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Unit tangent bundle
About: Unit tangent bundle is a research topic. Over the lifetime, 1056 publications have been published within this topic receiving 15845 citations.
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TL;DR: For every pointwise polynomial function on each fiber of the cotangent bundle of a Riemannian manifold M, a family of diffential operators is given, which acts on the space of smooth sections of a vector bundle on M as discussed by the authors.
Abstract: For every pointwise polynomial function on each fiber of the cotangent bundle of a Riemannian manifold M, a family of diffential operators is given, which acts on the space of smooth sections of a vector bundle on M. Such a correspondence may be considered as a rule to quantize classical systems moving in a Riemannian manifold or in a gauge field. Some applications of our construction are also given in this paper
35 citations
TL;DR: The main theorem of the present paper as mentioned in this paper states that any complex projective manifold of dimension 4 or more whose tangent bundle is big and 1-ample is equal either to a projective space or to a smooth quadric.
Abstract: A line bundle over a complex projective variety is called big and 1-ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1-dimensional fibers. A vector bundle is called big and 1-ample if the relative hyperplane line bundle over its projectivisation is big and 1-ample.
The main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more, whose tangent bundle is big and 1-ample, is equal either to a projective space or to a smooth quadric. Since big and 1-ample bundles are ?almost? ample, the present result is yet another extension of the celebrated Mori paper ?Projective manifolds with ample tangent bundles? (Ann. of Math. 110 (1979) 593?606).
The proof of the theorem applies results about contractions of complex symplectic manifolds and of manifolds whose tangent bundles are numerically effective. In the appendix we re-prove these results.
35 citations
TL;DR: In this paper, it was shown that the backward continued fraction map is a factor map of a special cross-section map for the geodesic flow on the unit tangent bundle of the modular surface.
Abstract: The ‘backward continued fraction’ map studied by A. Reyni is defined by y = g(x) where g(x) equals the fractional part of 1/(1−x) for 0 < x < 1. We show that it is a factor map of a special cross-section map for the geodesic flow on the unit tangent bundle of the modular surface. This gives an alternative derivation of the fact that this map preserves the infinite measure dx/x on the unit interval.
35 citations
TL;DR: The tangent groupoid of (M,H) as mentioned in this paper is a differentiable groupoid gHM encoding the smooth deformation of M × M to GM, which is a Lie group bundle of graded 2-step nilpotent Lie groups.
Abstract: As a step towards proving an index theorem for hypoelliptic operators on Heisenberg manifolds, including for those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes? tangent groupoid of a manifold. As is well known for a Heisenberg manifold (M,H) the relevant notion of tangent bundle is rather that of a Lie group bundle of graded 2-step nilpotent Lie groups GM. We define the tangent groupoid of (M,H) as a differentiable groupoid gHM encoding the smooth deformation of M × M to GM. In particular, this construction makes a crucial use of a refined notion of privileged coordinates and of a tangent-approximation result for Heisenberg diffeomorphisms.
35 citations