Publications

We introduce the notion of a flat extension of a connection \(\theta\) on a principal bundle. Roughly speaking, \(\theta\) admits a flat extension if it arises as the pull-back of a component of a Maurer--Cartan form. For trivial bundles over closed oriented \(3\)-manifolds, we relate the existence of certain flat extensions to the vanishing of the Chern–Simons invariant associated with \(\theta\). As an application, we recover the obstruction of Chern--Simons for the existence of a conformal immersion of a Riemannian \(3\)-manifold into Euclidean \(4\)-space. In addition, we obtain corresponding statements for a Lorentzian \(3\)-manifold, as well as a global obstruction for the existence of an equiaffine immersion into \(\mathbb{R}^4\) of a \(3\)-manifold that is equipped with a torsion-free connection preserving a volume form.

Related project(s):
68Minimal Lagrangian connections and related structures

In earlier work we have shown that for certain geometric structures on a smooth manifold \(M\) of dimension \(n\), one obtains an almost para-Kähler--Einstein metric on a manifold \(A\) of dimension \(2n\) associated to the structure on \(M\). The geometry also associates a diffeomorphism between \(A\) and \(T^*M\) to any torsion-free connection compatible with the geometric structure. Hence we can use this construction to associate to each compatible connection an almost para-Kähler--Einstein metric on \(T^*M\). In this short article, we discuss the relation of these metrics to Patterson--Walker metrics and derive explicit formulae for them in the cases of projective, conformal and Grassmannian structures.

Related project(s):
68Minimal Lagrangian connections and related structures

We establish a one-to-one correspondence between Finsler structures on the \(2\)-sphere 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 $$\mathbb{CP}(a_1,a_2)\to \mathbb{CP}(a_1,(a_1+a_2)/2,a_2)$$ of weighted projective spaces provide examples of Finsler \(2\)-spheres of constant curvature and all geodesics closed.

Related project(s):
68Minimal Lagrangian connections and related structures

We associate a flow \(\phi\) to a solution of the vortex equations on a closed oriented Riemannian 2-manifold \((M,g)\) of negative Euler characteristic and investigate its properties. We show that \(\phi\) always admits a dominated splitting and identify special cases in which \(\phi\) is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of \((M,g)\).

Related project(s):
68Minimal Lagrangian connections and related structures

Given a parabolic geometry on a smooth manifold \(M\), we study a natural affine bundle \(A \to M\), whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic geometry defines a reductive Cartan geometry on \(A\), which induces an almost bi-Lagrangian structure on \(A\) and a compatible linear connection on \(TA\). We prove that the split-signature metric given by the almost bi-Lagrangian structure is Einstein with non-zero scalar curvature, provided the parabolic geometry is torsion-free and \(|1|\)-graded. We proceed to study Weyl structures via the submanifold geometry of the image of the corresponding section in \(A\). For Weyl structures satisfying appropriate non-degeneracy conditions, we derive a universal formula for the second fundamental form of this image. We also show that for locally flat projective structures, this has close relations to solutions of a projectively invariant Monge-Ampere equation and thus to properly convex projective structures.

Related project(s):
68Minimal Lagrangian connections and related structures