Canonical Sasakian Metrics

Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L ²-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that only the standard polarization can be represented by a Sasaki-Einstein metric. During the preparation of this work, the first two authors were partially supported by NSF grant DMS-0504367.     

Extremal Sasakian horizons

We point out a simple construction of an infinite class of Einstein near-horizon geometries in all odd dimensions greater than five. Cross-sections of the horizons are inhomogeneous Sasakian metrics (but not Einstein) on S³×S²$S^3\times S^2$ and more generally on S³$S^3$-bundles over any compact positive Kähler–Einstein manifold. They are all consistent with the known topology and symmetry constraints for asymptotically flat or globally AdS black holes.     

Sasakian and Parabolic Higgs Bundles

Let M be a quasi-regular compact connected Sasakian manifold, and let N = M/S ¹ be the base projective variety. We establish an equivalence between the class of Sasakian G–Higgs bundles over M and the class of parabolic (or equivalently, ramified) G–Higgs bundles over the base N.     

Magnetic curves in Sasakian manifolds

In this paper we classify the magnetic trajectories corresponding to contact magnetic fields in Sasakian manifolds of arbitrary dimension. Moreover, when the ambient is a Sasakian space form, we prove that the codimension of the curve may be reduced to 2. This means that the magnetic curve lies on a 3-dimensional Sasakian space form, embedded as a totally geodesic submanifold of the Sasakian space form of dimension (2n+1).     

Spectral geometry of eta-Einstein Sasakian manifolds

We extend a result of Patodi for closed Riemannian manifolds to the context of closed contact manifolds by showing the condition that a manifold is an η$\eta$-Einstein Sasakian manifold is spectrally determined. We also prove that the condition that a Sasakian space form has constant ?$?$-sectional curvature c$c$ is spectrally determined.     

On holomorphic maps and Generalized Complex Geometry

We introduce a natural notion of holomorphic map between generalized complex manifolds and we prove some related results on Dirac structures and generalized Kähler manifolds.     

On Summability of Distributions and Spectral Geometry

Modulo the moment asymptotic expansion, the Cesáro and parametric behaviours of distributions at infinity are equivalent. On the strength of this result, we construct the asymptotic analysis for spectral densities arising from elliptic pseudodifferential operators. We show how Cesáro developments lead to efficient calculations of the expansion coefficients of counting number functionals and Green functions. The bosonic action functional proposed by Chamseddine and Connes can more generally be validated as a Cesáro asymptotic development. Received: 10 February 1997 / Accepted: 8 May 1997     

On the Noncommutative Geometry of Twisted Spheres

We describe noncommutative geometric aspects of twisted deformations, in particular of the spheres of Connes and Landi and of Connes and Dubois Violette, by using the differential and integral calculus on these spaces that is covariant under the action of their corresponding quantum symmetry groups. We start from multiparametric deformations of the orthogonal groups and related planes and spheres. We show that only in the twisted limit of these multiparametric deformations the covariant calculus on the plane gives, by a quotient procedure, a meaningful calculus on the sphere. In this calculus, the external algebra has the same dimension as the classical one. We develop the Haar functional on spheres and use it to define an integral of forms. In the twisted limit (differently from the general multiparametric case), the Haar functional is a trace and we thus obtain a cycle on the algebra. Moreover, we explicitly construct the *-Hodge operator on the space of forms on the plane and then by quotient on the sphere. We apply our results to even spheres and compute the Chern–Connes pairing between the character of this cycle, i.e. a cyclic 2n-cocycle, and the instanton projector defined in math.QA/0107070.     

Liouville theorem for pseudoharmonic maps from Sasakian manifolds

In this paper, we derive a sub-gradient estimate for pseudoharmonic maps from noncompact complete Sasakian manifolds which satisfy the CR sub-Laplace comparison property, to simply-connected Riemannian manifolds with nonpositive sectional curvature. As its application, we obtain some Liouville theorems for pseudoharmonic maps. In the Appendix, we modify the method and apply it to harmonic maps from noncompact complete Sasakian manifolds.     

