Given a manifold M and a proper sub-bundle ∆ ⊂ TM , we study homotopy properties of the horizontal base-point free loop space Λ, i.e. the space of absolutely continuous maps γ : S → M whose velocities are constrained to ∆ (for example: legendrian knots in a contact manifold). A key technical ingredient for our study is the proof that the base-point map F : Λ → M (the map associating to every lo...

A Riemannian manifold is said to be uniformly secure if there is a finite number s such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked by s point obstacles. We prove that the number of geodesics with length ≤ T between every pair of points in a uniformly secure manifold grows polynomially as T → ∞. By results of Gromov and Mañé, the fundamental group of...

Human action recognition remains a challenging task due to the various sources of video data and large intraclass variations. It thus becomes one of the key issues in recent research to explore effective and robust representation to handle such challenges. In this paper, we propose a novel representation approach by constructing mid-level words in videos and encoding them on Riemannian manifold...

We use Bott’s equivariant Morse theory to show that compact Riemannian three-manifolds with negative sectional curvature possess closed minimal surfaces of arbitrarily high Morse index. 1 Sources of noncompactness for parametrized minimal surfaces In this article, we apply the theory of parametrized two-dimensional minimal surfaces in a compact n-dimensional Riemannian manifold (M, 〈·, ·〉) to t...

We first derive the relation between the eikonal approximation to the Maxwell wave equations in an inhomogeneous anisotropic medium and geodesic motion in a three dimensional Riemannian manifold using a method which identifies the symplectic structure of the corresponding mechanics. We then apply an analogous method to the five dimensional generalization of Maxwell theory required by the gauge ...