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We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points., Comment: 23 pages

We prove a version of Topogonov's triangle comparison theorem with surfaces of revolution as model spaces. Given a model surface and a Riemannian manifold with a fixed base point, we give necessary and sufficient conditions under which every geodesic triangle in the manifold with a vertex at the base point has a corresponding Alexandrov triangle in the model. Under these conditions we also prove a version of the Maximal Radius Theorem and a Grove--Shiohama type Sphere Theorem.

We present a new strategy for proving the Ambrose conjecture, a global version of the Cartan local lemma. A linking curve is defined as a curve in the tangent space whose composition with the exponential map is tree formed. This key idea is used to define sutured manifolds. We prove first that any sutured manifold satisfies the Ambrose conjecture. We then prove that the set of sutured Riemmanian manifolds contains a residual set of the metrics on a given smooth manifold of dimension 3 by explicitely constructing a special type of linking curves that follow the "conjugate descending flow"., Comment: first part improved in second version

We show that for any point $p$ in a closed Riemannian manifold $M$, there exists at least one point $q\in M$ such that $p$ is critical for the distance function from $q$. We also show that such a point $q$ cannot always be reached with geodesic loops based at $q$ with midpoint $p$., Comment: 7 pages, 2 figures. Minor changes. Accepted for publication in Calculus of Variations and Partial Differential Equations

In this paper we study 1/k-geodesics, those closed geodesics that minimize on any subinterval of length $l(\gamma)/k$. We employ energy methods to provide a relationship between the 1/k-geodesics and what we define as the balanced points of the uniform energy. We show that classes of balanced points of the uniform energy persist under the Gromov-Hausdorff convergence of Riemannian manifolds. Additionally, we relate half-geodesics (1/2-geodesics) to the Grove-Shiohama critical points of the distance function. This relationship affords us the ability to study the behavior of half-geodesics via the well developed field of critical point theory. Along the way we provide a complete characterization of the differentiability of the Riemannian distance function., Comment: This second version includes a new section (now Section 5) which studies the uniform energy functional under Gromov-Hausdorff convergence

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics., Comment: 20 pages