Holomorphicity and the Walczak formula on Sasakian manifolds

The Walczak formula is a very nice tool for understanding the geometry of a Riemannian manifold equipped with two orthogonal complementary distributions. M. Svensson [Holomorphic foliations, harmonic morphisms and the Walczak formula, J. London Math. Soc. (2) 68 (3) (2003) 781–794] has shown that this formula simplifies to a Bochner-type formula when we are dealing with Kähler manifolds and holomorphic (integrable) distributions. We show in this paper that such results have a counterpart in Sasakian geometry. To this end, we build on a theory of (contact) holomorphicity on almost contact metric manifolds. Some other applications for (pseudo-)harmonic morphisms on Sasaki manifolds are outlined.     

Holomorphicity and the Walczak formula on Sasakian manifolds

The Walczak formula is a very nice tool for understanding the geometry of a Riemannian manifold equipped with two orthogonal complementary distributions. M. Svensson [Holomorphic foliations, harmonic morphisms and the Walczak formula, J. London Math. Soc. (2) 68 (3) (2003) 781–794] has shown that this formula simplifies to a Bochner-type formula when we are dealing with Kähler manifolds and holomorphic (integrable) distributions. We show in this paper that such results have a counterpart in Sasakian geometry. To this end, we build on a theory of (contact) holomorphicity on almost contact metric manifolds. Some other applications for (pseudo-)harmonic morphisms on Sasaki manifolds are outlined.     

On the Integral Geometry of Liouville Billiard Tables

The notion of a Radon transform is introduced for completely integrable billiard tables. In the case of Liouville billiard tables of dimension 3 we prove that the Radon transform is one-to-one on the space of continuous functions K on the boundary which are invariant with respect to the corresponding group of symmetries. We prove also that the frequency map associated with a class of Liouville billiard tables is non-degenerate. This allows us to obtain spectral rigidity of the corresponding Laplace-Beltrami operator with Robin boundary conditions.     

On the Geometry of Variational Principles of Physics

The essentials of the invariant mathematical apparatus used for geometrization of basic variational principles of physics and mechanics are presented. An important connection between the geometry of action functionals and the theory of fiber spaces that provides the mathematical basis for modern gauge theories of fundamental interactions is established.     

On the Geometry of Generalized Robertson-Walker Spacetimes: Geodesics

The geometry and, especially, the geodesics of a class of spacetimes generalizing Robertson-Walker ones (without any assumption on the fiber) is studied, under a global point of view. Our study covers geodesic connectedness, geodesic completeness and stability of completeness.     

On the Finite Spectral Triple of an Almost-Commutative Geometry

In this short communication, we examine the relevance of the signature of the space-time metric in the construction of the product of a pseudo-Riemannian spectral triple with a finite triple describing the internal geometry. We obtain arguments favoring the appearance of SU(2) and U(1) as gage groups in the standard model.     

On Semi-Classical States of Quantum Gravity and Noncommutative Geometry

We construct normalizable, semi-classical states for the previously proposed model of quantum gravity which is formulated as a spectral triple over holonomy loops. The semi-classical limit of the spectral triple gives the Dirac Hamiltonian in 3+1 dimensions. Also, time-independent lapse and shift fields emerge from the semi-classical states. Our analysis shows that the model might contain fermionic matter degrees of freedom.     

On the Geometry of Diffusion Processes in Continuized Dislocated Crystals

Nonequilibrium diffusion processes of point defects in continuized dislocated crystals are considered. A generalized, stochastically motivated gauge procedure to introduce the geometry of a material space describing the influence of dislocations on the free diffusion process, is used. The dependence of diffusing processes and their steady states on the curvature of the material space of edge dislocations, on the scalar density of these dislocations, and on the interaction energy between dislocations and a diffusing atom, is analysed. An equation defining the interaction energy is deduced, using statistical arguments, from the material space geometry.