An equivalence relation on the tangent bundle of a manifold is defined in order to extend a structure (modulated or not) onto it. This extension affords a representation of a structure in any tangent space and that in another tangent space can easily be derived. Euclidean symmetry operations associated with the tangent bundle are generalized and their usefulness for the determination of the int...

Let Sn be the n-sphere of constant positive curvature. For n ≥ 2, we will show that a measure on the unit tangent bundle of S, which is even and invariant under the geodesic flow, is not uniquely determined by its projection to S. Mathematics Subject Classification (2000). 53D25.

We investigate the distribution of orbits of a non-elementary discrete hyperbolic subgroup Γ acting on Hn and its geometric boundary ∂∞(H). In particular, we show that if Γ admits a finite Bowen-Margulis-Sullivan measure (for instance, if Γ is geometrically finite), then every Γ-orbit in ∂∞(H) is equidistributed with respect to the Patterson-Sullivan measure supported on the limit set Λ(Γ). The...

1 The Cotangent Complex: General Theory 4 1.1 Stable Envelopes and Tangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Relative Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 The Tangent Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 The Relative Cotangent Complex...

From this result one can easily deduce I (H) = n 1. For a detailed discussion of Cheeger’s constant and related results, one can consult [Cha, Chapter 6]. In general, it is very di¢ cult to know if the isoperimetric constant is positive or not and it is almost impossible to compute it explicitly if it is known to be positive. In this short note, we prove that the isoperimetric constant is posit...

For uniformly chosen random α ∈ [0, 1], it is known the probability the nth digit of the continued-fraction expansion, [α]n converges to the Gauss-Kuzmin distribution P([α]n = k) ≈ log2(1 + 1/k(k + 2)) as n → ∞. In this paper, we show the continued fraction digits of √ d, which are eventually periodic, also converge to the Gauss-Kuzmin distribution as d → ∞ with bounded class number, h(d). The ...