Complex geometry and real geometry

There is a well-known way to generalize the Riemann-Roch operator for Kähler manifold to that for Hermitian manifold. In this paper we show a slightly different way to get a generalized Riemann-Roch operator, which is just the Dirac operator. The difference between the two operators is that the latter one enables the so-called Pythagoras equalities.     

Functional determinants and geometry

Summary In this paper we study the zeta-function determinant in the context of elliptic boundary value problems. Our main technique is to relate the determinant of an operator, or a ratio of determinants, to the boundary values of the solutions of the operator. This has the advantage of restricting attention to the solutions of the operator, which do not depend on the boundary conditions and can often be written down explicitly, rather than the eigenvalues, which are usually difficult to work with. In addition, the problem is reduced to a calculation over the boundary of the manifold which is a closed manifold of dimension one less than the original manifold. This has special significance in the case that the manifold is a finite interval. In this case the boundary is a pair of points and the determinant of an ordinary differential operator is expressed in terms of the determinant of a finite matrix.The results are then applied to some geometric operators. In Sect. 4 we study the Jacobi operator acting along a geodesic segment and the covariant derivative operator acting along a loop. In Sect. 2 we calculate the determinant of the Laplacian acting on sections of a flat bundle over a flat torus. This can be related to an Eisenstein series and thus we have presented a new geometric method of summing such series. This sum is known as Kroneker's Second Limit formula. We then consider operators on a product manifoldM×S ¹.     

SUB-SIGNATURE OPERATORS, η-INVARIANTS AND A RIEMANN-ROCH THEOREM FOR FLAT VECTOR BUNDLES

The author presents an extension of the Atiyah-Patodi-Singer invariant for unitary representations [2, 3] to the non-unitary case, as well as to the case where the base manifold admits certain finer structures. In particular, when the base manifold has a fibration structure, a Riemann-Roch theorem for these invariants is established by computing the adiabatic limits of the associated η-invariants.     

Spectral geometry for Riemannian foliations

Let F be a Riemannian foliation on a Riemannian manifold (M, g), with bundle-like metric g. Aside from the Laplacian △_g associated to the metric g, there is another differential operator, the Jacobi operator J▽, which is a second order elliptic operator acting on sections of the normal bundle. Its spectrum is discrete as a consequence of the compactness of M. Hence one has two spectra, spec (M, g) = spectrum of △_g (acting on functions), and spec (F, J▽) = spectrum of J▽. We discuss the following problem: Which geometric properties of a Riemannian foliation F on a Riemannian manifold (M, g) are determined by the two types of spectral invariants?     

Cusp geometry and the cobordism invariance of the index

The cobordism invariance of the index on closed manifolds is reproved using the calculus Ψ_c of cusp pseudodifferential operators on a manifold with boundary. More generally, on a compact manifold with corners, the existence of a symmetric cusp differential operator of order 1 and of Dirac type near the boundary implies that the sum of the indices of the induced operators on the hyperfaces is null.     

Riemann-Roch spaces of the Hermitian function field with applications to algebraic geometry codes and low-discrepancy sequences

This paper is concerned with two applications of bases of Riemann-Roch spaces. In the first application, we define the floor of a divisor and obtain improved bounds on the parameters of algebraic geometry codes. These bounds apply to a larger class of codes than that of Homma and Kim (J. Pure Appl. Algebra 162 (2001) 273). Then we determine explicit bases for large classes of Riemann-Roch spaces of the Hermitian function field. These bases give better estimates on the parameters of a large class of m-point Hermitian codes. In the second application, these bases are used for fast implementation of Xing and Niederreiter's method (Acta. Arith. 72 (1995) 281) for the construction of low-discrepancy sequences.     

Affine symplectic geometry I: Applications to geometric inequalities

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     

C0 coarse geometry and scalar curvature

In this paper we introduce an alternative form of coarse geometry on proper metric spaces, which is more delicate at infinity than the standard metric coarse structure. There is an assembly map from the K-homology of a space to the K-theory of the C∗-algebra associated to the new coarse structure, which factors through the coarse K-homology of the space (with the new coarse structure). A Dirac-type operator on a complete Riemannian manifold M gives rise to a class in K-homology, and its image under assembly gives a higher index in the K-theory group. The main result of this paper is a vanishing theorem for the index of the Dirac operator on an open spin manifold for which the scalar curvature κ(x) tends to infinity as x tends to infinity. This is derived from a spectral vanishing theorem for any Dirac-type operator with discrete spectrum and finite dimensional eigenspaces.     

An elliptic theory of indicial weights and applications to non-linear geometry problems

Given an elliptic operator P on a non-compact manifold (with proper asymptotic conditions), there is a discrete set of numbers called indicial roots. It's known that P is Fredholm between weighted Sobolev spaces if and only if the weight is not indicial. We show that an elliptic theory exists even when the weight is indicial. We also discuss some simple applications to Yang–Mills theory and minimal surfaces.     

