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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.

53B25 Local submanifolds
53C30 Differential geometry of homogeneous manifolds
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