Transgression on hyperkähler manifolds and generalized higher torsion forms

We propose a generalization of the Hodge dd_c-lemma to the case of hyperk?hler manifolds. As an application we derive a global construction of the fourth order transgression of the Chern character forms of hyperholomorphic bundles over compact hyperk?hler manifolds. In Section 3 we consider the fourth order transgression for the infinite-dimensional bundle arising from local families of hyperk?hler manifolds. We propose a local construction of the fourth order transgression of the Chern character form. We derive an explicit expression for the arising hypertorsion differential form. Its zero-degree part may be expressed in terms of the Laplace operators defined on the fibres of the local family.     

An index theorem for Toeplitz operators on odd-dimensional manifolds with boundary

We establish an index theorem for Toeplitz operators on odd-dimensional spin manifolds with boundary. It may be thought of as an odd-dimensional analogue of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. In particular, there occurs naturally an invariant of η type associated to K¹ representatives on even-dimensional manifolds, which should be of independent interests. For example, it gives an intrinsic interpretation of the so called Wess-Zumino term in the WZW theory in physics.     

A RANK THEOREM FOR NONLINEAR SEMI-FREDHOLM OPERATORS BETWEEN TWO BANACH MANIFOLDS

In this article the concept of local conjugation of a C1 mapping between two Banach manifolds is introduced. Then a rank theorem for nonlinear semi-Fredholm operators between two Banach manifolds and a finite rank theorem are established in global analysis.     

Exotic Expansions and Pathological Properties of <Emphasis Type="Italic">ζ</Emphasis>-Functions on Conic Manifolds

We give a complete classification and present new exotic phenomena of the meromorphic structure of ζ-functions associated to general self-adjoint extensions of Laplace-type operators over conic manifolds. We show that the meromorphic extensions of these ζ-functions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. The corresponding heat kernel and resolvent trace expansions also exhibit exotic behaviors with logarithmic terms of arbitrary positive and negative multiplicity. We also give a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities.     

Some remarks about 1-convex manifolds on which all holomorphic line bundles are trivial

We give an abridged proof of an example already considered in [M. Col?oiu, On 1-convex manifolds with 1-dimensional exceptional set, Rev. Roumaine Math. Pures et Appl. 43 (1998) 97-104] of a 1-convex manifold X of dimension 3 such that all holomorphic line bundles on X are trivial. We also point out several mistakes of [Vo Van Tan, On the quasiprojectivity of compactifiable strongly pseudoconvex manifolds, Bull. Sci. Math. 129 (2005) 501-522] concerning this topic.     

p-Dirac Operators

We introduce non-linear Dirac operators in associated to the p-harmonic equation and we extend to other contexts including spin manifolds and the sphere. This paper is dedicated to the memory of Jarolim Bureš Received: October, 2007. Accepted: February, 2008.     

Elliptic Symbols

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 = Spin^c(n) (spin^c 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 Spin^h(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.     

Spin Structures on Flat Manifolds with Cyclic Holonomy

We study spin structures on flat Riemannian manifolds. The main result is a necessary and sufficient condition for a flat manifold with cyclic holonomy to have a spin structure.     

Levi--Tanaka algebras and homogeneous C R manifolds

In this paper we take up the problem of discussing C R manifolds of arbitrary C R codimension. We closely follow the general method of N.~Tanaka, while concentrating our attention to the case of manifolds endowed with partial complex structures. This study required a deeper understanding of the structure of the Levi--Tanaka algebras, which are the canonical prolongation of pseudocomplex fundamental graded Lie algebras. These algebras enjoy special properties, the understanding of which provided also a way to build up several different examples and points to a rich field of investigations. Here we restrained further our consideration to the homogeneous models.     

KO-obstructions of non-abelian group action on spin manifolds

In this paper, we give some KO-obstructions of non-Abelian group action on spin manifolds. These are closely related to the existence of metrics of positive scalar curvature on spin manifolds.     

Dixmier traces on noncompact isospectral deformations

We extend the isospectral deformations of Connes, Landi and Dubois-Violette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group Rl. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of functions on the manifold. We show that this relation persists for actions of Rl, under mild restrictions on the geometry of the manifold which guarantee the Dixmier traceability of those operators.     

