Colleen Robles

On Randers spaces of constant flag curvature

This paper concerns a ubiquitous class of Finsler metrics on smooth manifolds of dimension n. These are the Randers metrics. They figure prominently in both the theory and the applications of Finsler geometry. For n ≥ 3, we consider only those with constant flag curvature. For n = 2, we focus on those whose flag curvature is a (possibly constant) function of position only. We characterize such metrics by three efficient conditions. With the help of examples in 2 and 3 dimensions, we deduce that the Yasuda-Shimada classification of Randers space forms actually addresses only a special case. The corrected classification for that special case is sharp, holds for n ≥ 2, and follows readily from our three necessary and sufficient conditions.

Geodesics in Randers spaces of constant curvature

Geodesies in Randers spaces of constant curvature are classified. Randers metrics have received much attention lately as solutions to Zermelos problem of navigation; largely because this navigation structure provides the frame work for a complete classification of constant flag curvature Randers spaces. (Flag curvature is the Finslerian analog of Riemannian sectional curvature. See (BR04).) Briefly, a Randers metric is of constant flag curvature if and only if it solves Zer melos problem of navigation on a Riemannian manifold of constant sectional cur vature under the influence of an infinitesimal homothety W. See Subsection 1.1 for a sketch of the navigation problem, and Theorem 3 for an explicit statement of the classification result. The aim of this paper is to develop a geometric description of the geodesies in these spaces of constant curvature. Intuitively, these paths minimize travel time across a Riemannian landscape under windy conditions. Presently we will show that these curves are given by composing geodesies of the Riemannian metric with the flow generated by W. This claim is formalized by Theorem 2. Geodesies on surfaces of constant, nonpositive curvature are illustrated in Section 3. We then turn, in Section 4, to the constant flag curvature K = 1 Randers metrics on Sn. The case of the sphere is especially interesting; it is possible to endow this closed manifold with a metric whose geodesies display distinctly non-Riemannian behaviors. For example: (1) A metric is projectively flat if every point admits coordinates in which the geodesies are straight lines. Belt r amis theorem assures us that a Riemannian metric is of constant sectional curvature if and only if it is projectively flat. In contrast few Randers spaces of constant flag curvature are projectively flat. There are infinitely many nonisometric Randers metrics of constant, positive

Quotients of non-classical ag domains are not algebraic

A flag domain D = G/V for G a simple real non-compact group G with compact Cartan subgroup is non-classical if it does not fiber holomorphically or anti-holomorphically over a Hermitian symmetric space. We prove that any two points in a non-classical domain D can be joined by a finite chain of compact subvarieties of D. Then we prove that for F an infinite, finitely generated discrete subgroup of G, the analytic space F\D does not have an algebraic structure.

Fubini's theorem in codimension two

Abstract We classify codimension two complex submanifolds of projective space Xn ⊂ having the property that any line through a general point x ∈ X having contact to order two with X at x automatically has contact to order three. We give applications to the study of the Debarre-de Jong conjecture and of varieties whose Fano variety of lines has dimension 2n – 4.

Classification of horizontal s

We classify the horizontal

Projective invariants of CR-hypersurfaces

\text{SL}(2)

Characterization of Calabi–Yau variations of Hodge structure over tube domains by characteristic forms

s and

Zermelo navigation on Riemannian manifolds

\mathbb{R}

Ricci and Flag Curvatures in Finsler Geometry

-split polarized mixed Hodge structures on a Mumford–Tate domain.

Schubert varieties as variations of Hodge structure

We study the equivalence problem under projective transformation for CR-hypersurfaces of complex projective space. A complete set of projective differential invariants for analytic hypersurfaces is given. The self-dual strongly ℂ-linearly convex hypersurfaces are characterized.