1 Introduction
We study the problem of reducing the \(G\)-structure associated to a certain type of Segre structure to a torsion-free substructure.
Segre - or closely related structures and their counterparts in the category of complex manifolds were studied under various names, including tensor product structure [13], generalised conformal structure [11], complex paraconformal structure [3], (almost) Grassmann structure [1, 9, 14, 16], Segre structure [6, 12], and in [4] as an example of a class of structures called almost symmetric hermitian manifolds.
Here, by an \((m,n)\)-Segre structure on a manifold \(M\) we mean a smoothly varying family of cones \(\mathcal{S}_p\subset T_pM\) in the tangent spaces of \(M\), each linearly isomorphic to the Segre cone of linear maps \(\mathbb{R}^m\to \mathbb{R}^n\) of rank one. The tangent planes to \(M\) which are contained in some Segre cone \(\mathcal{S}_p\) come in two types, called \(\alpha\)- and \(\beta\)-planes. An immersed submanifold \(\Sigma \subset M\) whose tangent planes are all \(\beta\)-planes and which is maximal in the sense of inclusion is called a \(\beta\)-surface. A Segre structure is called \(\beta\)-integrable, if every \(\beta\)-plane is tangent to a unique \(\beta\)-surface.
In [12] Grossman showed that the space of paths of a certain class of geodesically simple path geometries, which he calls torsion-free, inherits a Segre structure. Bryant observed in [8] that the space of oriented geodesics \(\Lambda\) of a geodesically simple Finsler structure of constant flag curvature (cfc) \(1\) inherits a Kähler structure and a torsion-free \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-structure satisfying a certain positivity condition. Conversely, he shows that every torsion-free \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-structure satisfying the positivity condition (and an integrability condition for \(n=2\)) arises via a (generalised) cfc \(1\) Finsler structure.
The main result of the article is that locally every \(\beta\)-integrable \((2,n)\)-Segre structure \(\mathcal{S}\) can be reduced to a torsion-free \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-structure. It follows with Bryant’s result, that locally every \(\beta\)-integrable \((2,n)\)-Segre structure admitting a \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-reduction satisfying the positivity condition of [8] arises via a (generalised) cfc \(1\) Finsler structure.
Note that an \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-reduction of a \(\beta\)-integrable \((2,n)\)-Segre structure \(\mathcal{S}\) equips the underlying manifold with an integrable almost complex structure which preserves \(\mathcal{S}\) and for which the \(\beta\)-surfaces are totally real.
This article is organised as follows. In 2 we review the construction of a ‘twistor bundle’ \(\rho : X_\mathcal{S}\to M\) over a manifold \(M\) which is equipped with a \(\beta\)-integrable \((2,n)\)-Segre structure and show in 3 that \(\rho\)-sections having holomorphic image are in one-to-one correspondence with reductions of \(\mathcal{S}\) to torsion-free \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-structures on \(M\). It follows that locally every \(\beta\)-integrable \((2,n)\)-Segre structure can be reduced to a torsion-free \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\) structure. In 4 we show that for the homogeneous \((2,n)\)-Segre structure on the oriented \(2\)-plane Grassmannian \(M=G^+_2(\mathbb{R}^{n+2})\), the reductions are in one-to-one correspondence with the smooth quadrics \(Q \subset \mathbb{CP}^{n+1}\) without real points.
Before this work begun Robert Bryant informed the author about his private notes regarding the generality of positive constant flag curvature (cfc) Finsler structures on the \(n\)-sphere. He shows that a positive cfc Finsler structure on the \(n\)-sphere all of whose geodesics are closed and of the same length gives rise to a \(D^2\)-bundle \(\rho : X \to \Lambda\), fibering over the space of oriented geodesics \(\Lambda\), whose total space is a complex manifold. This bundle is isomorphic to \(\rho_0 : \mathbb{CP}^{n+1}\setminus\mathbb{RP}^{n+1}\to G^+_2(\mathbb{R}^{n+2})\) in the case of a rectilinear Finsler structure. In addition, the Finsler structure induces a \(\rho\)-section having holomorphic image (isomorphic to a quadric in the rectilinear case) and conversely every such section satisfying a certain convexity condition gives rise to a Finsler structure on \(S^n\) sharing the same geodesics. Using Kodaira deformation theory this allows Bryant to determine the generality of such Finsler structures sharing the same unparametrised geodesics. Although being related, the results in this article were arrived at independently.
Acknowledgements
Thanks to Schweizerischer Nationalfonds for its support via the postdoctoral fellowship PBFRP2-133545, to the Mathematical Sciences Research Institute in Berkeley for its support via a postdoctoral fellowship in the period September 2011-June 2012, and to the Forschungsinstitut für Mathematik at ETH Zürich for its support in the period July-December 2012. The author would like to thank Robert L. Bryant for helpful discussions. The author is also grateful to the referees for pointing out a mistake in an earlier version of this article and for other valuable suggestions.
...