Some remarks on Finsler manifolds with constant flag curvature

By Robert L. Bryant. 2001

114 citations recorded by the source corpus.

This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. The second remark is that there is a natural way to construct a (not necessarily complete) Finsler n-manifold of constant positive flag curvature out of a hypersurface in suitably general position in CPn. The third remark is that there is a description of the Finsler metrics of constant curvature on S2 in terms of a Riemannian metric and 1-form on the space of its geodesics. In particular, this allows one to use any (Riemannian) Zoll metric of positive Gauss curvature on S2 to construct a global Finsler metric of constant positive curvature on S2. The fourth remark concerns the generality of the space of (local) Finsler metrics of constant positive flag curvature in dimension n+1 > 2. It is shown that such metrics depend on n(n+1) arbitrary functions of n+1 variables and that such metrics naturally correspond to certain torsion-free S1·GL(n,R)structures on 2n-manifolds. As a by-product, it is found that these groups do occur as the holonomy of torsion-free affine connections in dimension 2n, a hitherto unsuspected phenomenon. 1991 Mathematics Subject Classification. 53B40, 53C60, 58A15.

Subfields: Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/5c057d4d19c1bd778f1890c7fd244dee03e523eb

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Citations in the corpus

On a Class of Projectively Flat Metrics with Constant Flag Curvature (2008, background): “Bryant studied and characterized locally projectively flat Finsler metrics with constant flag curvature K = 1 [5]-[7].”
Zermelo navigation on Riemannian manifolds (2003, background): “Bryant for encouraging us to count isometry classes properly, and for bringing [14] to our attention.”
On the definition and examples of Finsler metrics (2011, background): “(1) Our definition of (standard) Finsler metric agrees with classical references as [1, 2, 5, 12, 34, 44, 45] and our definition of conic Finsler metric is equivalent to the notion of generalized Finsler metric by [10] (but we retain our nomenclature as we are also dealing with other generalizations”
On the Flag Curvature of Finsler Metrics of Scalar Curvature (2003, unknown): “Projectively flat Finsler metrics on a convex domain in Rn are regular solutions to Hilbert’s fourth problem [18].”
Nonreversible Finsler Metrics of Positive Flag Curvature (2004, methodology): “For example, in contrast to the Riemannian case, there are Finsler metrics of constant positive flag curvature on spheres that are not isometric to the standard Riemannian metric; see [Bao et al. 2003; Bryant 2002; Shen 2002] for their characterization.”
On a new class of Finsler metrics (2011, background): “Bryant’s metrics[4, 5, 6] are rectilinear Finsler metrics on S with flag curvature K = 1 and given in the following form with X ∈ S, Y ∈ TXS, F (X,Y ) = R { √ Q(X,X)Q(Y, Y )−Q(X,Y )2 Q(X,X) − i Q(X,Y ) Q(X,X) }”
Projectively flat Randers metrics with constant flag curvature (2003, background): “Very recently, Bryant and the author have independently discussed the classification problem of locally projectively Finsler metrics of constant flag curvature from different approaches [ Br3 ][Sh2].”