In this paper we study the geodesics and ends of compact surfaces satisfying the "uniform visibility" axiom. We are primarily though not exclusively interested in finitely connected surfaces, which topologically are compact Riemann surfaces with a finite number of punctures.
The first part of this paper constructs a class of naturally reductive metrics on compact Lie groups and shows that all naturally reductive left invariant metrics are of this type if the group is simple. The second part analyzes the question of when these metrics are Einstein an…
Simply connected Reimannian symmetric spaces were classified by E. Cartan. This paper examines an attempt to obtain an analogous classification when the metric has indefinite signature.
Blaschke rolling theorem in its original form dealt with the problem of finding sufficient conditions for a ball [italic]D₁ to roll freely inside a compact convex set [italic]D₂ with a non-empty interior. The paper is divided into two parts. Part I is primarily concerned with pr…
In this paper we formulate and prove an index theorem for minimal surfaces of higher topological type spanning one boundary contour. Our techniques carry over to surfaces with several boundary contours as well as to unoriented surfaces.
We study the behavior of maximal geodesics outside a sufficiently large compact set in a finitely connected, complete and noncompact 2-dimensional Riemannian manifold (possibly with boundary). One of our main results states that if the total curvature of [italic]M exists and is …
This text discusses the Euler class, the intersection number, the Lefschetz number, and the Thom class. The aim of discussing these well-known concepts is to describe them in a "computable" way. These will be expressed in differential or integral forms, giving formulae, generali…
One of the most cited books in mathematics, John Milnor's exposition of Morse theory has been the most important book on the subject for more than forty years. Morse theory was developed in the 1920s by mathematician Marston Morse. (Morse was on the faculty of the Institute for …
"In this book, Berger provides a survey of the main developments in Riemannian geometry in the last fifty years, focusing his main attention on the following five areas: Curvature and topology; the construction of and the classification of space forms; distinguished metrics, esp…
