共查询到20条相似文献,搜索用时 15 毫秒
1.
The aim of this work is to show that in any complete Riemannian
manifold M, without boundary, the curvature operator is nonnegative
if and only if the Dirac Laplacian D2 generates a C*-Markovian
semigroup (i.e. a strongly continuous, completely positive, contraction
semigroup) on the Cliord C*-algebra of Mor, equivalently, if
and only if the quadratic form $\mathcal{E}$D of D2
is a C*-Dirichlet form. 相似文献
2.
Maxim Braverman 《K-Theory》2002,27(1):61-101
Let D be a (generalized) Dirac operator on a noncompact complete Riemannian manifold M acted on by a compact Lie group G. Let v: M g = Lie G be an equivariant map, such that the corresponding vector field on M does not vanish outside of a compact subset. These data define an element of K-theory of the transversal cotangent bundle to M. Hence, by embedding of M into a compact manifold, one can define a topological index of the pair (D,v) as an element of the completed ring of characters of G. We define an analytic index of (D,v) as an index space of certain deformation of D and we prove that the analytic and topological indexes coincide. As a main step of the proof, we show that index is an invariant of a certain class of cobordisms, similar to the one considered by Ginzburg, Guillemin and Karshon. In particular, this means that the topological index of Atiyah is also invariant under this class of noncompact cobordisms. As an application, we extend the Atiyah–Segal–Singer equivariant index theorem to our noncompact setting. In particular, we obtain a new proof of this theorem for compact manifolds. 相似文献
3.
S. Mehdi 《Advances in Mathematics》2006,199(1):1-28
Let G be a real reductive Lie group and G/H a reductive homogeneous space. We consider Kostant's cubic Dirac operator D on G/H twisted with a finite-dimensional representation of H. Under the assumption that G and H have the same complex rank, we construct a nonzero intertwining operator from principal series representations of G into the kernel of D. The Langlands parameters of these principal series are described explicitly. In particular, we obtain an explicit integral formula for certain solutions of the cubic Dirac equation D=0 on G/H. 相似文献
4.
Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and bounded diameter, and almost non-negative scalar
curvature. Let r = 1 if n = 2,3 and r = 2[n/2]-1 + 1 if n ≥ 4. We show that if the square of the Dirac operator on such a manifold has r small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a non-trivial spin structure, then there exists a uniform lower bound on the r-th eigenvalue of the square of the Dirac operator. If a manifold with almost non-negative scalar curvature has one small
Dirac eigenvalue, and if the volume is not too small, then we show that the metric is close to a Ricci-flat metric on M with a parallel spinor. In dimension 4 this implies that M is either a torus or a K3-surface.
相似文献
5.
Peter Franek 《Advances in Applied Clifford Algebras》2007,17(3):469-480
In this paper, we show the existence of a sequence of invariant differential operators on a particular homogeneous model G/P of a Cartan geometry. The first operator in this sequence is locally the Dirac operator in 2 Clifford variables, D = (D
1, D
2), where D
i
= ∑
j
e
j
. ∂
ij
. It follows from the construction that this operator is invariant with respect to the action of the group G. There are 2 other G-invariant differential operators following it so that the sequence of operators is exact. We compute the local expression
of these operators and show that it coincides with the operators described in [2, 5, 6] by the tools of Clifford analysis.
However, it follows from our approach that the operators are invariant.
The work presented here was supported by the grants GAUK 447/2004 and GA ČR 201/05/H005. 相似文献
6.
S. Mehdi 《Journal of Functional Analysis》2003,198(2):536-557
Let G/H be a semisimple symmetric space. We consider a Dirac operator D on G/H twisted by a finite dimensional H-representation. We give an explicit integral formula for certain solutions of the equation D=0. In particular, some quotients of standard principal series representations are seen to occur in the kernel of D. 相似文献
7.
J?drzej ?niatycki 《Journal of Fixed Point Theory and Applications》2011,10(2):339-358
We consider a Dirac structure D on a manifold Q invariant under a proper action of a Lie group G on Q. Our aim is to describe the structure of the orbit space D/G in terms of the structure of Q/G. 相似文献
8.
《Differential Geometry and its Applications》2003,18(1):21-32
Let D be the Dirac operator on a compact spin manifold M. Assume that 0 is in the spectrum of D. We prove the existence of a lower bound on the first positive eigenvalue of D depending only on the spin structure and the conformal type. 相似文献
9.
Assume that the compact Riemannian spin manifold (Mn,g) admits a G-structure with characteristic connection ∇ and parallel characteristic torsion (∇T=0), and consider the Dirac operator D1/3 corresponding to the torsion T/3. This operator plays an eminent role in the investigation of such manifolds and includes as special cases Kostant's “cubic Dirac operator” and the Dolbeault operator. In this article, we describe a general method of computation for lower bounds of the eigenvalues of D1/3 by a clever deformation of the spinorial connection. In order to get explicit bounds, each geometric structure needs to be investigated separately; we do this in full generality in dimension 4 and for Sasaki manifolds in dimension 5. 相似文献
10.
Christian Bär 《Geometriae Dedicata》1992,41(1):103-107
We show that the fundamental tone of D
2 is zero, where D is the classical Dirac operator on the hyperbolic space. 相似文献
11.
