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Differential geometry of immersed surfaces in three-dimensional normed spaces. (English) Zbl 1475.53022

The paper presents an approach to the geometry of immersed surfaces in three-dimensional (normed or) Minkowski spaces from different points of views; Minkowski geometry, classical differential geometry and Finsler geometry. In view of Minkowski geometry, some concepts from Euclidean geometry such as orthogonality and curvature are extended and then their behavior are studied. From the viewpoint of Finsler geometry, the curvature of a Finsler manifold whose geometry is induced by the Minkowski geometry of an ambient space is studied.
Explicitly, for a given surface in a three-dimensional Minkowski space, the authors endow the surface with the transversal vector field obtained via the Birkhoff orthogonality associated to the norm, and then they get an analogue of the Gauss map, the so-called Gauss-Birkhoff map of the surface. Then they define concepts of principal, Gaussian, and mean curvatures in terms of the eigenvalues of the differential of this map. Also, they extend the notions of curvature lines, asymptotic directions, and umbilic points. In particular they prove that an immersed connected surface all whose points are umbilic is contained in a plane or in a Minkowski sphere (Section 4).
Furthermore, the authors extend the notion of normal curvature of a surface by considering the plane curvatures of curves given as plane sections of the surface (Section 5). They describe the relations between the normal curvature and the principal curvatures. Under an additional hypothesis on the (Minkowski) curvature lines of the surface they prove that a surface with constant (positive) Minkowski Gaussian curvature should be a Minkowski sphere (Section 6).

MSC:

53A35 Non-Euclidean differential geometry
53A15 Affine differential geometry
52A15 Convex sets in \(3\) dimensions (including convex surfaces)
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
46B20 Geometry and structure of normed linear spaces
51B20 Minkowski geometries in nonlinear incidence geometry
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