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1.
We generalize Turaev's definition of torsion invariants of pairs (M,&\xi;), where M is a 3-dimensional manifold and &\xi; is an Euler structure on M (a non-singular vector field up to homotopy relative to ∂M and modifications supported in a ball contained in Int(M)). Namely, we allow M to have arbitrary boundary and &\xi; to have simple (convex and/or concave) tangency circles to the boundary. We prove that
Turaev's H
1(M)-equivariance formula holds also in our generalized context. Using branched standard spines to encode vector fields we show
how to explicitly invert Turaev's reconstruction map from combinatorial to smooth Euler structures, thus making the computation
of torsions a more effective one. Euler structures of the sort we consider naturally arise in the study of pseudo-Legendrian
knots (i.e.~knots transversal to a given vector field), and hence of Legendrian knots in contact 3-manifolds. We show that torsion, as
an absolute invariant, contains a lifting to pseudo-Legendrian knots of the classical Alexander invariant. We also precisely
analyze the information carried by torsion as a relative invariant of pseudo-Legendrian knots which are framed-isotopic.
Received: 3 October 2000 / Revised version: 20 April 2001 相似文献
2.
I. G. Korepanov 《Journal of Mathematical Sciences》2007,144(5):4437-4445
We construct knot invariants on the basis of ascribing Euclidean geometric values to a triangulation of the sphere S
3, where the knot lies. Edges of the triangulation along which the knot goes are distinguished by a nonzero deficit angle,
in the terminology of the Regge calculus.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 105–117, 2005. 相似文献
3.
I. G. Korepanov 《Theoretical and Mathematical Physics》2009,158(3):344-354
We generalize the construction of invariants of three-dimensional manifolds with a triangulated boundary that we previously
proposed for the case where the boundary consists of not more than one connected component to any number of components. These
invariants are based on the torsion of acyclic complexes of geometric origin. An adequate tool for studying such invariants
turns out to be Berezin’s calculus of anticommuting variables; in particular, they are used to formulate our main theorem,
concerning the composition of invariants under a gluing of manifolds. We show that the theory satisfies a natural modification
of Atiyah’s axioms for anticommuting variables.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 405–418, March, 2009. 相似文献
4.
The goal of this paper is to establish a geometric program to study elliptic pseudodifferential boundary problems which arise naturally under cutting and pasting of geometric and spectral invariants of Dirac-type operators on manifolds with corners endowed with multi-cylindrical, or b-type, metrics and ‘b-admissible’ partitioning hypersurfaces. We show that the Cauchy data space of a Dirac operator on such a manifold is Lagrangian for the self-adjoint case, the corresponding Calderón projector is a b-pseudodifferential operator of order 0, characterize Fredholmness, prove relative index formulæ, and solve the Bojarski conjecture.
Mathematics Subject Classifications (2000): 58J28, 58J52. 相似文献
5.
M. Cencelj Yu. V. Muranov D. Repovš 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2006,76(1):35-55
The problem of splitting a homotopy equivalence along a submanifold is closely related to the surgery exact sequence and to
the problem of surgery of manifold pairs. In classical surgery theory there exist two approaches to surgery in the category
of manifolds with boundaries. In the rel ∂ case the surgery on a manifold pair is considered with the given fixed manifold
structure on the boundary. In the relative case the surgery on the manifold with boundary is considered without fixing maps
on the boundary. Consider a normal map to a manifold pair (Y, ∂Y) ⊂ (X, ∂X) with boundary which is a simple homotopy equivalence on the boundary∂X. This map defines a mixed structure on the manifold with the boundary in the sense of Wall. We introduce and study groups
of obstructions to splitting of such mixed structures along submanifold with boundary (Y, ∂Y). We describe relations of these groups to classical surgery and splitting obstruction groups. We also consider several geometric
examples. 相似文献
6.
