Page 345 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 345
DIFFERENTIAL GEOMETRY: OLD AND NEW (MS-15)
Differential geometry of submanifolds in flag varieties via differential
equations
Boris Doubrov, doubrov@bsu.by
Belarusian State University, Belarus
We give a unified method for the general equivalence problem osculating embeddings
ϕ : (M, f) → Flag(V, φ)
from a filtered manifold (M, f) to a flag variety Flag(V, φ). We establish an algorithm to obtain
the complete systems of invariants for the osculating maps which satisfy the reasonable regu-
larity condition of constant symbol of type (g−, gr V ). We show the categorical isomorphism
between the extrinsic geometries in flag varieties and the (weightedly) involutive systems of
linear differential equations of finite type. Therefore we also obtain a complete system of in-
variants for a general involutive systems of linear differential equations of finite type and of
constant symbol.
The invariants of an osculating map (or an involutive system of linear differential equations)
are proved to be controlled by the cohomology group H+1 (g−, gl(V )/ Prol(g−)), which is de-
fined algebraically from the symbol of the osculating map (resp. involutive system), and which,
in many cases (in particular, if the symbol is associated with a simple Lie algebra and its irre-
ducible representation), can be computed by the algebraic harmonic theory, and the vanishing
of which gives rigidity theorems in various concrete geometries.
J-trajectories in Sol40
Zlatko Erjavec, zlatko.erjavec@foi.hr
University of Zagreb, Croatia
Coauthor: Jun-ichi Inoguchi
J-trajectories are arc length parameterized curves in almost Hermitian manifold which satisfy
the equation ∇γ˙ γ˙ = qJγ˙ . J-trajectories are 4-dim analogon of 3-dim magnetic trajectories,
curves which satisfy the Lorentz equation ∇γ˙ γ˙ = qφγ˙ .
In this talk J-trajectories in the 4-dimensional solvable Lie group Sol40 are considered.
Moreover, the first and the second curvature of a non-geodesic J-trajectory in an arbitrary 4-
dimensional LCK manifold whose anti Lee field has constant length are examined. In particular,
the curvatures of non-geodesic J-trajectories in Sol40 are characterized.
Similarlity geometry revisited: Differential Geometry and CAGD
Jun-ichi Inoguchi, inoguchi@math.tsukuba.ac.jp
University of Tsukuba, Japan
Similarity geometry is a Klein geometry whose transformation group is the similarity transfor-
mation group. The similarity transformation group is generated by Euclidean isometries and
scalings.
One can develop differential geometry of plane curves under similarity transformation group.
343
Differential geometry of submanifolds in flag varieties via differential
equations
Boris Doubrov, doubrov@bsu.by
Belarusian State University, Belarus
We give a unified method for the general equivalence problem osculating embeddings
ϕ : (M, f) → Flag(V, φ)
from a filtered manifold (M, f) to a flag variety Flag(V, φ). We establish an algorithm to obtain
the complete systems of invariants for the osculating maps which satisfy the reasonable regu-
larity condition of constant symbol of type (g−, gr V ). We show the categorical isomorphism
between the extrinsic geometries in flag varieties and the (weightedly) involutive systems of
linear differential equations of finite type. Therefore we also obtain a complete system of in-
variants for a general involutive systems of linear differential equations of finite type and of
constant symbol.
The invariants of an osculating map (or an involutive system of linear differential equations)
are proved to be controlled by the cohomology group H+1 (g−, gl(V )/ Prol(g−)), which is de-
fined algebraically from the symbol of the osculating map (resp. involutive system), and which,
in many cases (in particular, if the symbol is associated with a simple Lie algebra and its irre-
ducible representation), can be computed by the algebraic harmonic theory, and the vanishing
of which gives rigidity theorems in various concrete geometries.
J-trajectories in Sol40
Zlatko Erjavec, zlatko.erjavec@foi.hr
University of Zagreb, Croatia
Coauthor: Jun-ichi Inoguchi
J-trajectories are arc length parameterized curves in almost Hermitian manifold which satisfy
the equation ∇γ˙ γ˙ = qJγ˙ . J-trajectories are 4-dim analogon of 3-dim magnetic trajectories,
curves which satisfy the Lorentz equation ∇γ˙ γ˙ = qφγ˙ .
In this talk J-trajectories in the 4-dimensional solvable Lie group Sol40 are considered.
Moreover, the first and the second curvature of a non-geodesic J-trajectory in an arbitrary 4-
dimensional LCK manifold whose anti Lee field has constant length are examined. In particular,
the curvatures of non-geodesic J-trajectories in Sol40 are characterized.
Similarlity geometry revisited: Differential Geometry and CAGD
Jun-ichi Inoguchi, inoguchi@math.tsukuba.ac.jp
University of Tsukuba, Japan
Similarity geometry is a Klein geometry whose transformation group is the similarity transfor-
mation group. The similarity transformation group is generated by Euclidean isometries and
scalings.
One can develop differential geometry of plane curves under similarity transformation group.
343
