arXiv research

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

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 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.

For odd-dimensional spheres, there's always a second short geodesic.

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…

It is known that for every smooth great circle fibration of the 3-sphere, the distribution of tangent 2-planes orthogonal to the fibres is a contact structure, in fact a tight one, but we show here that, beginning with the 5-sphere, there exist smooth great circle fibrations of all odd-dimensional spheres for which the…

In this paper, we first give a direct proof for two recurrence relations of the heat kernels for hyperbolic spaces in \cite{DM}. Then, by similar computation, we give two similar recurrence relations of the heat kernels for spheres. Finally, as an application, we compute the diagonal of heat kernels for odd dimensional…

We compute the analytic torsion of a cone over a sphere of dimension 1, 2, and 3, and we conjecture a general formula for the cone over an odd dimensional sphere.

Study verifies Homogeneity Conjecture for three odd-dimensional spheres in positive curvature.

We prove the existence of exotic but homotopically trivial contact structures on spheres of dimension 8k-1. Together with previous results of Eliashberg and the second author this establishes the existence of such structures on all odd-dimensional spheres (of dimension at least 3).

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…

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…

Study magnetic geodesics on odd spheres, computing critical energy values.

Using the higher analytic torsion form of Bismut and Lott we construct a characteristic class for smooth sphere bundles. We calculate this class in the case where the sphere bundle comes from a complex vector bundle. Related to these characteristic classes we define nontrivial continuous group cohomology classes of the…

We present a boundary version of a theorem about solenoidal unit vector fields with minimum energy on a spherical domain of an odd dimensional Euclidean sphere.

We prove the existence of an abundance of new Einstein metrics on odd dimensional spheres including exotic spheres, many of them depending on continuous parameters. The number of families as well as the number of parameter grows double exponentially with the dimension. Our method of proof uses Brieskorn-Pham singularit…

A natural oriented (2k+2)-chain in CP^{2k+1} with boundary twice RP^{2k+1}, its complex shade, is constructed. Via intersection numbers with the shade, a new invariant, the shade number of k-dimensional subvarieties with normal vector fields along their real part, is introduced. For an even-dimensional real variety, th…

A theory of signatures for odd-dimensional links in rational homology spheres is studied via their generalized Seifert surfaces. The jump functions of signatures are shown invariant under appropriately generalized concordance and a special care is given to accommodate 1-dimensional links with mutual linking. Furthermor…

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…

Study CR-geometry analog of conformal volume for spheres' submanifolds.

We give explicit formulas for all odd order differential intertwinors on the subbundle of the bundle of spinor-$k$-forms that are annihilated by the Clifford multiplication over the odd dimensional standard sphere. The Dirac and Rarita-Schwinger operators appear in the case of $k=0$ and $k=1$, respectively.

This paper is based on a talk presented by the first author at the Short Program on Riemannian Geometry that took place at the Centre de Recherche Mathématiques, Université de Montréal, during the period June 28-July 16, 2004. It is a report on our joint work with János Kollár concerning the existence of an abundance o…

Unique symplectic fillings of odd spheres' cotangent bundles proven.

Here we study geodesics connecting two given points on odd-dimensional spheres respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of 3-dimensional sphere and some partial results are obtained in the general case. The Carnot-Carathéodory distance is calculated. We also p…

This paper surveys some of the known results on $δ$-ideal CR submanifolds in complex space forms, the nearly Kähler $6$-sphere and odd dimensional unit spheres. In addition, the relationship between $δ$-ideal CR submanifolds and critical points of the $λ$-bienergy is mentioned. Some topics on variational problem for th…

We give a dynamical characterisation of odd-dimensional balls within the class of all contact manifolds whose boundary is a standard even-dimensional sphere. The characterisation is in terms of the non-existence of short periodic Reeb orbits.

For two complex vector bundles admitting a homomorphism, whose singularity locates in the disjoint union of some odd--dimensional spheres, we give a formula to compute the relative Chern characteristic number of these two complex vector bundles. In particular, for a spin manifold admitting some sphere bundle structure,…

The integral of the top dimensional term of the multiplicative sequence of Pontryagin forms associated to an even formal power series is calculated for special Riemannian metrics on the unit ball of a hermitean vector space. Using this result we calculate the generating function of the reduced Dirac and signature eta-i…

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 …

The study constructs K-contact manifolds with minimal closed Reeb orbits and provides conditions for their homeomorphism to spheres.

A construction of Kaehler-Einstein metrics using Galois coverings, studied by Arezzo-Ghigi-Pirola, is generalized to orbifolds. By applying it to certain orbifold covers of P^n which are trivial set theoretically, one obtains new Einstein metrics on odd-dimensional spheres. The method also gives Kaehler-Einstein metric…

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…

New invariant computed for Brieskorn homology spheres.

The present paper studies globally defined Kropina metrics as solutions of the Zermelo's navigation problem. Moreover, we characterize the Kropina metrics of constant flag curvature showing that up to local isometry, there are only two model spaces of them: the Euclidean space and the odd-dimensional spheres.

A numerical expression in the form of an integral is given for the determinant of the scalar GJMS operator on an odd--dimensional sphere. Manipulation yields a curious sum formula for the logdet in terms of the logdets of the ordinary conformal Laplacian for other dimensions. A few graphs are drawn.

We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a parallelizable manifold. Furthermore, it is shown that on such homotopy spheres $\script…

We construct the first examples of non-Kähler complex structures on $R^4$. These complex surfaces have some analogies with the complex structures constructed in early Fifties by Calabi and Eckmann on the products of two odd-dimensional spheres. However, our construction is quite different from that of Calabi and Eckman…

We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi-Kahler quotient, for constructing Kahler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi-Kahler quotients of toric CR manifolds, and in particular, products of odd …

We describe several families of Lagrangian submanifolds in the complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendrian submanifolds of th…

New Sasaki metrics with constant scalar curvature on sphere bundles are constructed.

We characterize the Zoll Riemannian metrics on a given simply connected spin closed manifold as those Riemannian metrics for which two suitable min-max values in a finite dimensional loop space coincide. We also show that on odd dimensional Riemannian spheres, when certain pairs of min-max values in the loop space coin…

The question of whether a Sasakian metric can admit an additional compatible (K-)contact structure is addressed. In the complete case if the second structure is also assumed Sasakian, works of Tachibana-Yu and Tanno show that the manifold must be 3-Sasakian or an odd dimensional sphere with constant curvature. Some ext…

In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations are homeomorphic to the product of spheres. As a corollary, we obtain the chara…

Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal …

A free action of a finite group on an odd-dimensional sphere is said to be almost linear if the action restricted to each cyclic or 2-hyperelementary subgroup is conjugate to a free linear action. We begin this survey paper by reviewing the status of almost linear actions on the 3-sphere. We then discuss almost linear …

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

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

Proves another theorem for odd dimensional manifolds with boundary.