Let M be a C∞ manifold (without corners) and let E ⊆ TM be a subbundle of the tangent bundle. In class we discussed the notion of integral manifolds for E in M (as well as maximal ones), essentially as a generalization of the theory of integral curves for vector fields. Roughly speaking, in the special case that E is a trivial line bundle we are in the setup of integral curves for (non-vanishin...

We exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy. Introduction Consider a distribution on a manifold M, i.e. subbundle of the tangent bundle Π ⊂ TM . Non-holonomic Riemannian metric is a Riemannian metric g ∈ SΠ on this bundle. We call the pair (Π, g) sub-Riemannian structure. A curve γ : [0, 1] → M is called horizontal if γ̇ is a sect...

We study the reduced symplectic cohomology of disk subbundles in negative line bundles. show that this theory “sees” spectrum a quantum action on cohomology. Precisely, decomposes into generalized eigenspaces first Chern class by cup product. The bundle radius R sees all whose eigenvalues have size less than R, up to rescaling fixed constant. Similarly, we an annulus subbundle between radii R1 ...

We construct a model space C(sp(R 2n)) for the variety of Abelian simply transitive groups of affine transformations of type Sp(R 2n). The model is stratified and its principal stratum is a Zariski-open subbundle of a natural vector bundle over the Grassmannian of Lagrangian subspaces in R 2n. Next we show that every flat special Kähler manifold may be constructed locally from a holomorphic fun...

We extend the definition of involutivity to non-Lipschitz tangent subbundles using generalized functions. prove Frobenius Theorem with sharp regularity estimate when subbundle is log-Lipschitz: if $${\mathcal {V}}$$ a log-Lipschitz involutive rank r, then for any $$\varepsilon >0$$ , locally there homeomorphism $$\Phi (u,v)$$ such that ,\frac{\partial \Phi }{\partial u^1},\dots u^r}\in C^{0,1-\...

We show in this paper that every bijective linear isometry between the continuous section spaces of two non-square Banach bundles gives rise to a Banach bundle isomorphism. This is to support our expectation that the geometric structure of the continuous section space of a Banach bundle determines completely its bundle structures. We also describe the structure of an into isometry from a contin...

In this article we show that when the structure group of the reducible principal bundle P is SU(r) and Q ⊂ P is an SO(r)-subbundle of P , the rank of the holonomy group of a connection which is gauge equivalent to its conjugate connection is less than or equal to [ r 2 ] , and use the estimate to show that for all odd prime r, if the holonomy group of the irreducible connection as above is simp...

This paper considers control systems defined on Lie algebroids. After deriving basic controllability tests for general control systems, we specialize our discussion to the class of mechanical control systems on Lie algebroids. This class of systems includes mechanical systems subject to holonomic and nonholonomic constraints, mechanical systems with symmetry and mechanical systems evolving on s...

The main theorem of this paper asserts that the inclusion of the space of projective Lagrangian planes into the space of Lagrangian submanifolds of complex projective space induces an injective homomorphism of fundamental groups. We introduce three invariants of exact loops of Lagrangian submanifolds that are modelled on invariants introduced by Polterovich for loops of Hamiltonian symplectomor...