Comparison geometry referred to warped product models

Generalized Alexandrov-Toponogov Theorems for Radially Curved Manifolds and Their Applications

“…The Clairaut relation gives restrictions to the behavior of geodesics on models and plays an essential role for our study. The following proposition is valid for all the warped product models as stated in the classification [9]. = (γ (s), ∇t (γ (s)…”

“…The following theorem has been established in [9] and valid for pointed manifolds referred to model surfaces of revolution.…”

Rays and souls in von Mangoldt planes

“…We say that M m is a von Mangoldt plane if its sectional curvature G m := − m ′′ m is a non-increasing function of r . The Toponogov comparison theorem was extended in [IMS03] to open complete manifolds with radial sectional curvature bounded below by the curvature of a von Mangoldt plane, leading to various applications in [ST02, KO07,KT] and generalizations in [MS06,KT10,Mac10]. A point q in a Riemannian manifold is called a critical point of infinity if each unit tangent vector at q makes angle ≤ π 2 with a ray that starts at q .…”

“…The Toponogov comparison theorem was extended in [Itokawa et al 2003] to open complete manifolds with radial sectional curvature bounded below by the curvature of a von Mangoldt plane, leading to various applications in [Shiohama and Tanaka 2002;Kondo and Ohta 2007;Kondo and Tanaka 2011] and generalizations in [Mashiko and Shiohama 2006;Kondo and Tanaka 2010;Machigashira 2010].…”

The Axiom of Plane on Warped Product Models and Its Application

“…Here C(N) is the cut locus to N. A pair (M, N) is by definition a warped product model if and only if ρ ± = ± (or ρ = ) is satisfied and the radial curvature of (M, N) depends only on the oriented (or usual) distance to N. With this notation the characterization of the warped product models has been established in [8] as follows.…”