Consider a closed, smooth manifold M of non-positive curvature. Write p:UM→M for the unit tangent bundle over M and let � > denote the subset consisting of all vectors of higher rank. This subset is closed and invariant under the geodesic flow � on UM. We define the structured dimension s-dim � > which, essentially, is the dimension of the set p(� > ) of base points of � > . The main result of ...

Let $M$ be an orientable hypersurface in the Euclidean space $R^{2n}$ with induced metric $g$ and $TM$ be its tangent bundle. It is known that the tangent bundle $TM$ has induced metric $overline{g}$ as submanifold of the Euclidean space $R^{4n}$ which is not a natural metric in the sense that the submersion $pi :(TM,overline{g})rightarrow (M,g)$ is not the Riemannian submersion. In this paper...

vision and then adopted after an extensive design process. OISD may serve to be a more powerful model of change in that it capitalizes on powerful system concepts such as positive feedback and key leverage points, while still retaining a pragmatic orientation. The six phases of OISD are described below. The phases are loose and overlapping; they naturally would probably occur in sequential orde...

We establish upper and lower bounds on the number of connected components of lines tangent to four triangles in . We show that four triangles in may admit at least 88 tangent lines, and at most 216 isolated tangent lines, or an infinity (this may happen if the lines supporting the sides of the triangles are not in general position). In the latter case, the tangent lines may form up to 216 conne...

1. Write the formula for the Liouville measure Λ on the unit tangent bundle T 1 H n. Prove that Λ is invariant under the actions by Isom(H n) and the geodesic flow g t on T 1 H n. 2. Let δ be a geodesic in the hyperbolic space H n and let π : H n → δ denote the nearest point projection map. Suppose that for some x, y ∈ H n and some r > 0 we have d(x, δ), d(y, δ) ≥ r. Prove that d(π(x), π(y)) ≤ ...

This paper has three objectives. First to recall the link between the classical Legendre-Fenschel transformation and a useful isomorphism between 1-jets of functions on a vector bundle and on its dual. As a particular consequence we obtain the classical isomorphism between the cotangent bundle of the tangent bundle T ∗TM and the tangent bundle of the cotangent bundle TT ∗M of any manifold M. Se...

Treating matrices as points in n2-dimensional space, we apply geometry to study and explain algorithms for the numerical determination of the Jordan structure of a matrix. Traditional notions such as sensitivity of subspaces are replaced with angles between tangent spaces of manifolds in n2-dimensional space. We show that the subspace sensitivity is associated with a small angle between complem...

Constructing an efficient parametrization of a large, noisy data set of points lying close to a smooth manifold in high dimension remains a fundamental problem. One approach consists in recovering a local parametrization using the local tangent plane. Principal component analysis (PCA) is often the tool of choice, as it returns an optimal basis in the case of noise-free samples from a linear su...

— We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a p-adic field from the metric, differential and measure geometric point of view. We prove that the local density of such sets at each of their points does exist. We then introduce the notion of distinguished tangent cone with respect to some open subgroup with finite index in the multiplicative group of our ...