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Let X be a conformal vector field on the standard sphere $S^n$ with standard metric $g_{S^n}$, then there exists a unique conformal vector fields in the unit ball $B_1(0)\subset \mathbb{R}^{n+1}$, such that $Y|_{S^n}=X$. As we know, the vector fields $X=\nabla_{g_{S^n}}x^i,1\leq i\leq n+1$ are conformal on $(S^n,g_{S^n})$, where $x^i$ are coordinate functions. Then, is it possible to write down the explicit expressions of such extensions $Y$ for such conformal vector fields $X$ on spheres?

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  • $\begingroup$ Does "conformal vector field" mean the flow is conformal? If so, then yes they have closed form. The group of conformal diffeos is finite-dimensional and has a well-understood description as a matrix group. $\endgroup$ – Ryan Budney Feb 9 '16 at 6:27
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    $\begingroup$ See mathoverflow.net/questions/60687/the-conformal-group-of-sn?rq=1 $\endgroup$ – Vít Tuček Feb 9 '16 at 7:46
  • $\begingroup$ @njucxz: $X$ is the projection of the gradient of $x_i$ w.r.t. the standard metric of $R^{n+1}$ to the unit sphere, isn'it?. Such gradients are the vectors $e_1,e_2,....,e_{n+1}$ of the canonical orthonormal frame. Are you asking how to project such constant vector fields to the unit sphere? $\endgroup$ – Holonomia Feb 9 '16 at 10:53
  • $\begingroup$ @ Holonomai Not yet! The gradient $x^i$ is with respect to the standard spherical metric. $\endgroup$ – njucxz Feb 10 '16 at 22:01
  • $\begingroup$ @njucxz and such gradient is the orthogonal projection of the gradient of $x^i$ w.r.t. the standard metric of $R^{n+1}$ . So you are indeed asking how to project vectors form the ambient Euclidean space to the sphere. $\endgroup$ – Holonomia Feb 11 '16 at 8:50

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