Christian Lange
Mathematisches Institut
Universität Köln
Weyertal 8690
50931 Köln, Germany
Raum 103
Telefon: +49 221 470 3716 (momentan nur per Email erreichbar)
Email: clange at math dot unikoeln dot de
Preprints:

A BoothbyWang theorem for Besse contact forms (with M. Kegel)
Abstract. We prove a BoothbyWang type theorem for Besse Reeb flows that are not necessarily Zoll, i.e. Reeb flows all of whose orbits are periodic, but possibly with different periods. More precisely, we characterize contact manifolds whose Reeb flows are Besse as principal circleorbibundles over integral symplectic orbifolds satisfying some cohomological condition. As a corollary of this and of a result by CristofaroGardiner and Mazzucchelli we obtain a complete classification of closed Besse contact 3manifolds up to strict contactomorphism.
arXiv pdf, (2020).

Closed geodesics on compact orbifolds and on noncompact manifolds (with C. Zwickler)
Abstract. We study the existence of closed geodesics on compact Riemannian orbifolds, and on noncompact Riemannian manifolds in the presence of a cocompact, isometric group action. We show that every noncontractible Riemannian manifold which admits such an action, and every odddimensional, compact Riemannian orbifold has a nontrivial closed geodesic.
arXiv pdf, (2019).

A diameter gap for quotients of the unit sphere (with C. Gorodski, A. Lytchak and R. Mendes)
Abstract. We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimensionindependent positive constant.
arXiv pdf, (2019).

Deformations of the Veronese embedding and Finsler 2spheres of constant curvature (with T. Mettler)
Abstract. We establish a onetoone correspondence between Finsler structures on the 2sphere with constant curvature 1 and all geodesics closed on the one hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and all of whose geodesics closed on the other hand. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding CP(a1,a2)>CP(a1,(a1+a2)/2,a2) of weighted projective spaces provide examples of Finsler 2spheres of constant curvature and all geodesics closed.
arXiv pdf, (2019).

Odddimensional orbifolds with all geodesics closed are covered by manifolds (with M. Amann and M. Radeschi)
Abstract. Manifolds all of whose geodesics are closed have been studied a lot, but there are only few examples known. The situation is different if one allows in addition for orbifold singularities. We show, nevertheless, that the abundance of new examples is restricted to even dimensions. As one key ingredient we provide a characterization of orientable manifolds among orientable orbifolds in terms of characteristic classes.
arXiv pdf, (2018).
Refereed articles:

On the rigidity of Zoll magnetic systems on surfaces (with L. Asselle),
accepted for publication by Nonlinearity.
Abstract. We study the existence of closed geodesics on compact Riemannian orbifolds, and on noncompact Riemannian manifolds in the presence of a cocompact, isometric group action. We show that every noncontractible Riemannian manifold which admits such an action, and every odddimensional, compact Riemannian orbifold has a nontrivial closed geodesic.
arXiv pdf, (2019).

Orbifolds from a metric viewpoint,
Geom. Dedicata (2020), doi: 10.1007/s1071102000521x.
Abstract. We characterize Riemannian orbifolds and their coverings in terms of metric geometry. In particular, we show that the metric double of a Riemannian orbifold along the closure of its codimension one stratum is a Riemannian orbifold and that the natural projection is an orbifold covering.
Journal version, arXiv.

When is the underlying space of an orbifold a manifold?,
Trans. Amer. Math. Soc. 372 (2019), no. 4, 27992828.
Abstract. We classify orthogonal actions of finite groups on Euclidean vector spaces for which the corresponding quotient space is a topological, homological or Lipschitz manifold, possibly with boundary. In particular, our results answer the question of when the underlying space of an orbifold is a manifold.
Journal version, arXiv.

A Hamiltonian version of a result of Gromoll and Grove (with U. Frauenfelder und S. Suhr),
Ann. Inst. Fourier (Grenoble) 69 (2019), no. 1, 409419.
Abstract. The theorem that if all geodesics of a Riemannian twosphere are closed they are also simple closed is generalized to real Hamiltonian structures on RP^3. For reversible Finsler 2spheres all of whose geodesics are closed this implies that the lengths of all geodesics coincide.
Journal version, arXiv.

On the existence of closed geodesics on 2orbifolds,
Pacific J. Math. 294 (2), (2018), 453472.
Abstract. We show that on every compact Riemannian 2orbifold there exist infinitely many closed geodesics of positive length.
Journal version, arXiv.

Seifert fibrations of lens spaces (with H. Geiges),
Abh. Math. Semin. Univ. Hambg. 88 (1), (2018), 122.
Abstract. We classify the Seifert fibrations of any given lens space L(p,q). We give an algorithmic construction of a Seifert fibration of L(p,q) over the base orbifold S^2(m,n) with the coprime parts of m and n arbitrarily prescribed. This algorithm produces all possible Seifert fibrations, and the equivalences between the resulting Seifert fibrations are described completely. Also, we show that all Seifert fibrations are equivalent to certain standard models.
Journal version, arXiv.

On metrics on 2orbifolds all of whose geodesics are closed,
J. Reine Angew. Math. 758 (2020), 6794.
Abstract. We show that the geodesic period spectrum of a Riemannian 2orbifold all of whose geodesics are closed depends, up to a constant, only on its orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll, Grove and Pries that all prime geodesics have the same length. In the appendix we partly strengthen our result in terms of conjugacy of contact forms and explain how to deduce rigidity on the real projective plane based on a systolic inequality due to Pu.
Journal version, pdf, arXiv.

Affine functions on Alexandrov spaces (with S. Stadler),
Math. Z. 289 (2018), no. 12, 455469.
Abstract. We show that every finitedimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.
Journal version, arXiv.

Equivariant smoothing of piecewise linear manifolds,
Math. Proc. Camb. Philos. Soc. 164 (2), (2018), 369380.
Abstract. We characterize finite groups G generated by orthogonal transformations in a finitedimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient piecewise linear structure.
Journal version, arXiv.

Characterization of finite groups generated by reflections and rotations,
J. Topol. 9 (4), (2016), 11091129.
Abstract. We characterize finite groups G generated by orthogonal transformations in a finitedimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient piecewise linear structure.
Journal version, pdf.

Classification of finite groups generated by reflections and rotations (with M. Mikhailova),
Transform. Groups 21 (4), (2016), 11551201.
Abstract. We classify finite groups generated by orthogonal transformations in a finitedimensional Euclidean space whose fixed point subspace has codimension one or two. These groups naturally arise in the study of the quotient of a Euclidean space by a finite orthogonal group and hence in the theory of orbifolds.
Journal version
My preprints are also available on the arXiv.
Theses
 Some results on orbifold quotients and related objects, thesis, Universität zu Köln, (2016).
 A geometric approach to the structure theory of semisimple Lie algebras and symmetric spaces, diploma thesis, Universität zu Köln, (2012).
 Highly transmissive windows for low noise terahertz receivers, bachelor thesis, Universität zu Köln, (2011).
C. Lange, 15.3.20