On Kähler-Norden manifolds

This paper is concerned with the problem of the geometry of Norden manifolds. Some properties of Riemannian curvature tensors and curvature scalars of Kähler-Norden manifolds using the theory of Tachibana operators is presented.     

Spectral triples and manifolds with boundary

We investigate manifolds with boundary in noncommutative geometry. Spectral triples associated to a symmetric differential operator and a local boundary condition are constructed. We show that there is no tadpole for classical Dirac operators with a chiral boundary condition on spin manifolds.     

Killing vector fields of constant length on Riemannian manifolds

We study the nontrivial Killing vector fields of constant length and the corresponding flows on complete smooth Riemannian manifolds. Various examples are constructed of the Killing vector fields of constant length generated by the isometric effective almost free but not free actions of S ¹ on the Riemannian manifolds close in some sense to symmetric spaces. The latter manifolds include “almost round” odd-dimensional spheres and unit vector bundles over Riemannian manifolds. We obtain some curvature constraints on the Riemannian manifolds admitting nontrivial Killing fields of constant length.     

Spectral bounds for Dirac operators on open manifolds

We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich’s estimate for manifolds with positive scalar curvature as well as the author’s estimate on surfaces.      

Scalar Curvature Rigidity for Asymptotically Locally Hyperbolic Manifolds

Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asymptotic Killing spinors is related to the spin structure on the end. The expression for the mass is calculated and proven to vanish for conformally compact Einstein manifolds with conformal boundary a spherical space form, giving rigidity. In the four dimensional case, the signature of the manifold is related to the spin structure on the end and explicit formulas for the relevant invariants are given.     

A Callias-Type Index Theorem with Degenerate Potentials

A generalization of Callias’ index theorem for self adjoint Dirac operators with skew adjoint potentials on asymptotically conic manifolds is presented in which the potential term may have constant rank nullspace at infinity. The index obtained depends on the choice of a family of Fredholm extensions, though as in the classical version it depends only on the data at infinity.     

Microbundles, manifolds and metrisability

The notion of a microbundle was introduced in the 1960s but the theory came to an abrupt halt when it was shown that for a metrisable manifold, microbundles are equivalent to fibre bundles. In this paper we consider microbundles over non-metrisable manifolds. In some cases microbundles are equivalent to fibre bundles but in others they are not. In particular, we show that a manifold is metrisable if and only if its tangent microbundle is equivalent to a fibre bundle. We also illustrate that for some non-metrisable manifolds every trivial microbundle contains a trivial fibre bundle whereas other manifolds may support a trivial microbundle not containing a trivial fibre bundle.

     

Gerbes,Clifford Modules and the Index Theorem

The use of bundle gerbes and bundle gerbe modules is considered as a replacement for the usual theory of Clifford modules on manifolds that fail to be spin. It is shown that both sides of the Atiyah-Singer index formula for coupled Dirac operators can be given natural interpretations using this language and that the resulting formula is still an identity.     

Conformally invariant powers of the Dirac operator in Clifford analysis

The paper deals with conformally invariant higher‐order operators acting on spinor‐valued functions, such that their symbols are given by powers of the Dirac operator. A general classification result proves that these are unique, up to a constant multiple. A general construction for such an invariant operators on manifolds with a given conformal spin structure was described in (Conformally Invariant Powers of the Ambient Dirac Operator. ArXiv math.DG/0112033, preprint), generalizing the case of powers of the Laplace operator from (J. London Math. Soc. 1992; 46 :557–565). Although there is no hope to obtain explicit formulae for higher powers of the Laplace or Dirac operator on a general manifold, it is possible to write down an explicit formula on Einstein manifolds in case of the Laplace operator (see Laplacian Operators and Curvature on Conformally Einstein Manifolds. ArXiv: math/0506037, 2006). Here we shall treat the spinor case on the sphere. We shall compute the explicit form of such operators on the sphere, and we shall show that they coincide with operators studied in (J. Four. Anal. Appl. 2002; 8 (6):535–563). The methods used are coming from representation theory combined with traditional Clifford analysis techniques. Copyright © 2010 John Wiley & Sons, Ltd.