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Michael Hardy
Michael Hardy
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Hyperkahler Hyperkähler structure on $TS^2$

What are explicit expressions for the operators $I,J$ and $K$ on the hyperkahlerhyperkähler manifold $TS^2$ (or $TRP^2$), say, in the coordinates $(x_1,x_2, \alpha_1, \alpha_2)$ where a line $m\in TS^2$ is given parametrically by $(x_1+\alpha_1t,x_2+\alpha_2t,t)$? Are they known? I did not find them in the book "The geometry and dynamics of magnetic monopoles" by Atiyah and Hitchin, and could not extract them from it.

Hyperkahler structure on $TS^2$

What are explicit expressions for the operators $I,J$ and $K$ on the hyperkahler manifold $TS^2$ (or $TRP^2$), say, in the coordinates $(x_1,x_2, \alpha_1, \alpha_2)$ where a line $m\in TS^2$ is given parametrically by $(x_1+\alpha_1t,x_2+\alpha_2t,t)$? Are they known? I did not find them in the book "The geometry and dynamics of magnetic monopoles" by Atiyah and Hitchin, and could not extract them from it.

Hyperkähler structure on $TS^2$

What are explicit expressions for the operators $I,J$ and $K$ on the hyperkähler manifold $TS^2$ (or $TRP^2$), say, in the coordinates $(x_1,x_2, \alpha_1, \alpha_2)$ where a line $m\in TS^2$ is given parametrically by $(x_1+\alpha_1t,x_2+\alpha_2t,t)$? Are they known? I did not find them in the book "The geometry and dynamics of magnetic monopoles" by Atiyah and Hitchin, and could not extract them from it.

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sgvl
sgvl
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Hyperkahler structure on $TS^2$

What are explicit expressions for the operators $I,J$ and $K$ on the hyperkahler manifold $TS^2$ (or $TRP^2$), say, in the coordinates $(x_1,x_2, \alpha_1, \alpha_2)$ where a line $m\in TS^2$ is given parametrically by $(x_1+\alpha_1t,x_2+\alpha_2t,t)$? Are they known? I did not find them in the book "The geometry and dynamics of magnetic monopoles" by Atiyah and Hitchin, and could not extract them from it.