We construct examples of complete Riemannian manifolds having the property that every geodesic lies in a totally geodesic hyperbolic plane. Despite the abundance of totally geodesic hyperbolic planes, these examples are not locally homogenous.

Let M be a Hadamard manifold, that is, a complete simply connected riemannian manifold with non-positive sectional curvatures. Then every geodesic segment α : [0, a] → M from α(0) to α(a) can be extended to a geodesic ray α : [0,∞) → M . We say then that the Hadamard manifold M is geodesically complete. Note that, in this case, all geodesic rays are proper maps. CAT(0) spaces are generalization...

For two vertices X,Y ∈ V (G), a cycle is called a geodesic cycle with X and Y if a shortest path joining X and Y lies on the cycle. A graph G is called to be geodesic k-pancyclic if any two vertices X,Y on G have such geodesic cycle of length l that 2dG(X,Y ) + k ≤ l ≤ |V (G)|. In this paper, we show that the n-dimensional Möbius cube MQn is geodesic 2-pancyclic for n ≥ 3. This result is optima...

In this paper, a geodesic α-invex subset of a Riemannian manifold is introduced. Geodesic α-invex and α-preinvex functions on a geodesic α-invex set with respect to particular maps are also defined. Further, we study the relationships between geodesic α-invex and α-preinvex functions on Riemannian manifolds. Some results of a non smooth geodesic α-preinvex function are also discussed using prox...

On arbitrary spacetimes, we study the characteristic Cauchy problem for Dirac fields on a light-cone. We prove the existence and uniqueness of solutions in the future of the light-cone inside a geodesically convex neighbourhood of the vertex. This is done for data in L 2 and we give an explicit definition of the space of data on the light-cone producing a solution in H 1. The method is based on...

We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.

Since 2001, work has been underway to create coupled atmosphere, ocean, sea ice, and land surface models in a unified framework. The project involves Colorado State University (CSU), the University of California at Los Angeles (UCLA), the Naval Postgraduate School (NPGS), and the Los Alamos National Laboratory. (LANL). All of the sub-models use geodesic grids, as explained in Section 2. In addi...

Two Riemannian manifolds are said to have C-conjugate geodesic flows if there exist an C diffeomorphism between their unit tangent bundles which intertwines the geodesic flows. We obtain a number of rigidity results for the geodesic flows on compact 2-step Riemannian nilmanifolds: For generic 2-step nilmanifolds the geodesic flow is C rigid. For special classes of 2-step nilmanifolds, we show t...

We propose a new class of distances for the purpose of data clustering, called the geodesic distance, and introduce a geodesic extension of K-medoids algorithm. We analyze the theoretical properties of the geodesic distance within a clustering framework and prove that the geodesic K-medoids algorithm converges to the correct clustering assignment in the asymptotic regime, even in the presence o...