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On the geometry of constant angle surfaces in $$Sol_{3}$$. (English) Zbl 1236.53011
Let $$Sol_3$$ be the 3-dimensional unimodular, solvable but not nilpotent Lie group which can be represented by $$\mathbb{R}^3$$ with the group operation $(x,y,z)\ast (x',y',z')=(x+e^{-z}x',y+e^{z}y',z+z')$ and can be equipped with the left invariant metric $\widetilde g=e^{2z}dx^2+e^{-2z}dy^2+dz^2$ where $$(x,y,z)$$ are canonical coordinates of $$\mathbb{R}^3$$. An orthonormal basis of left-invariant vector fields is given by $e_1=e^{-z}\frac{\partial}{\partial x},\;\;e_2=e^{z}\frac{\partial}{\partial y},\;\;e_3=\frac{\partial}{\partial z}.$ In $$Sol_3$$, $${\mathcal{H}}^1=\{dy\equiv0\}$$ and $${\mathcal{H}}^2=\{dx\equiv0\}$$ are totally geodesic foliations whose leaves are the hyperbolic plane (thought as the upper half plane model) and whose unit normal are $$e_1$$ and $$e_2$$ respectively.
In the present paper, the authors define an oriented surface $$M$$, isometrically immersed in $$Sol_3$$, as constant angle surface with respect to $${\mathcal{H}}^1$$ (resp. $${\mathcal{H}}^2$$) if the angle between its normal and $$e_1$$ (resp. $$e_2$$) is constant in each point of the surface $$M$$.
They classify such surfaces. The main result in this direction (Theorem 4.2) states that general constant angle surface in $$Sol_3$$ can be parametrized as $$F(u,v)=\gamma_1(v)*\gamma_2(u)$$, where $\gamma_1(v)=\Big(\sin\theta\int\limits^v\xi(\tau)e^{-\zeta(\tau)}d\tau,~ \pm\cos\theta\int\limits^v\xi(\tau)e^{\zeta(\tau)}d\tau,~\zeta(v)\Big)$ and $\gamma_2(u)=\Big(\sin\theta ~I(u),~\pm\cos\theta ~J(u),~-\frac12\log\cosh\bar u\Big)$ and $$\zeta$$, $$\xi$$ are arbitrary functions depending on $$v$$. The curve $$\gamma_2$$ is parametrized by arclength.

##### MSC:
 53B25 Local submanifolds 53C30 Differential geometry of homogeneous manifolds
##### Keywords:
constant angle surfaces; homogeneous spaces
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