On the geodesic flow of tori without conjugate points

Supported by “Deutsche Forschungsgemeinschaft”     

Riemannian tori without conjugate points are flat

The first author was supported by the Deutsche Forschungsgemeinschaft fellowship and enjoyed the hospitality of the University of Freiburg.     

Riemannian manifolds without conjugate points

Summary A generalisation of Theorems ofHadamard andMyers on spaces without conjugate points. To Enrico Bompiani on his scientific Jubilee.     

Symplectic twist maps without conjugate points

For sequences of symplectic twist maps without conjugate points, an invariant Lagrangian subbundle is constructed. This allows one to deduce that absence of conjugate points is a rare property in some classes of maps. This work was partially supported by EPSRC grant GR/M11349.     

Asymptotic behavior of manifolds without conjugate points

This research was partially supported by GARC-KOSEF.     

Convexity of spheres in a manifold without conjugate points

For a non-compact, complete and simply connected manifoldM without conjugate points, we prove that if the determinant of the second fundamental form of the geodesic spheres inM is a radial function, then the geodesic spheres are convex. We also show that ifM is two or three dimensional and without conjugate points, then, at every point there exists a ray with no focal points on it relative to the initial point of the ray. The proofs use a result from the theory of vector bundles combined with the index lemma.     

On extreme points in separable conjugate spaces

It is proved that every bounded closed and convex subset of an arbitrary conjugate separable Banach space is the closed convex hull of its extreme points.     

Finsler manifolds without conjugate points and with integral Ricci curvature

We prove that the integral of the Ricci curvature on the unit tangent bundle SM of a complete Finsler manifold M without conjugate points is nonpositive and vanishes only if M is flat, provided that the Ricci curvature on SM has an integrable positive or negative part.     

On conjugate points and the Leitmann equivalent problem approach

This article extends the Leitmann equivalence method to a class of problems featuring conjugate points. The class is characterised by the requirement that the set of indifference points of a given problem forms a finite stratification.     

A note on the divergence of geodesic rays in manifolds without conjugate points

Let (M, g) be a compact Riemannian manifold without conjugate points and let be its universal covering endowed with the pullback of the metric g by the covering map. We show that geodesic rays in which meet an axis of a covering isometry diverge from this axis. This result generalizes well known results by Morse and Hedlund in the context of globally minimizing geodesics in the universal covering of compact surfaces.      

On meromorphically harmonic starlike functions with respect to symmetric conjugate points

In a previous paper M.P. Chen, Z.-R. Wu and Z.-Z. Zou [M.P. Chen, Z.-R. Wu, Z.-Z. Zou, On functions α-starlike with respect to symmetric conjugate points, J. Math. Anal. Appl. 201 (1996) 25-34] developed a method, using some operators, to deal with functions analytic and starlike with respect to symmetric conjugate points in the unit disc. Then, the same method is employed to functions meromorphic by Z.Z. Zou and Z.-R. Wu [Zhong Zhu Zou, Zhuo-Ren Wu, On meromorphically starlike functions and functions meromorphically starlike with respect to symmetric conjugate points, J. Math. Anal. Appl. 261 (2001) 17-27]. Now, the method can be employed to functions meromorphic harmonic in the punctured disc 0<|z|<1. Especially, a sharp coefficient estimate and a structural representation of such functions are obtained.