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Special coordinate systems are constructed in a neighborhood of a point or of a curve. Taylor expansions can then be easily inferred for the metric, the connection, or the Finsler Lagrangian in terms of curvature invariants. These coordinates circumvent the difficulties of the normal and Fermi coordinates in Finsler geometry, which in general are not sufficiently differentiable. They are obtained applying the usual constructions to the pullback of a horizontally torsionless connection. The results so obtained are easily specialized to the Berwald or Chern–Rund connections and have application in the study of the equivalence principle in Finslerian extensions of general relativity. [ABSTRACT FROM AUTHOR]

Abstract: Let be a four-dimensional complete noncompact Bach-flat Riemannian manifold with positive Yamabe constant. In this paper, we show that has a constant curvature if it has a nonnegative constant scalar curvature and sufficiently small -norm of trace-free Riemannian curvature tensor. Moreover, we get a gap theorem for with positive scalar curvature. [Copyright &y& Elsevier]

Abstract: We define the notion of extrinsic symplectic symmetric spaces and exhibit some of their properties. We construct large families of examples and show how they fit in the perspective of a complete classification of these manifolds. We also build a natural -quantization on a class of examples. [Copyright &y& Elsevier]

Abstract: Hitchin shows in [N. Hitchin, The geometry of three-forms in six and seven dimensions, J. Differ. Geom. 55 (2000) 547–576. math.DG/0010054v1; N. Hitchin, Stable forms and special metrics, Global Differential Geometry: The Mathematical Legacy of Alfred Gray, in: Contemp. Math., volume 288, American Math. Soc., 2001, pp. 70–89. math.DG/0107101v1] that half-flat -structures on a six-dimensional manifold can be lifted to parallel -structure on the product . We show that Hitchin’s approach can also be used to construct nearly parallel -structures by lifting so-called nearly half-flat structures. These -structures are described by pairs of stable 2- and 3-forms with , for some . [Copyright &y& Elsevier]

Abstract: By means of the orbit method we show that, for a compact Lie group, the Blattner–Kostant–Sternberg pairing map, with the constants being appropriately fixed, is unitary. Along the way we establish a holomorphic Peter–Weyl theorem for the complexification of a compact Lie group. Among our crucial tools is Kirillov’s character formula. The basic observation is that the Weyl vector is lurking behind the Kirillov character formula, as well as behind the requisite half-form correction on which the Blatter–Kostant–Sternberg-pairing for the compact Lie group relies, and thus produces the appropriate shift which, in turn, controls the unitarity of the BKS-pairing map. Our methods are independent of heat kernel harmonic analysis, which is used by B. C. Hall to obtain a number of these results [B.C. Hall, The Segal–Bargmann Coherent State Transform for compact Lie groups, J. Funct. Anal. 122 (1994) 103–151; B.C. Hall, Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type, Comm. Math. Phys. 226 (2002) 233–268, quant.ph/0012015]. [Copyright &y& Elsevier]

Abstract: In this paper, we carry a detailed study of mechanical systems with configuration space for which the base variables are being controlled. The overall system’s motion is considered to be induced from the base one due to the presence of general non-holonomic constraints. It is shown that the solution can be factorized into dynamical and geometrical parts. Moreover, under favorable kinematical circumstances, the dynamical part admits a further factorization since it can be reconstructed from an intermediate (body) momentum solution, yielding a reconstruction phase formula. Finally, we apply this results to the study of concrete mechanical systems. [Copyright &y& Elsevier]

Abstract: Suppose that the sphere has initially a homogeneous distribution of mass and let be the Lie group of orientation preserving projective diffeomorphisms of . A projective motion of the sphere, that is, a smooth curve in , is called force free if it is a critical point of the kinetic energy functional. We find explicit examples of force free projective motions of and, more generally, examples of subgroups of such that a force free motion initially tangent to remains in for all time (in contrast with the previously studied case for conformal motions, this property does not hold for ). The main tool is a Riemannian metric on , which turns out to be not complete (in particular not invariant, as happens with non-rigid motions), given by the kinetic energy. [Copyright &y& Elsevier]

Abstract: We show that every bad orbifold vector bundle can be realized as the restriction of a good orbifold vector bundle to a suborbifold of the base space. We give an explicit construction of this result in which the Chen–Ruan orbifold cohomologies of the two base spaces are isomorphic (as additive groups). This construction is used to indicate an extension of the Chern–Weil construction of characteristic classes to bad orbifold vector bundles. In particular, we apply this construction to the orbifold Euler class and demonstrate that it acts as an obstruction to the existence of nonvanishing sections. [Copyright &y& Elsevier]

Abstract: In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogeneous geodesics on the compact semi-simple Lie group is established. We introduce the notion of a naturally reductive homogeneous Finsler space. As a special case, we study homogeneous geodesics in homogeneous Randers spaces. Finally, we study some curvature properties of homogeneous geodesics. In particular, we prove that the S-curvature vanishes along the homogeneous geodesics. [Copyright &y& Elsevier]

Abstract: The local structure of a 3-dimensional essentially weakly para-cosymplectic manifold is described in two ways: using special adapted local frames and special coordinate systems. This enables a description of the curvature of such manifolds. Local isometries and Killing vector fields are also investigated. It is proved that if a 3-dimensional weakly para-cosymplectic manifold is locally homogeneous as a Riemannian manifold, then it is para-cosymplectic or locally flat. Then a classification of such manifolds is given. [Copyright &y& Elsevier]

Abstract: We construct infinitely many seven-dimensional Einstein metrics of weak holonomy . These metrics are defined on principal SO(3) bundles over four-dimensional Bianchi IX orbifolds with the Tod–Hitchin metrics. The Tod–Hitchin metric has an orbifold singularity parametrized by an integer, and is shown to be similar near the singularity to the Taub-NUT de Sitter metric with a special charge. We show, however, that the seven-dimensional metrics on the total space are actually smooth. The geodesics on the weak manifolds are discussed. It is shown that the geodesic equation is equivalent to the Hamiltonian equation of an interacting rigid body system. We also discuss M-theory on the product space of AdS4 and the seven-dimensional manifolds, and the dual gauge theories in three dimensions. [Copyright &y& Elsevier]