This paper is connected with the problem of describing path metric spaces which are homeomorphic to manifolds and biLipschitz homogeneous, i.e., whose biLipschitz homeomorphism group acts transitively. Our main result is the following. LetX = G/H be a homogeneous space of a Lie group G, and let d be a geodesic distance on X inducing the same topology. Suppose there exists a subgroup GS of G whi...

There is a canonical exact symplectic structure on the unit tangent bundle of a Riemannian manifold M given by pulling-back the symplectic two form ω and Liouville one form λ from the cotangent bundle T ∗M using the Riemannian metric. The pull-back of λ gives a contact form on level-sets of the length function on TM . The geodesic flow of M is given by the Reeb vectorfield of this contact struc...

Which smooth compact 4-manifolds admit an Einstein metric with non-negative Einstein constant? A complete answer is provided in the special case of 4-manifolds that also happen to admit either a complex structure or a symplectic structure. A Riemannian manifold (M, g) is said to be Einstein if it has constant Ricci curvature, in the sense that the function v −→ r(v, v) on the unit tangent bundl...

The well–known notions from tangent bundle geometry, like nonlinear connections and semisprays, are extended to bundle–type tangent manifolds. Also, new objects interesting from a dynamical point of view, like symmetries of nonlinear connections, are introduced. AMS Mathematics Subject Classification (2000): 58A30, 34A26, 37C10, 53C15

LetM be an n-dimensional Riemannian manifold. ThenM is a differentiable manifold with a smoothly varying inner product 〈, 〉x on each tangent space TxM . An orthonormal frame at the point x ∈ M is an orthonormal basis {e1, . . . , en} of the tangent space TxM . The set of orthonormal frames at each point is isomorphic to the Lie group O(n), and the set of orthonormal frames on M forms a principa...

In this paper we study two different topics. The first topic is the applications of the geometric quantization scheme of Witten introduced in [2] and [15] to the problem of the quantum background independence in string theory. The second topic is the introduction of a Z structure on the tangent space of the moduli space of polarized CY threefolds M(M). Based on the existence of a Z structure on...

We study a new kind of Courant algebroid on Poisson manifolds, which is a variant of the generalized tangent bundle in the sense that the roles of tangent and the cotangent bundle are exchanged. Its symmetry is a semidirect product of β-diffeomorphisms and βtransformations. It is a starting point of an alternative version of the generalized geometry based on the cotangent bundle, such as Dirac ...

On extrinsic geometry of unit normal vector fields of Riemannian hyperfoliations. Abstract We consider a unit normal vector field of (local) hyperfoliation on a given Riemannian manifold as a submanifold in the unit tangent bundle with Sasaki metric. We give an explicit expression of the second fundamental form for this submanifold and a rather simple condition its totally geodesic property in ...

Consider a one parameter family of diffeomorphisms fε such that f0 is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties. Let νε be any u-Gibbs state for fε. We prove (Theorem 1) that if A is a C ∞ function then the map A → νε(A) is differentiable at ε = 0. This implies (Corollary 1) that the difference of Birkhoff averages of the perturbed and un...

An almost contact metric structure is parametrized by a section σ of an associated homogeneous fibre bundle, and conditions for σ to be a harmonic section, and a harmonic map, are studied. These involve the characteristic vector field ξ, and the almost complex structure in the contact subbundle. Several examples are given where the harmonic section equations for σ reduce to those for ξ, regarde...