On Geodesic Segments in the Infinitesimal Asymptotic Teichmüller Spaces

LetAZ(R)be the infinitesimal asymptotic Teichmüller space of a Riemann surfaceRof infinite type. It is known thatAZ(R)is the quotient Banach space of the infinitesimal Teichmüller spaceZ(R), whereZ(R)is the dual space of integrable quadratic differentials. The purpose of this paper is to study the nonuniqueness of geodesic segment joining two points inAZ(R). We prove that there exist infinitely many geodesic segments between the basepoint and every nonsubstantial point in the universal infinitesimal asymptotic Teichmüller spaceAZ(D)by constructing a special degenerating sequence.

A Note on Starshaped Sets in 2-Dimensional Manifolds without Conjugate Points

LetWnbeC∞complete, simply connectedn-dimensional Riemannian manifolds without conjugate points. Assume thatS⊂W2is starshaped wherekerS≠S. For every pointx∈S∖kerS, defineA(x)={y:ylies on some geodesic segment inSfromxto a point of kerS}. There is a finite collectionAof all maximalAsets whose union isS. Further, kerS=∩{A:AinA}.

On Asymmetric Distances

In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

GEODESICS AND OPERATOR MEANS IN THE SPACE OF POSITIVE OPERATORS

The set A+ of positive invertible elements of a C*-algebra has a natural structure of reductive homogeneous manifold with a Finsler metric. Because pairs of points can be joined by uniquely determined geodesics and geodesics are "short" curves, there is a natural notion of convexity: C ⊂ A+ is convex if the geodesic segment joining a, b ∈ C is contained in C. We show that this notion is related to the classical convexity of real and operator valued functions. Several results about convexity are proved in this paper. The expressions of these results are closely related to the operator means of Kubo and Ando, in particular to the geometric mean of Pusz and Woronowicz, and they produce several norm estimations and operator inequalities.

A Note on Hyperconvexity in Riemannian Manifolds

Let M denote a connected Riemannian manifold of class C3, with positive definite C2 metric. The curvature tensor then exists, and is continuous.By a classical theorem of J. H. C. Whitehead (1), every point x of M has the property that all sufficiently small spherical neighbourhoods V of x are convex; that is, (i) to every y,z ∈ V there is one and only one geodesic segment yz in M which is the shortest path joining them:f:([0, 1]) → M,f(0) = y, f(1) = z; and (ii) this segment yz lies entirely in V:f([0, 1]) V; (iii) if f is parametrized proportional to arc length, then f(t) is a C2 function of y, t, and z.Let V be a convex set in M; and let y1 y2, Z1, z2 ∈ V.