Pogorelov, A.V.

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We have [A, A] = 0. Now, the extended framing F (λ) is given by F (λ) = exp λzA + λ−1 zA ∈ ΛGτ . The reason for that relating to the Iwasawa decomposition of the twisted loop group stated in the former section is that exp(λzA) ∈ ΛGC τ and exp(λzA) = exp λzA + λ−1 zA · exp −λ−1 zA ∈ ΛGτ · Λ− GC τ . Letting λ = 1, we see that F = exp zA + zA is a SO0 (2, 2)-framing for f and the immersion f is given by the first raw of F . We may set exp(zA) = ϕ0 I + ϕ1 A + ϕ2 A2 + ϕ3 A3 , where  √ √ √ 1 √2z   + e− 2z + ei 2z + e−i 2z , e ϕ0 =   4  √ √ √ √  1    ϕ1 = √ e 2z − e− 2z − iei 2z + ie−i 2z , 4 2 √ √ √ 1 √2z  − 2z i 2z −i 2z  e + e − e − e , ϕ = 2   8  √ √ √ √  1    ϕ3 = √ e 2z − e− 2z + iei 2z − ie−i 2z .

September 4, 2013 17:10 WSPC - Proceedings Trim Size: 9in x 6in main Extrinsic circular trajectories on geodesic spheres in a complex projective space 45 We call two smooth curves γ1 , γ2 on a Riemannian manifold which are parameterized by their arclength congruent to each other if there exist an isometry ϕ of the base manifold and a constant t0 satisfying γ2 (t) = ϕ◦γ1 (t+t0 ) for all t. On a real space form, which is one of a standard sphere, a Euclidean space and a real hyperbolic space, two circles are congruent to each other if and only if they have the same geodesic curvature.

F. Alday and J. 4707 [hep-th]. hepth. 2. L. F. Alday and J. 0663 [hep-th]. 3. V. Balan and J. Dorfmeister, Birkhoff decompositions and Iwasawa decompositions for loop groups, Tohoku Math. J. 53(2000), 593-615. 4. H. 0934 [hep-th]. 5. H. Dorn, G. Jorjadze and S. 0977 [hep-th]. 6. P. Kellersch, The Iwasawa decomposition for the untwisted group of loops in semisimple Lie groups, Ph. D. Thesis, Technische Universit¨ at M¨ unich, 1999. Received March 29, 2013. jp Dedicated to Professor Sadahiro MAEDA on the occasion of his 60th birthday In this paper we explain circles of positive geodesic curvature on a complex projective space as extrinsic shapes of trajectories for Sasakian magnetic fields on geodesic spheres in this space.

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