Welcome to the research group
Differential Geometry and Geometric Structures

FB3, photo by Narges Lali

Members & friends of the group in Jul 2021
photograph © by Narges Lali

Differential geometry has been a thriving area of research for more than 200 years, employing methods from analysis to investigate geometric problems. Typical questions involve the shape of smooth curves and surfaces and the geometry of manifolds and Lie groups. The field is at the core of theoretical physics and plays an important role in applications to engineering and design.

Finite and infinite geometric structures are ubiquitous in mathematics. Their investigation is often intimately related to other areas, such as algebra, combinatorics or computer science.

These two aspects of geometric research stimulate and inform each other, for example, in the area of "discrete differential geometry", which is particularly well suited for computer aided shape design.

Gallery of some research interests and projects

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Symmetry breaking in geometry: We discuss a geometric mechanism that may, in analogy to similar notions in physics, be considered as "symmetry breaking" in geometry. (Fuchs, Hertrich-Jeromin, Pember; Fig ©Nimmervoll)

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Cyclic coordinate systems: an integrable discretization in terms of a discrete flat connection is discussed. Examples include systems with discrete flat fronts or with Dupin cyclides as coordinate surfaces (Hertrich-Jeromin, Szewieczek)

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We study surfaces with a family of spherical curvature lines by evolving an initial spherical curve through Lie sphere transformations, e.g., the Wente torus (Cho, Pember, Szewieczek)

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Discrete Weierstrass-type representations are known for a wide variety of discrete surfaces classes. In this project, we describe them in a unified manner, in terms of the Omega-dual transformation applied to to a prescribed Gauss map. (Pember, Polly, Yasumoto)

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Billiards: The research addresses invariants of trajectories of a mass point in an ellipse with ideal physical reflections in the boundary. Henrici's flexible hyperboloid paves the way to transitions between isometric billiards in ellipses and ellipsoids (Stachel).

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Spreads and Parallelisms: The topic of our research are connections among spreads and parallelisms of projective spaces with areas like the geometry of field extensions, topological geometry, kinematic spaces, translation planes or flocks of quadrics. (Havlicek)

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This is a surface of (hyperbolic) rotation in hyperbolic space that has constant Gauss curvature, a recent classification project. (Hertrich-Jeromin, Pember, Polly)

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Singularity Closeness of Stewart-Gough Platforms: This project is devoted to evaluating the closeness of Stewart-Gough platforms to singularities. (Nawratil)

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Geometric shape generation: We aim to understand geometric methods to generate and design (geometric) shapes, e.g., shape generation by means of representation formulae, by transformations, kinematic generation methods, etc. (Hertrich-Jeromin, Fig Lara Miro)

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Affine Differential Geometry: In affine differential geometry a main point of research is the investigation of special surfaces in three dimensional affine space. (Manhart)

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Transformations & Singularities: We aim to understand how transformations of particular surfaces behave (or fail to behave) at singularities, and to study how those transformations create (or annihilate) singularities. The figure shows the isothermic dual of an ellipsoid, which is an affine image of a minimal Scherk tower. (Hertrich-Jeromin)

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Stewart Gough Platforms with Self-Motions: The main aim of the project is the systematic determination, investigation and classification of Steward-Gough platforms with self-motions. (Nawratil)

News

21 Mar 2024: Geometry seminar
Gudrun Szewieczek (TU Munich): Discrete isothermic nets with a family of spherical parameter lines from holomorphic maps

Abstract

Smooth surfaces with a family of planar or spherical curvature lines are an active area of research, driven by both purely differential geometric aspects and practical applications such as architectural design. In integrable geometry it is a natural question to ask which of these surfaces admit a conformal curvature line parametrization and are therefore isothermic surfaces.

It is an open problem to explicitly describe all those smooth isothermic surfaces. However, over time, prominent examples were found in this rich integrable surface class: above all Wente's torus. More recently, further specific examples have led to the discovery of compact Bonnet pairs and to free boundary solutions for minimal and CMC-surfaces.

This talk covers a discrete version of the problem: we shall generate all discrete isothermic nets with a family of spherical curvature lines from special discrete holomorphic maps via the concept of "lifted-folding". In particular, we point out how this novel approach leads to quasi-periodic solutions and to topological tori with symmetries.

This is joint work with Tim Hoffmann.

14 Mar 2024: Geometry seminar (Sem.R. DB gelb 03)
David Sykes (TU Wien): CR Hypersurface Geometry, an Introduction

Abstract

CR geometry concerns structures on real submanifolds in complex spaces that are preserved under biholomorphisms. This talk will present a light introduction to CR geometry of real hypersurfaces. We will survey some of the area's major classical results, namely solutions to local equivalence problems of E Cartan, Tanaka, and Chern-Moser and their applications. And we will preview some of the area's current-day research trends related to Levi degenerate structures.
29 Nov 2023: Geometry seminar 16:15
Martin Kilian (TU Wien): Meshes with Spherical Faces

Abstract

A truly Möbius invariant discrete surface theory must consider meshes where the transformation group acts on all of its elements, including edges and faces. We therefore systematically describe so called sphere meshes with spherical faces and circular arcs as edges. Driven by aspects important for manufacturing, we provide the means to cluster spherical panels by their radii. We investigate the generation of sphere meshes which allow for a geometric support structure and characterize all such meshes with triangular combinatorics in terms of non-Euclidean geometries. We generate sphere meshes with hexagonal combinatorics by intersecting tangential spheres of a reference surface and let them evolve - guided by the surface curvature - to visually convex hexagons, even in negatively curved areas. Furthermore, we extend meshes with circular faces of all combinatorics to sphere meshes by filling its circles with suitable spherical caps and provide a re-meshing scheme to obtain quadrilateral sphere meshes with support structure from given sphere congruences. By broadening polyhedral meshes to sphere meshes we exploit the additional degrees of freedom to minimize intersection angles of neighboring spheres enabling the use of spherical panels that provide a softer perception of the overall surface.
22 Nov 2023: Geometry seminar
Felix Dellinger (TU Wien): Orthogonal structures

Abstract

In this talk we introduce a definition for orthogonal quadrilateral nets based on equal diagonal length in every quad. This definition can be motivated through Ivory's Theorem and rhombic bi-nets. We find that non-trivial orthogonal multi-nets exist, i.e., nets where the orthogonality condition holds for every combinatorial rectangle and present a method to construct them. The orthogonality condition is well suited for numerical optimization. Since the definition does not depend on planar quadrilaterals it can be paired with common discretizations of conjugate nets, asymptotic nets, geodesic nets, Chebyshev nets or principal symmetric nets. This gives a way to numerically compute prinicipal nets, minimal surfaces, developable surfaces and cmc-surfaces.

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