806 research outputs found
Perestroikas of Shocks and Singularities of Minimum Functions
The shock discontinuities, generically present in inviscid solutions of the
forced Burgers equation, and their bifurcations happening in the course of time
(perestroikas) are classified in two and three dimensions -- the
one-dimensional case is well known. This classification is a result of
selecting among all the perestroikas occurring for minimum functions depending
generically on time, the ones permitted by the convexity of the Hamiltonian of
the Burgers dynamics. Topological restrictions on the admissible perestroikas
of shocks are obtained. The resulting classification can be extended to the
so-called viscosity solutions of a Hamilton--Jacobi equation, provided the
Hamiltonian is convex.Comment: 20 pages, 8 figures, 3 tables; my e-mail: [email protected]
Maslov class and minimality in Calabi-Yau manifolds
Generalizing the construction of the Maslov class for a Lagrangian embedding
in a symplectic vector space, we prove that it is possible to give a consistent
definition of this class for any Lagrangian submanifold of a Calabi-Yau
manifold. Moreover, we prove that this class can be represented by the
contraction of the Kaehler form associated to the Calabi-Yau metric, with the
mean curvature vector field of the Lagrangian embedding. Finally, we suggest a
possible generalization of the Maslov class for Lagrangian submanifolds of any
symplectic manifold, via the mean curvature representation.Comment: 16 pages To be published in Journal of Geometry and Physic
Toward a general theory of linking invariants
Let N_1, N_2, M be smooth manifolds with dim N_1 + dim N_2 +1 = dim M$ and
let phi_i, for i=1,2, be smooth mappings of N_i to M with Im phi_1 and Im phi_2
disjoint. The classical linking number lk(phi_1,phi_2) is defined only when
phi_1*[N_1] = phi_2*[N_2] = 0 in H_*(M).
The affine linking invariant alk is a generalization of lk to the case where
phi_1*[N_1] or phi_2*[N_2] are not zero-homologous. In arXiv:math.GT/0207219 we
constructed the first examples of affine linking invariants of
nonzero-homologous spheres in the spherical tangent bundle of a manifold, and
showed that alk is intimately related to the causality relation of wave fronts
on manifolds.
In this paper we develop the general theory. The invariant alk appears to be
a universal Vassiliev-Goussarov invariant of order < 2. In the case where
phi_1*[N_1] and phi_2*[N_2] are 0 in homology it is a splitting of the
classical linking number into a collection of independent invariants.
To construct alk we introduce a new pairing mu on the bordism groups of
spaces of mappings of N_1 and N_2 into M, not necessarily under the restriction
dim N_1 + dim N_2 +1 = dim M. For the zero-dimensional bordism groups, mu can
be related to the Hatcher-Quinn invariant. In the case N_1=N_2=S^1, it is
related to the Chas-Sullivan string homology super Lie bracket, and to the
Goldman Lie bracket of free loops on surfaces.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper42.abs.htm
Lagrangian concordance of Legendrian knots
In this article we define Lagrangian concordance of Legendrian knots, the
analogue of smooth concordance of knots in the Legendrian category. In
particular we study the relation of Lagrangian concordance under Legendrian
isotopy. The focus is primarily on the algebraic aspects of the problem. We
study the behavior of the classical invariants under this relation, namely the
Thurston-Bennequin number and the rotation number, and we provide some examples
of non-trivial Legendrian knots bounding Lagrangian surfaces in . Using
these examples, we are able to provide a new proof of the local Thom
conjecture.Comment: 18 pages, 4 figures. v2: Minor corrections and a proof of conjecture
6.4 of version 1 (now Theorem 6.4). v3: Several substantial changes notably
the proof of theorems 1.2 and 5.1, this is the version accepted for
publication in "Algebraic and Geometric Topology" published in January 201
The homology of the Milnor fiber for classical braid groups
In this paper we compute the homology of the braid groups, with coefficients
in the module Z[q^+-1] given by the ring of Laurent polynomials with integer
coefficients and where the action of the braid group is defined by mapping each
generator of the standard presentation to multiplication by -q.
The homology thus computed is isomorphic to the homology with constant
coefficients of the Milnor fiber of the discriminantal singularity.Comment: This is the version published by Algebraic & Geometric Topology on 14
November 200
Cyclicity in families of circle maps
In this paper we will study families of circle maps of the form xâŠx+2Ïr+af(x)(mod2Ï) and investigate how many periodic trajectories maps from this family can have for a âtypicalâ function f provided the parameter a is small
Geodesic flow, left-handedness, and templates
We establish that, for every hyperbolic orbifold of type (2, q, ) and
for every orbifold of type (2, 3, 4g+2), the geodesic flow on the unit tangent
bundle is left-handed. This implies that the link formed by every collection of
periodic orbits (i) bounds a Birkhoff section for the geodesic flow, and (ii)
is a fibered link. We also prove similar results for the torus with any flat
metric. Besides, we observe that the natural extension of the conjecture to
arbitrary hyperbolic surfaces (with non-trivial homology) is false.Comment: Version accepted for publication (Algebraic & Geometric Topology), 60
page
On Integrability of spinning particle motion in higher-dimensional black hole spacetimes
We study the motion of a classical spinning particle (with spin degrees of
freedom described by a vector of Grassmann variables) in higher-dimensional
general rotating black hole spacetimes with a cosmological constant. In all
dimensions n we exhibit n bosonic functionally independent integrals of
spinning particle motion, corresponding to explicit and hidden symmetries
generated from the principal conformal Killing--Yano tensor. Moreover, we
demonstrate that in 4-, 5-, 6-, and 7-dimensional black hole spacetimes such
integrals are in involution, proving the bosonic part of the motion integrable.
We conjecture that the same conclusion remains valid in all higher dimensions.
Our result generalizes the result of Page et. al. [hep-th/0611083] on complete
integrability of geodesic motion in these spacetimes.Comment: Version 2: revised version, added references. 5 pages, no figure
On the viability of local criteria for chaos
We consider here a recently proposed geometrical criterion for local
instability based on the geodesic deviation equation. Although such a criterion
can be useful in some cases, we show here that, in general, it is neither
necessary nor sufficient for the occurrence of chaos. To this purpose, we
introduce a class of chaotic two-dimensional systems with Gaussian curvature
everywhere positive and, hence, locally stable. We show explicitly that chaotic
behavior arises from some trajectories that reach certain non convex parts of
the boundary of the effective Riemannian manifold. Our result questions, once
more, the viability of local, curvature-based criteria to predict chaotic
behavior.Comment: 10 page
Multiplicative properties of Morin maps
In the first part of the paper we construct a ring structure on the rational
cobordism classes of Morin maps (i. e. smooth generic maps of corank 1). We
show that associating to a Morin map its singular strata defines a ring
homomorphism to \Omega_* \otimes \Q, the rational oriented cobordism ring.
This is proved by analyzing multiple-point sets of product immersion. Using
these homomorphisms we are able to identify the ring of Morin maps.
In the second part of the paper we compute the oriented Thom polynomial of
the singularity type with \Q coefficients. Then we provide a
product formula for the and the singularities.Comment: Corrected some small misprints and made lot of minor (mainly
grammatical) alterations. 10 page
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