The existence of a full asymptotic expansion for the heat content asymptotics of an operator of Laplace type with classical
Zaremba boundary conditions on a smooth manifold is established. The first three coefficients in this asymptotic expansion
are determined in terms of geometric invariants; partial information is obtained about the fourth coefficient.
相似文献
7.
Given a compact smooth manifold M with non-empty boundary and a Morse function, a pseudo-gradient Morse-Smale vector field adapted to the boundary allows one
to build a Morse complex whose homology is isomorphic to the (absolute or relative to the boundary) homology of M with integer coefficients. Our approach simplifies other methods which have been discussed in more specific geometric settings. 相似文献
8.
This paper is concerned with the relations between the differential invariants of a smooth manifold embedded in the Euclidean
space and the square of the distance function from the manifold. In particular, we are interested in curvature invariants
like the mean curvature vector and the second fundamental form. We find that these invariants can be computed in a very simple
way using the third order derivatives of the squared distance function. Moreover, we study a general class of functionals
depending on the derivatives up to a given order γ of the squared distance function and we find an algorithm for the computation
of the Euler equation. Our class of functionals includes as particular cases the well-known area functional (γ = 2), the integral
of the square of the quadratic norm of the second fundamental form (γ = 3), and the Willmore functional. 相似文献
9.
We introduce a notion of weak isospectrality for continuous deformations. Consider the Laplace–Beltrami operator on a compact Riemannian manifold with Robin boundary conditions. Given a Kronecker invariant torus Λ of the billiard ball map with a Diophantine vector of rotation we prove that certain integrals on Λ involving the function in the Robin boundary conditions remain constant under weak isospectral deformations. To this end we construct continuous families of quasimodes associated with Λ. We obtain also isospectral invariants of the Laplacian with a real-valued potential on a compact manifold for continuous deformations of the potential. These invariants are obtained from the first Birkhoff invariant of the microlocal monodromy operator associated to Λ. As an application we prove spectral rigidity of the Robin boundary conditions in the case of Liouville billiard tables of dimension two in the presence of a (?/2?)2 group of symmetries. 相似文献
10.
Maria Hermínia de Paula Leite Mello Mário Olivero Marques da Silva 《manuscripta mathematica》2000,101(2):191-198
In this paper we consider the question of the existence of a nonstable vector bundle monomorphism u:α→β over a closed, connected and smooth manifold M, when dimension of α= 3, dimension of β= dimension of M=n≡ 0(4). The singularity method provides the full obstruction to this problem and under some homological hypothesis we can
compute it in terms of well known invariants.
Received: 31 May 1999 相似文献
11.
Let M be a compact smooth Riemannian manifold with smooth boundary. We establish the existence of an asymptotic series for the heat content of M with a perfect conducting boundary and show that the coefficients in the series are non-local invariants which are recursively determined by the coefficients for the series with corresponding zero Dirichlet boundary condition. 相似文献
12.
In this article, we study the limiting behavior of the Brown–York mass and Hawking mass along nearly round surfaces at infinity
of an asymptotically flat manifold. Nearly round surfaces can be defined in an intrinsic way. Our results show that the ADM
mass of an asymptotically flat three manifold can be approximated by some geometric invariants of a family of nearly round
surfaces, which approach to infinity of the manifold. 相似文献
13.
Simon A. King 《Topology and its Applications》2007,154(6):1141-1156
Turaev-Viro invariants are defined via state sum polynomials associated to a special spine or a triangulation of a compact 3-manifold. By evaluation of the state sum at any solution of the so-called Biedenharn-Elliott equations, one obtains a homeomorphism invariant of the manifold (“numerical Turaev-Viro invariant”). The Biedenharn-Elliott equations define a polynomial ideal. The key observation of this paper is that the coset of the state sum polynomial with respect to that ideal is a homeomorphism invariant of the manifold (“ideal Turaev-Viro invariant”), stronger than the numerical Turaev-Viro invariants. Using computer algebra, we obtain computational results on several examples of ideal Turaev-Viro invariants, for all closed orientable irreducible manifolds of complexity at most 9. 相似文献
14.
