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Welcome to the research group
Differential Geometry and Geometric Structures
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Members & friends of the group in Jul 2021
photograph © by Narges Lali
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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.
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Gallery of some research interests and projects
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)
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)
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)
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)
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).
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)
This is a surface of (hyperbolic) rotation in hyperbolic
space that has constant Gauss curvature,
a
recent classification project.
(Hertrich-Jeromin, Pember, Polly)
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)
Affine Differential Geometry:
In affine differential geometry a main point of research is
the investigation of special surfaces in three dimensional
affine space.
(Manhart)
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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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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