Proposition 1.2 (Splitting Principle). Let E → M be a complex vector bundle of rank n over a manifold X. Then there is a manifold F (E) and a smooth fibration π : F (E) → M such that • The homomorphism π∗ : H∗(M, Z) → H∗(F (E), Z) is injective. • The bundle π∗E splits into a direct sum of complex line bundles, i.e. π∗E = l1 ⊕ · · ·⊕ ln Proof. Consider the projectivisation p : PC(E) → M and the ...

For a uniformly hyperbolic diffeomorphism f , the induced volume deformation φu in the unstable subbundle over a compact f -invariant set significantly characterizes the geometry of the set as well as the dynamics in its neighborhood. Under the hypothesis of uniform hyperbolicity, and particularly for hyperbolic surface diffeomorphisms, a large number of dynamical quantifiers such as, for examp...

This paper studies the construction of geometric integrators for nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange equations in a setting which permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold on it. Additionally, it ...

Let (M,ω0) be a compact Kähler manifold of dimension n and D a divisor of M with at most normal crossing singularities . Let D = ∪i=1Di , where the Di are smooth divisors of M . Denote M \ D by M , called a quasi-compact Kähler manifold. j : M → M is the inclusion mapping. According to [3], one can construct a complete Kähler metric ω on M which is a Poincaré-like metric near the divisor and of...

In this note it is shown that the Maslov Index for pairs of Lagrangian Paths as introduced by Cappell, Lee and Miller ([1]) appears by parallel transporting elements of (a certain complex line-subbundle of) the symplectic spinorbundle over Euclidean space, when pulled back to an (embedded) Lagrangian submanifold L, along closed or non-closed paths therein. More precisely, the CLM-Index mod 4 de...

It is shown that every bundle ! M of complex, irreducible and faithful modules over the Cliiord bundle of an even-dimensional Riemannian space (M; g) with local model (V; h) is associated with a cpin (\Cliiord") structure on M, this being an extension of the SO(h)-bundle of orthonormal frames on M to the Cliiord group Cpin(h) = (C Spin(h))=Z 2. An explicit construction is given of the total spa...

A Lorentzian manifold is defined here as a smooth pseudoRiemannian manifold with a metric tensor of signature (2n + 1, 1). A Robinson manifold is a Lorentzian manifold M of dimension > 4 with a subbundle N of the complexification of TM such that the fibers of N → M are maximal totally null (isotropic) and [SecN,SecN ] ⊂ SecN . Robinson manifolds are close analogs of the proper Riemannian, Hermi...

Let C be a nonsingular irreducible projective curve of genus g ≥ 2 defined over the complex numbers. Suppose that 1 ≤ n ′ ≤ n − 1 and n ′ d − nd ′ = n ′ (n − n ′)(g − 1). It is known that, for the general vector bundle E of rank n and degree d, the maximal degree of a subbundle of E of rank n ′ is d ′ and that there are finitely many such subbundles. We obtain a formula for the number of these ...

abstract : horizontal root fractures in permanent teeth are uncommon injuries among dental traumas. the principle of treating horizontal root fractures of permanent teeth is repositioning and fixation of fractured segment. diagnosis of tooth fractures based on radiographic findings and the fracture healing process are affected by factors before and after injury. the present case reports the tre...

Let $(M,g^{TM})$ be a noncompact complete Riemannian manifold of dimension $n$, and let $F\subseteq TM$ an integrable subbundle $TM$. $g^F=g^{TM}|_{F}$ the restricted metric on $F$ $k^F$ associated leafwise scalar curvature. $f:M\to S^n(1)$ smooth area decreasing map along $F$, which is locally constant near infinity non-zero degree. We show that if $k^F> {\rm rk}(F)({\rm rk}(F)-1)$ support ${\...