Alexander G. Reznikov 《Israel Journal of Mathematics》1992,80(1-2):207-224
We use the integral geometric formulas in the symplectic space of geodesics of a Riemannian manifold to derive various inequalities
of isoperimetric type. We give a sharp lower bound for the area of the minimal bubble spanning a spherical curve in ℝ3. We also present an “inverse Croke inequality” relating the area of the boundary of a complex domain in a Riemannian manifold
to the injectivity radius and the volume of the domain. We prove a sharp lower bound for the ground state of the harmonic
oscillator operator inL
2(M), whereM is a Hadamard manifold.
This article is dedicated to my dear friend Julia Rashba 相似文献
15.
D. Burghelea T. Kappeler P. McDonald L. Friedlander 《Geometric And Functional Analysis》1996,6(5):751-859
For a closed Riemannian manifold (M, g) we extend the definition of analytic and Reidemeister torsion associated to a unitary representation of 1 (M) on a finite dimensional vector space to a representation on aA-Hilbert moduleW of finite type whereA is a finite von Neumann algebra. If (M,W) is of determinant class we prove, generalizing the Cheeger-Müller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, theL
2-analytic andL
2-Reidemeister torsions are equal.The first three authors were supported by NSF. The first two authors wish to thank the Erwin-Schrödinger-Institute, Vienna, for hospitality and support during the summer of 1993 when part of this work was done. 相似文献
16.
In this work, we describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented
punctured surface Σ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of
a triple of flags in the complex hyperbolic plane
H2\mathbb C{\bf H^2_{\mathbb {C}}} . We establish a bijection between a set of decorations of an ideal triangulation of Σ and a subset of the PU(2,1)-representation
variety of π
1(Σ). 相似文献
17.
A vector field on Riemannian manifold is called conformal Killing if it generates oneparameter group of conformal transformation. The class of conformal Killing symmetric tensor fields of
an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometric
and physical problems. In this paper, we prove that a trace-free conformal Killing tensor field is identically zero if it
vanishes on some hypersurface. This statement is a basis of the theorem on decomposition of a symmetric tensor field on a
compact manifold with boundary to a sum of three fields of special types. We also establish triviality of the space of trace-free
conformal Killing tensor fields on some closed manifolds. 相似文献
18.
Tim D. Cochran 《Commentarii Mathematici Helvetici》1985,60(1):291-311
A geometric notion of a “derivative” is defined for 2-component links ofS
n inS
n+2 and used to construct a sequenceβ
i
,i=1,2,... of abelian concordance invariants which vanish for boundary links. Forn>1, these generalize the only heretofore known invariant, the Sato-Levine invariant. Forn=1, these invariants are additive under any band-sum and consequently provide new information about which 1-links are concordant
to boundary links. Examples are given of concordance classes successfully distinguished by theβ
i
but not by their
, Murasugi 2-height, Sato-Levine invariant or Alexander polynomial.
Supported in part by a grant from the National Science Foundation. 相似文献
19.
We study some special almost complex structures on strictly pseudoconvex domains in ℝ2
n
. They appear naturally as limits under a nonisotropic scaling procedure and play a role of model objects in the geometry
of almost complex manifolds with boundary. We determine explicitely some geometric invariants of these model structures and
derive necessary and sufficient conditions for their integrability. As applications we prove a boundary extension and a compactness
principle for some elliptic diffeomorphisms between relatively compact domains. 相似文献
20.
Thomas Lorenz 《Set-Valued Analysis》2008,16(1):1-50
The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order
geometric evolutions, however, the topological boundary need not be continuous in the sense of Painlevé–Kuratowski. So this
paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional
analysis beyond the traditional border of vector spaces: Distribution-like solutions are introduced in a set just supplied
with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler
approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological
vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of ℝ
N
evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity
assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions
for the uniqueness of these solutions.
相似文献