Margarita Kraus 《Annals of Global Analysis and Geometry》2001,19(3):235-257
We consider the Dirac operator on fibrations overS
1 which have up to holonomy a warped product metric. Wegive lower bounds for the eigenvalues on M and if the Diracoperator on the typical fibre F has a kernel, we calculatethe corresponding part of the spectrum on M explicitly.Moreover, we discuss the dependence of the spectrum of theholonomy and obtain bounds for the multiplicity of the eigenvalues. 相似文献
12.
Gong Guihua 《偏微分方程通讯》2013,38(1-2):341-362
In this paper, we discuss the following conjecture raised by Baum and Douglas: For any first order elliptic differential operator D on a smooth manifold M with boundary ?M D possesses a (local) elliptic boundary condition if and only if ?[D]=0 in K1(?M), where [D] is the relative K-cycle in Ko(M,?M) corresponding to D. We prove the “if” part of this conjecture for dim(M)≠4,5,6,7 and the “only if” part of the conjecture for arbitrary dimension. 相似文献
13.
We give a local proof of an index theorem for a Dirac-type operator that is invariant with respect to the action of a foliation groupoid G. If M denotes the space of units of G then the input is a G-equivariant fiber bundle P→M along with a G-invariant fiberwise Dirac-type operator D on P. The index theorem is a formula for the pairing of the index of D, as an element of a certain K-theory group, with a closed graded trace on a certain noncommutative de Rham algebra Ω*B associated to G. The proof is by means of superconnections in the framework of noncommutative geometry. 相似文献
14.
Multiplication operators on sobolev disk algebra 总被引:2,自引:0,他引:2
WANG Zongyao & LIU Yiqiang Department of Mathematics East China University of Science Technology Shanghai China 《中国科学A辑(英文版)》2005,48(10):1395-1410
In this paper,we study the algebra consisting of analytic functions in the Sobolev space W~(2,2) (D) (D is the unit disk),called the Sobolev disk algebra,explore the properties of the multiplication operators M_f on it and give the characterization of the corn- mutant algebra A′(M_f) of M_f.We show that A′(M_f) is commutative if and only if M_f~* is a Cowen-Douglas operator of index 1. 相似文献
15.
Christian Br 《Mathematische Nachrichten》1999,201(1):7-35
If G is the structure group of a manifold M it is shown how a certain ideal in the character ring of G corresponds to the set of geometric elliptic operators on M. This provides a simple method to construct these operators. For classical structure groups like G = O(n) (Riemannian manifolds), G = SO(n) (oriented Riemannian manifolds), G = U(m) (almost complex manifolds), G = Spin(n) (spin manifolds), or G = Spinc(n) (spinc manifolds) this yields well known classical operators like the Euler—deRham operator, signature operator, Cauchy—Riemann operator, or the Dirac operator. For some less well studied structure groups like Spinh(n) or Sp(q)Sp(1) we can determine the corresponding operators. As applications, we obtain integrality results for such manifolds by applying the Atiyah—Singer Index Theorem to these operators. Finally, we explain how immersions yield interesting structure groups to which one can apply this method. This yields lower bounds on the codimension of immersions in terms of topological data of the manifolds involved. 相似文献
16.
Lingli Wang 《Frontiers of Mathematics in China》2010,5(1):179-190
Let G be a finite group, and let π
e
(G) be the spectrum of G, that is, the set of all element orders of G. In 1987, Shi Wujie put forward the following conjecture. If G is a finite group and M is a non-abelian simple group, then G ≅ M if and only if |G| = |M| and π
e
(G) = π
e
(M). In this short paper, we prove that if G is a finite group, then G ≅ M if and only if |G| = |M| and π
e
(G) = π
e
(M), where M = D
n
(2) and n is even. 相似文献
17.
We consider the one-dimensional Dirac operator on a finite interval G = (a, b). We analyze the uniform componentwise equiconvergence of expansions in root vector functions of this operator with the trigonometric Fourier series on a compact set. Theorems on the componentwise equiconvergence on a compact set and the componentwise localization principle are proved. 相似文献
18.
A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext1(G, M) = 0 (resp. Tor1(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given. 相似文献
19.
Zdravka Božikov 《Archiv der Mathematik》2006,86(1):11-15
According to a classical result of Burnside, if G is a finite 2-group, then the Frattini subgroup Φ(G) of G cannot be a nonabelian group of order 8. Here we study the next possible case, where G is a finite 2-group and Φ(G) is nonabelian of order 16. We show that in that case Φ(G) ≅ M × C2, where M ≅ D8 or M ≅ Q8 and we shall classify all such groups G (Theorem A).
Received: 16 February 2005; revised: 7 March 2005 相似文献
20.
A graph G is minimal harmoniously colorable if it has a proper vertex coloring in which each pair of colors occurs exactly once on an edge. In particular, if D is a 2-design we consider the graph G whose vertices are the points and blocks of D and where two vertices of G are adjacent if and only if the corresponding elements of D are incident. It will be shown that if D is symmetric then G is minimal harmoniously colorable if and only if D is a Hadamard design with corresponding Hadamard matrix of a certain form. We obtain some results if D is nonsymmetric, and construct two classes of nonsymmetric minimal harmoniously colorable designs. © 1994 John Wiley & Sons, Inc. 相似文献