A fixed point formula of Lefschetz type in Arakelov geometry I: statement and proof

We consider arithmetic varieties endowed with an action of the group scheme of n-th roots of unity and we define equivariant arithmetic K ₀-theory for these varieties. We use the equivariant analytic torsion to define direct image maps in this context and we prove a Riemann-Roch theorem for the natural transformation of equivariant arithmetic K ₀-theory induced by the restriction to the fixed point scheme; this theorem can be viewed as an analog, in the context of Arakelov geometry, of the regular case of the theorem proved by P. Baum, W. Fulton and G. Quart in [BaFQ]. We show that it implies an equivariant refinement of the arithmetic Riemann-Roch theorem, in a form conjectured by J.-M. Bismut (cf. [B2, Par. (l), p. 353] and also Ch. Soulé’s question in [SABK, 1.5, p. 162]). Oblatum 22-I-1999 & 20-II-2001?Published online: 4 May 2001     

On the spectral geometry for the Jacobi operators of harmonic maps into a quaternionic projective space

We characterize both invariant and totally real immersions into the quaternionic projective space by the spectra of the Jacobi operator. Also, we study spectral characterization of harmonic submersions when the target manifold is the quaternionic projective space.     

Nonstrictly hyperbolic systems of conservation laws of the conjugate type

We consider an inverse boundary problem for a general second order self-adjoint elliptic differential operator on a compact differential manifold with boundary. The inverse problem is that of the reconstruction of the manifold and operator via all but finite number of eigenvalues and traces on the boundary of the corresponding eigenfunctions of the operator. We prove that the data determine the manifold and the operator to within the group of the generalized gauge transformations. The proof is based upon a procedure of the reconstruction of a canonical object in the orbit of the group. This object, the canonical Schrödinger operator, is uniquely determined via its incomplete boundary spectral data.     

Dirichlet to Neumann operator on differential forms

We define the Dirichlet to Neumann operator on exterior differential forms for a compact Riemannian manifold with boundary and prove that the real additive cohomology structure of the manifold is determined by the DN operator. In particular, an explicit formula is obtained which expresses Betti numbers of the manifold through the DN operator. We express also the Hilbert transform through the DN map. The Hilbert transform connects boundary traces of conjugate co-closed forms.     

Relations Between the Data of the Dynamical and Spectral Inverse Problems

As is known, the boundary spectral data of a compact Riemannian manifold with boundary are determined by its dynamical boundary data (the response operator of the wave equation) corresponding to any time interval: the response operator is represented in the form of a series over spectral data. The converse is true in the following sense: the response operator determines the manifold and, thus, its spectral data. To find these latter, one can reconstruct the manifold and then solve the (direct) boundary spectral problem. Obviously, such a way is not efficient and the question arises of whether one can extract the spectral data from the response operator without solving the inverse problem (without reconstructing the manifold). In the paper, a positive answer is given and a direct time-optimal procedure of extracting the spectral data from the response operator based on a variational principle is proposed. Bibliography: 9 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 30–48.     

Pontrjagin numbers and pseudo-riemannian geometry

Summary An integral formula for the Pontrjagin numbers of a compact orientable real 4k dimensional differentiable manifold which has a pseudo-Riemannian metric is derived. This formula allows the Pontrjagin numbers to be expressed in terms of the index, or signature, of the differentiable manifold. The application of these formulae to the four dimensional Lorentzian manifolds of the general theory of relativity is discussed. A corresponding formula for the Chern numbers of a complex differentiable manifold with a Hermitian metric is also given.     

On Riemann-Roch Formulas for Multiplicities

A theorem of Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups on compact Kähler manifolds says that the dimension of the -invariant subspace is equal to the Riemann-Roch number of the symplectic quotient. Combined with the shifting-trick, this gives explicit formulas for the multiplicities of the various irreducible components. One of the assumptions of the theorem is that the reduction is regular, so that the reduced space is a smooth symplectic manifold. In this paper, we prove a generalization of this result to the case where the reduced space may have orbifold singularities. The result extends to non-Kählerian settings, if one defines the representation by the equivariant index of the -Dirac operator associated to the quantizing line bundle.

     

Canonical Lorentzian spin structure and twistor spinors on the Fefferman space of a contact Riemannian manifold

The Fefferman space of a contact Riemannian manifold carries a Lorentzian spin structure canonically. On the Lorentzian spin manifold, we investigate the Dirac operator and the twistor operator closely. In particular, we show that, if the contact Riemannian manifold is integrable, then there exist non-zero global solutions of the twistor equation.     

Puzzle geometry and rigidity: The Fibonacci cycle is hyperbolic

We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and ``complex bounds', two generalized polynomial-like maps which admit a topological conjugacy, quasiconformal outside the filled-in Julia set, are indeed quasiconformally conjugate. The proof uses a new abstract removability-type result for quasiconformal maps, following ideas of Heinonen and Koskela and of Kallunki and Koskela, optimized for applications in complex dynamics. We prove, as the first application of this new method, that, for even criticalities distinct from two, the period two cycle of the Fibonacci renormalization operator is hyperbolic with -dimensional unstable manifold.