arXiv research

Paper extends index theorem to odd-dimensional manifolds with even-dimensional boundaries.

Proves another theorem for odd dimensional manifolds with boundary.

No contact Anosov diffeomorphisms exist on odd-dimensional manifolds.

We prove for closed, odd-dimensional GKM$_3$ manifolds of non-negative sectional curvature that both the equivariant and the ordinary rational cohomology split off the cohomology of an odd-dimensional sphere.

Proves two theorems on odd-dimensional manifolds with boundary.

A theorem on odd dimensional noncompact manifolds shows curvature bounds.

The study finds infinitely many different geometries for odd-dimensional manifolds with positive Ricci curvature.

In this note, we find the conditions on an odd-dimensional Riemannian manifolds under which its twistor space is eta-Einstein.

In this talk, we review the heat kernel approach to the Atiyah-Singer index theorem for Dirac operators on closed manifolds, as well as the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. We also discuss the odd dimensional counterparts of the above results. In particular, we describe…

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…

Odd-dimensional Riemannian manifolds admit pure spin-c Killing spinors if and only if they are α-Sasakian.

A notion of equivariant spectral flows for families of self-dual elliptic operators on Riemannian manifolds is purposed. As a consequence, a local version of a Lefschetz fix point theorem is proved for Toeplitz operators on odd-dimensional spin manifolds.

Paper solves long neck problem on odd-dimensional spin manifolds.

Constructs modular forms and proves divisibility results for odd-dimensional manifolds.

Odd-dimensional manifolds have contact maps of non-zero degree.

The study finds conditions for Sasakian manifolds and generalised Ricci solitons.

In a 1967 paper, Banchoff stated that a certain type of polyhedral curvature, that applies to all finite polyhedra, was zero at all vertices of an odd-dimensional polyhedral manifold; one then obtains an elementary proof that odd-dimensional manifolds have zero Euler characteristic. In a previous paper, the author defi…

Efficient multisections found for odd-dimensional tori.

We compute the transgressed forms of some modularly invariant characteristic forms,which are related to the twisted elliptic genera. We study the modularity properties of these secondary characteristic forms and relations among them. We also get some twisted anomaly cancellation formulas on some odd dimensional manifol…

The aim of this paper is to study Sasakian immersions of compact Sasakian manifolds into the odd-dimensional sphere equipped with the standard Sasakian structure. We obtain a complete classification of such manifolds in the Einstein and $η$-Einstein cases when the codimension of the immersion is $4$. Moreover, we exhib…

We establish several Witten type rigidity and vanishing theorems for twisted Toeplitz operators on odd dimensional manifolds. We obtain our results by combining the modular method, modular transgression and some careful analysis of odd Chern classes for cocycles in odd $K$-theory. Moreover we discover that in odd dimen…

We prove a homological stability theorem for the moduli spaces of manifolds diffeomorphic to $\#^{g}(S^{n+1}\times S^{n})$, provided $n \geq 4$. This is an odd dimensional analogue of a recent homological stability result of S. Galatius and O. Randal Williams for the moduli space of manifolds diffeomorphic to $\#^{g}(S…

New rigidity theorems for spin^c manifolds using modular invariance.

This paper is a continuation of our work on theta and zeta functions In the previous papers we considered the case of even dimensional rank one symmetric spaces of non-compact type. The present is concerned with the odd-dimensional case, i.e. with odd-dimensional real hyperbolic manifolds. It is the natural appearence …

Study finds infinite families of Sasaki-Einstein metrics on spheres.

The paper defines invariants for positive scalar curvature metrics on manifolds with boundary.

We give an exhaustive description of all simply connected odd dimensional cohomogeneity one manifolds that can possibly support an invariant metric with positive sectional curvature. Among the known examples of odd dimensional manifolds with positive curvature, apart from spheres, there are two infinite families among …

We study the analytic torsion of the cone over an orientable odd dimensional compact connected Riemannian manifold W. We prove that the logarithm of the analytic torsion of the cone decomposes as the sum of the logarithm of the root of the analytic torsion of the boundary of the cone, plus a topological term, plus a fu…

Defines new two-variable elliptic genera for manifolds and derives modular forms.

Odd-dimensional orbifolds' Euler characteristic equals half of their boundary's.

New nonacyclic Reidemeister torsions defined for odd-dimensional manifolds.

Study shows non-existence of certain contact structures on odd-dimensional manifolds.

We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions its eigenvalues are the electromagnetic oscillation frequencies in vacuum without external sources. In…

Study of invariants on manifolds with boundary involving equivariant spectral flow and η-invariants.

We develop isometry and inversion formulas for the Segal--Bargmann transform on odd-dimensional hyperbolic spaces that are as parallel as possible to the dual case of odd-dimensional spheres.

We prove a new rigidity result for an open manifold M with nonnegative sectional curvature whose soul S is odd-dimensional. Specifically, there exists a geodesic in S and a parallel vertical plane field along it with constant vertical curvature and vanishing normal curvature. Under the added assumption that the Sharafu…

Homotopy theory for $(2n+1)$-dimensional manifold triads with fixed boundary.

We study local Lie algebras of pairs of functions which generate infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds.

D.Freed has formulated and proved an index theorem on odd dimensional spin manifolds with boundary. The proof is based on analysis by Calderon and Seeley. In this note we are going to give a proof of this theorem using the heat kernels methods for boundary conditions of Dirichlet and Von Neumann type. Moreover we consi…

A Riemann-Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous space, the notion of homogeneous Riemann-Cartan space is introduced in a natura…

It is well known that the Euler characteristic of an odd dimensional compact manifold is zero. An Euler complex is a combinatorial analogue of a compact manifold. We present here an elementary proof of the corresponding result for Euler complexes.

Algorithm decides if odd-dimensional maps can be immersed.

We prove an asymptotic bound on the eta invariant of a family of coupled Dirac operators on an odd dimensional manifold. In the case when the manifold is the unit circle bundle of a positive line bundle over a complex manifold, we obtain precise formulas for the eta invariant.

We study Selberg zeta functions $Z(s,σ)$ associated to locally homogeneous vector bundles over the unit-sphere bundle of a complete odd-dimensional hyperbolic manifold of finite volume. We assume a certain condition on the fundamental group of the manifold. A priori, the Selberg zeta functions are defined only for s in…

We prove the equality of the $L^2$-analytic torsion and the intersection R torsion of the even dimensional finite metric cone over an odd dimensional compact manifold.

For even dimensional manifolds, we prove some twisted anomaly cancellation formulas which generalize some well-known cancellation formulas. For odd dimensional manifolds, we obtain some modularly invariant characteristic forms by the Chern-Simons transgression and we also get some twisted anomaly cancellation formulas.

We study the existence of closed geodesics on compact Riemannian orbifolds, and on noncompact Riemannian manifolds in the presence of a cocompact, isometric group action. We show that every noncontractible Riemannian manifold which admits such an action, and every odd-dimensional, compact Riemannian orbifold has a nont…

In this paper, we use the flag curvature formula for homogeneous Finsler spaces in our previous work to classify odd dimensional smooth coset spaces admitting positively curved reversible homogeneous Finsler metrics. We will show that the most features of L. Bérard-Bergery's classification results for odd dimensional p…