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Gudlaugur Thorbergsson

Publications and source records attributed to Gudlaugur Thorbergsson.

14 recordsLinked to original sources
arXiv2016

Tits Geometry and Positive Curvature

There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical buildings of rank at least three. We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type M associated with any polar action of cohomogeneity at least two on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type M, we prove that in all cases but two that its universal Tits cover indeed is a building. We construct a topology on this universal cover making it into a compact spherical building in the sense of Burns and Spatzier. Using this structure we classify up to equivariant diffeomorphism all polar actions on (simply connected) positively curved manifolds of cohomogeneity at least two.

math.DG↗
arXiv2016

Rank three geometry and positive curvature

An axiomatic characterization of buildings of type $\CC_3$ due to Tits is used to prove that any cohomogeneity two polar action of type $\CC_3$ on a positively curved simply connected manifold is equivariantly diffeomorphic to a polar action on a rank one symmetric space. This includes two actions on the Cayley plane whose associated $\CC_3$ type geometry is not covered by a building.

math.DG↗
arXiv2009

On the Funk transform on compact symmetric spaces

We prove that a function on an irreducible compact symmetric space M, which is not a sphere, is determined by its integrals over the shortest closed geodesics in M. We also prove a support theorem for the Funk transform on rank one symmetric spaces which are not spheres.

math.DG↗
arXiv2007

Curvature explosion in quotients and applications

We prove that the quotient space of a variationally complete group action is a good Riemannian orbifold. The result is generalized to singular Riemannian foliations without horizontal conjugate points.

math.DG↗
arXiv2007

On the Geometry of the Orbits of Hermann Actions

We investigate the submanifold geometry of the orbits of Hermann actions on Riemannian symmetric spaces. After proving that the curvature and shape operators of these orbits commute, we calculate the eigenvalues of the shape operators in terms of the restricted roots. As applications, we get a formula for the volumes of the orbits and a new proof of a Weyl-type integration formula for Hermann actions.

math.DG↗
arXiv2006

Inflection points and double tangents on anti-convex curves in the real projective plane

A simple closed curve $γ$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(γ)-2δ(γ)=3$ for anti-convex curves, where $i(γ)$ is the number of independent (true) inflection points and $δ(γ)$ the number of independent double tangents. This formula is a refinement of the classical Möbius theorem. We shall also show that there are three inflection points on a given anti-convex curve such that the tangent lines at these three inflection points cross the curve only once. Our approach is axiomatic and can be applied in other situations. For example, we prove similar results for curves of constant width as a corollary.

math.DG↗
arXiv2002

Representations of compact Lie groups and the osculating spaces of their orbits

Several classes of irreducible orthogonal representations of compact Lie groups that are of importance in Differential Geometry have the property that the second osculating spaces of all of their nontrivial orbits coincide with the representation space. We say that representations with this property are of class O^2. Our approach in the present paper will be to find restrictions on the class O^2 and then apply them to classify variationally complete and taut representations. The known classifications of cohomogeneity one and two orthogonal representations and more generally of polar representations will also follow easily.

math.DG↗
arXiv2001

Completely integrable curve flows on Adjoint orbits

It is known that the Schrödinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schrödinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the $n$-wave equation. In this paper, we give a systematic method to construct integrable geometric curve flows on Adjoint $U$-orbits from flows in the soliton hierarchy associated to a compact Lie group $U$. There are natural geometric bi-Hamiltonian structures on the space of curves on Adjoint orbits, and they correspond to the order two and three Hamiltonian structures on soliton equations under our construction. We study the Hamiltonian theory of these geometric curve flows and also give several explicit examples.

math.DG↗
arXiv2001

A global theory of flexes of periodic functions

For a real valued periodic smooth function u on R, $n\ge 0$, one defines the osculating polynomial $ϕ_s$ (of order 2n+1) at a point $s\in R$ to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex (resp. clean minimal flex) of the function u on $S^1$ if and only if $ϕ_s\ge u$ (resp. $ϕ_s\le u$) and the preimage $(ϕ-u)^{-1}(0)$ is connected. We prove that any smooth periodic function u has at least n+1 clean maximal flexes of order 2n+1 and at least n+1 clean minimal flexes of order 2n+1. The assertion is clearly reminiscent of Morse theory and generalizes the classical four vertex theorem for convex plane curves.

math.DG↗
arXiv2001

Cycles of Bott-Samelson type for taut representations

Bott and Samuelson constructed explicit cycles representing a basis of the Z_2-homology of the orbits of variationally complete representations of compact Lie groups. As a consequence, all those orbits are taut. We were able to show that an irreducible representation of a compact Lie group, all of whose orbits are taut, is either variationally complete or it is one of the following orthogonal representations (n bigger than or equal to 2): the (standard) x_R (spin) representation of SO(2)xSpin(9); or the (standard) x_C (standard) representation of U(2)xSp(n); or the (standard)^3 x_ H (standard) representation of SU(2)xSp(n). In this paper we will show how to adapt the construction of the cycles of Bott and Samelson to the orbits of these three representations. As a result, they also admit explicit cycles representing a basis of their Z_2-homology and, in particular, this provides another proof of their tautness.

math.DG↗
arXiv2000

Sextactic points on a simple closed curve

We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method is axiomatic and can be applied in other situations.

math.DG↗
arXiv2000

Polar and coisotropic actions on Kaehler manifolds

The main result of this paper is that a polar action on a compact irreducible homogeneous Kaehler manifold is coisotropic. This is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics.

math.DG↗
arXiv1997

Tight immersions and local differential geometry

An immersion of a compact manifold is tight if it admits the minimal total absolute curvature over all immersions of the manifold. A prominent result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which characterizes minimal total absolute curvature immersions, and tight immersions, of spheres into a Euclidean space. In this paper we examine tight immersions of highly connected manifolds; i.e., 2k-dimensional manifolds that are (k-1)-connected by not k-connected, and characterize the immersions of highest codimension.

dg-ga↗