Page 347 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 347
DIFFERENTIAL GEOMETRY: OLD AND NEW (MS-15)
Cauchy-Riemann geometry of Legendrian curves in the 3-dimensional
Sphere
Emilio Musso, emilio.musso@polito.it
Politecnico di Torino, Italy
Let S3 be the unit 3-sphere with its standard Cauchy–Riemann (CR) structure. We consider
the CR geometry of Legendrian curves in S3, thought of as a 3-dimensional homogeneous
CR manifold. We introduce the two main local invariants : a line element (the cr-infinitesimal
strain) and the cr-bending. Integrating the invariant line element we get the simplest cr-invariant
variational problem for Legendrian curves in S3 . We discuss Liouville integrability and the
existence of closed critical curves.
On the Bishop frame of a partially null curve in Minkowski spacetime
Emilija Nešovic´, emilija@kg.ac.rs
University of Kragujevac, Faculty of Science, Department of mathematics and informatics,
Serbia
The Bishop frame {T, N1, N2} (relatively parallel adapted frame) of a regular curve in Eu-
clidean space E3 contains the tangent vector field T of the curve and two relatively parallel
vector fields N1 and N2 whose derivatives in arc length parameter s make minimal rotations
along the curve. In Minkowski spaces E13 and E14, the Bishop frame of a non-null curve and a
null Cartan curve has analogous property.
In this talk, we present a method for obtaining the Bishop frame (rotation minimizing frame)
of a partially null curve α lying in the lightlike hyperplane of Minkowski spacetime. We show
that α has two possible Bishop frames, one of which coincides with its Frenet frame. By using
spacetime geometric algebra, we derive the Darboux bivectors of Frenet and Bishop frame and
give geometric interpretation of the Frenet and the Bishop curvatures in terms of areas obtained
by projecting the Darboux bivector onto a spacelike or a lightlike plane.
Topologically Embedded Pseudospherical Surfaces
Lorenzo Nicolodi, lorenzo.nicolodi@unipr.it
Università di Parma, Italy
It is known that the class of traveling wave solutions of the sine-Gordon equation is in 1-1
correspondence with the class of (necessarily singular) pseudospherical helicoids, i.e., pseudo-
spherical surfaces in Euclidean space with screw-motion symmetry. We illustrate our solution
to the problem of explicitly describing all pseudospherical helicoids posed by A. Popov in
[Lobachevsky Geometry and Modern Nonlinear Problems, Birkhäuser, Cham, 2014]. As an
application, countably many continuous families of topologically embedded pseudospherical
helicoids are constructed. This is joint work with Emilio Musso.
345
Cauchy-Riemann geometry of Legendrian curves in the 3-dimensional
Sphere
Emilio Musso, emilio.musso@polito.it
Politecnico di Torino, Italy
Let S3 be the unit 3-sphere with its standard Cauchy–Riemann (CR) structure. We consider
the CR geometry of Legendrian curves in S3, thought of as a 3-dimensional homogeneous
CR manifold. We introduce the two main local invariants : a line element (the cr-infinitesimal
strain) and the cr-bending. Integrating the invariant line element we get the simplest cr-invariant
variational problem for Legendrian curves in S3 . We discuss Liouville integrability and the
existence of closed critical curves.
On the Bishop frame of a partially null curve in Minkowski spacetime
Emilija Nešovic´, emilija@kg.ac.rs
University of Kragujevac, Faculty of Science, Department of mathematics and informatics,
Serbia
The Bishop frame {T, N1, N2} (relatively parallel adapted frame) of a regular curve in Eu-
clidean space E3 contains the tangent vector field T of the curve and two relatively parallel
vector fields N1 and N2 whose derivatives in arc length parameter s make minimal rotations
along the curve. In Minkowski spaces E13 and E14, the Bishop frame of a non-null curve and a
null Cartan curve has analogous property.
In this talk, we present a method for obtaining the Bishop frame (rotation minimizing frame)
of a partially null curve α lying in the lightlike hyperplane of Minkowski spacetime. We show
that α has two possible Bishop frames, one of which coincides with its Frenet frame. By using
spacetime geometric algebra, we derive the Darboux bivectors of Frenet and Bishop frame and
give geometric interpretation of the Frenet and the Bishop curvatures in terms of areas obtained
by projecting the Darboux bivector onto a spacelike or a lightlike plane.
Topologically Embedded Pseudospherical Surfaces
Lorenzo Nicolodi, lorenzo.nicolodi@unipr.it
Università di Parma, Italy
It is known that the class of traveling wave solutions of the sine-Gordon equation is in 1-1
correspondence with the class of (necessarily singular) pseudospherical helicoids, i.e., pseudo-
spherical surfaces in Euclidean space with screw-motion symmetry. We illustrate our solution
to the problem of explicitly describing all pseudospherical helicoids posed by A. Popov in
[Lobachevsky Geometry and Modern Nonlinear Problems, Birkhäuser, Cham, 2014]. As an
application, countably many continuous families of topologically embedded pseudospherical
helicoids are constructed. This is joint work with Emilio Musso.
345
