# Differential geometry of immersed surfaces in three-dimensional normed spaces

@article{Balestro2020DifferentialGO, title={Differential geometry of immersed surfaces in three-dimensional normed spaces}, author={Vitor Balestro and Horst Martini and Ralph Teixeira}, journal={Abhandlungen aus dem Mathematischen Seminar der Universit{\"a}t Hamburg}, year={2020}, volume={90}, pages={111-134} }

In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff orthogonality, we get an analogue of the Gauss map. Then we can define concepts of principal, Gaussian, and mean curvatures in terms of the eigenvalues of the differential of this map. Considering planar sections containing the normal field, we also define normal…

## 10 Citations

### On curvature of surfaces immersed in normed spaces

- MathematicsMonatshefte für Mathematik
- 2020

The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary…

### Surface immersions in normed spaces from the affine point of view

- MathematicsGeometriae Dedicata
- 2018

The aim of this paper is to investigate the differential geometry of immersed surfaces in three-dimensional normed spaces from the viewpoint of affine differential geometry. We endow the surface with…

### Some topics in differential geometry of normed spaces

- Mathematics
- 2017

Abstract For a surface immersed in a three-dimensional space endowed with a smooth norm instead of an inner product, one can define analogous concepts of curvature and metric. With such concepts in…

### Curvature types of planar curves for gauges

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- 2020

In this paper results from the differential geometry of curves are extended from normed planes to gauge planes which are obtained by neglecting the symmetry axiom. Based on the gauge analogue of the…

### On the Existence of Almost Affinely Flat Structure Induced by Hypersurface Immersion on Connected Compact Manifold

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- 2018

Given a hypersurface immersion and a transversal vector field, the formula of Gauss leads to an induced connection and a symmetric bilinear function called affine fundamental form. We define the norm…

### Rotational surfaces in a $3$-dimensional normed space

- Mathematics
- 2021

We study rotational surfaces with constant Minkowski Gaussian curvature and rotational surfaces with constant Minkowski mean curvature in a 3-dimensional normed space with rotationally symmetric…

### Singularities of generic line congruences

- MathematicsJournal of the Mathematical Society of Japan
- 2022

Line congruences are 2-dimensional families of lines in 3space. The singularities that appear in generic line congruences are folds, cusps and swallowtails ([7]). In this paper we give a geometric…

### Stability of Hypersurfaces in Minkowsky Normed Spaces

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- 2020

Abstract. We extend to Minkowski spaces the classical result of Barbosa and do Carmo [1] that characterizes the euclidean sphere as the unique compact stable CMC hypersurface of Rn. More precisely,…

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The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary…

### Surface immersions in normed spaces from the affine point of view

- MathematicsGeometriae Dedicata
- 2018

The aim of this paper is to investigate the differential geometry of immersed surfaces in three-dimensional normed spaces from the viewpoint of affine differential geometry. We endow the surface with…

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Abstract For a surface immersed in a three-dimensional space endowed with a smooth norm instead of an inner product, one can define analogous concepts of curvature and metric. With such concepts in…

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