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...

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 ...

In previous work n-symplectic geometry on the adapted frame bundle λ : LπE → E of an n = (m + k)-dimensional fiber bundle π : E → M has been used to forumulate covariant Lagrangian field theory that is standardly formulated on the bundle J1π of 1 jets of sections of π. In this paper we set up an n-symplectic Hamilton-Jacobi equation in order to identify the analogue of a polarization that plays...

Let X be a smooth projective curve of genus g ≥ 2 and S(r, d) the moduli scheme of all rank r stable vector bundles of degree d on X. Fix an integer k with 0 < k < r. H. Lange introduced a natural stratification of S(r, d) using the degree of a rank k subbundle of any E ∈ S(r, d) with maximal degree. Every non-dense stratum, say W (k, r − k, a, d − a), has in a natural way a fiber structure h :...

For an oriented isometric immersion f : M → S the spherical Gauss map is the Legendrian immersion of its unit normal bundle UM⊥ into the unit sphere subbundle of TS, and the geodesic Gauss map γ projects this into the manifold of oriented geodesics in S (the Grassmannian of oriented 2-planes in R), giving a Lagrangian immersion of UM⊥ into a Kähler-Einstein manifold. We give expressions for the...

Let M denote a compact, orientable 3–manifold and let a denote a smooth 1–form on M such that a^ da is nowhere zero. Such a 1–form is called a contact form. The associated Reeb vector field is the section, v , of TM that generates the kernel of da and pairs with a to give 1. The generalized Weinstein conjecture in dimension three asserts that v has at least one closed integral curve (see Weinst...

In the study of submanifolds in a homogeneous space X = G/P of a Lie group G, the method of moving frames is both a unifying concept and an effective tool, which is the version of the method of equivalence applied to submanifold geometry. Let φ be the left invariant, Lie algebra g-valued Maurer-Cartan form of G. The local equivalence problem for a submanifold f : M →֒ X is solved on a canonical ...

We consider the category of anchored Banach vector bundles and we discuss the notion of semispray. Adding on the set of sections of an anchored Banach vector bundle a Lie bracket with some properties one gets the notion of Lie algebroid. We prove that the Lie algebroids form also a category. A Dirac structure on a Banach manifold M is defined as a subbundle of the big tangent bundle TM ⊕ T ∗M t...

The Lagrange-d’Alembert equations of a non-holonomic system with symmetry can be reduced to the Lagrange-d’Alembert-Poincaré equations. In a previous contribution we have shown that both sets of equations fall in the category of so-called ‘Lagrangian systems on a subbundle of a Lie algebroid’. In this paper, we investigate the special case when the reduced system is again invariant under a new ...

In recent joint work with András Vasy [19], we analyze the low energy behavior of differential form-valued waves on black hole spacetimes. In order to deduce asymptotics and decay from this, one in addition needs high energy estimates for the wave operator acting on sections of the form bundle. The present paper provides these on perturbations of Schwarzschild-de Sitter spaces in all spacetime ...