Explicit immersions of surfaces in $${{\mathbb {R}}}^4$$ R 4 with arbitrary constant Jordan angles
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Explicit immersions of surfaces in R4 with arbitrary constant Jordan angles J. Monterde1 · R. C. Volpe1 Received: 9 November 2018 / Accepted: 3 March 2020 © Springer Nature B.V. 2020
Abstract An immersed surface in R4 is said to has constant Jordan angles (CJA) if the angles between its tangent planes and a fixed plane do not depend on the choice of the point. The constant Jordan angles surfaces in R4 has been proved to exist, Bayard et al. (Geom Dedicata 162:153– 176, 2013), but there are only explicit examples of non planar surfaces for the extremal angles 0 and π2 as the Clifford torus. In this work, an alternative proof of the existence has been obtained that is based on the solution of a hyperbolic partial differential equation. Finally, after a study of the known solutions of the hyperbolic equation, an explicit expression with arbitrary CJA is provided for a family of immersions with an additional geometric property written in terms of the local invariants. Keywords Constant Jordan angles surfaces · Jordan angles · Helix surfaces · Local invariants Mathematics Subject Classification (2000) 53C42
1 Introduction Lancret’s theorem is a well-known result which characterizes curves in 3D-space making a constant angle with a fixed direction. When studying surfaces immersed in R4 , the natural generalization consists in computing some kind of angle between a fixed plane and the tangent planes to the surface. The difference is that instead of computing angles between lines, this is, between one dimensional subspaces, one has to compute angles between higher dimensional subspaces. The corresponding notion is called the set of Jordan angles between subspaces.
The fist was partially supported by the Spanish Ministry of Economy and Competitiviness DGICYT grant MTM2015-64013 and the second author was supported by the Generalitat Valenciana VALi+D grant ACIF/2016/342.
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R. C. Volpe [email protected] J. Monterde [email protected]
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Departamento de Matemàtiques, Universitat de València, 46100 Burjassot, Spain
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Geometriae Dedicata
In the case of two dimensional subspaces, the position of a plane with respect to the other is determined by a pair of angles. In the reference [1], an existence result is shown: for any pair of Jordan angles, φ1 and φ2 , there exists a surface immersed in R4 making constant Jordan angles (CJA) with a fixed plane. The proof there is as follows: the authors start with an immersed surface of the kind (u, v, f (u, v), g(u, v)), this is, a kind of graph surface for a pair of arbitrary functions f and g. The condition of the surface having CJA with a fixed plane can be translated then into a system of partial differential equations where f and g are the unknowns. Existence theorems of solutions of PDE can be applied then to conclude that the desired surface with CJA must exists. Nevertheless, explicit examples of non planar immersed surfaces with CJA are only shown for some particular cases. The main goal of our contribution is to provide, given any pair of angles 0 < φ1 < φ2 < π2 , explicit a
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