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.

Structure of symmetric tensors of type (0,2) and tensors of type (1,1) on the tangent bundle

Abstract: The concepts of M-tensor and M-connection on the tangent bundle TM of a smooth manifold M are used in a study of symmetric tensors of type (0, 2) and tensors of type (1, 1) on TM. The constructions make use of certain local frames adapted to an M-connection. They involve extending known results on TM using tensors on M to cases in which these tensors are replaced by M-tensors. Particular attention is devoted to (pseudo-) Riemannian metrics on TM, notably those for which the vertical distribution on TM is null or nonnull, and to the construction of almost product and almost complex structures on TM.

A characterization of seven compact Kaehler submanifolds by holomorphic pinching

Abstract: From the results of Simons [9] and Chern, Do Carmo and Kobayashi [2], we know that, in the class of compact minimal submanifolds of a sphere, the totally geodesic submanifolds are isolated and that some simple minimal submanifolds can be characterized by suitable pinching on their curvatures. These ideas are extended naturally to compact Kaehler submanifolds of a complex projective space. The problem of the best pinching for the above submanifolds was studied later by several authors, e.g. Yau [11] and Ogiue [4]. For surfaces and hypersurfaces, the problem is completely resolved. However in the general case we have only partial results. Let M' be a compact Kaehler submanifold, of complex dimension n, immersed in the complex projective space CPtm(1) endowed with the Fubini-Study metric of constant holomorphic sectional curvature 1. Let H and K be the holomorphic sectional curvature and the sectional curvature of Mn respectively. Ogiue conjectured the following: (1) If H > , or (2) If n ? 2 and K> , or (3) If m-n n(n + ) and K > 0, 2 then Mn is a linear subvariety of CPm(1). Recently, using natural arguments at the minimum of the function H defined on the unit tangent bundle of MW, the author [7] and Verstraelen and the author [8] resolved the conjectures (1) and (2) respectively. In this paper we obtain the following complete solution of the pinching problem in the Kaehlerian case.

An alternative approach to the Cartan form in Lagrangian field theories

Abstract: An operator analogous to the almost tangent structure on a tangent bundle is defined on jet bundles of a fibred manifold. This operator is used to construct a Cartan form. The construction is unique for first-order Lagrangians and is also unique when restricted to higher-order mechanics.

Equidistribution of kronecker sequences along closed horocycles

Abstract: It is well known that (i) for every irrational number α the Kronecker sequence mα (m =1 ,...,M ) is equidistributed modulo one in the limit M →∞ , and (ii) closed horocycles of lengthbecome equidis- tributed in the unit tangent bundle T1M of a hyperbolic surface M of finite area, as � →∞ . In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant ν> 0 the Kronecker sequence embedded in T1M along a long closed horocycle becomes equidistributed in T1M for almost all α ,p rovided that � = M ν →∞ . This equidistribution result holds in fact under explicit diophantine conditions on α (e.g. for α = √ 2) provided that ν< 1, or ν< 2 with additional assumptions on the Fourier coefficients of certain automorphic forms. Finally, we show that for ν =2 , our equidistribution theorem implies a recent result of Rudnick and Sar- nak on the uniformity of the pair correlation density of the sequence n 2 α modulo one.

Unit Tangent Sphere Bundles with Constant Scalar Curvature

Abstract: As a first step in the search for curvature homogeneous unit tangent sphere bundles we derive necessary and sufficient conditions for a manifold to have a unit tangent sphere bundle with constant scalar curvature. We give complete classifications for low dimensions and for conformally flat manifolds. Further, we determine when the unit tangent sphere bundle is Einstein or Ricci-parallel.