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There are two vector fields in $\mathbb{R}^3$, $F1=x\frac{\partial}{\partial y} - y\frac{\partial }{\partial x},F2=z\frac{\partial}{\partial x} - x\frac{\partial }{\partial z}$ and find a manifold includes point $(x_0,y_0,z_0)$, the tangent space of which is spanned by these two vector fields.

My understanding is: let's assume the manifold is $f(x,y,z)=0$, then we have$f(x_0,y_0,z_0)=0$. For a point P on the manifold, the tangent space $T_pM$ is the tangent plane on P, and $(F1\times F2)\bot T_pM$. Isn't the form of vector fileds here should be like $(\fbox{$\phantom{5}$},\fbox{$\phantom{5}$},\fbox{$\phantom{5}$})$?

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Consider $f(x,y,z)=x^2+y^2+z^2=c$ $df=2xdx+2ydy+2zdz$, you remark that $df_{(x,y,z)}(-y,x,0)=-2xy+2yx=0$ and $df_{(x,y,z)}(-z,0,x)=-2xz+2zx=0$, thus if $(x,y,z)\neq (0,0,0)$, $f$ is a submersion on a neighborhood of $(x,y,z)$ and the tangent space at $(x,y,z)$ is generated by $(-y,x,0)$ and $(-z,0,x)$. The manifold is the sphere wich radius is $\sqrt{x_0^2+y_0^2+z_0^2}$.

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  • $\begingroup$ Thank you! That explains $\endgroup$
    – XJY95
    Apr 13, 2018 at 9:04

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