Gravity in non-commutative geometry

We study general relativity in the framework of non-commutative differential geometry. As a prerequisite we develop the basic notions of non-commutative Riemannian geometry, including analogues of Riemannian metric, curvature and scalar curvature. This enables us to introduce a generalized Einstein-Hilbert action for non-commutative Riemannian spaces. As an example we study a space-time which is the product of a four dimensional manifold by a two-point space, using the tools of non-commutative Riemannian geometry, and derive its generalized Einstein-Hilbert action. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.Dedicated to H. ArakiSupported in part by the Swiss National Foundation (SNF)     

7-Dimensional compact Riemannian manifolds with Killing spinors

Using a link between Einstein-Sasakian structures and Killing spinors we prove a general construction principle of odd-dimensional Riemannian manifolds with real Killing spinors. In dimensionn=7 we classify all compact Riemannian manifolds with two or three Killing spinors. Finally we classify nonflat 7-dimensional Riemannian manifolds with parallel spinor fields.     

Cohomological tautness for Riemannian foliations

In this paper, we present new results on the tautness of Riemannian foliations in their historical context. The first part of the paper gives a short history of the problem. For a closed manifold, the tautness of a Riemannian foliation can be characterized cohomologically. We extend this cohomological characterization to a class of foliations which includes the foliated strata of any singular Riemannian foliation of a closed manifold.     

A Geometric Approach to the Classification of the Equilibrium Shapes of Self-Gravitating Fluids

The classification of the equilibrium shapes that a self-gravitating fluid can take in a Riemannian manifold is a classical problem in Mathematical Physics. In this paper it is proved that the equilibrium shapes are isoparametric submanifolds. Some geometric properties of them are also obtained, e.g. classification and existence for some Riemannian spaces and relationship with the isoperimetric problem and the group of isometries of the manifold. Our approach to the problem is geometrical and allows to study the equilibrium shapes on general Riemannian spaces.     

Post-Newtonian limit of Finsler space theories of gravity and solar system tests

Finsler geometry is considered as a wider framework for analysing solar system tests of theories of gravity than is afforded by Riemannian geometry. The post-Newtonian limit for the spherically symmetric one-body problem is examined by expanding the Finsler metric about the Minkowski space of Special Relativity for those Finsler spaces whose null surface is Riemannian. In such a framework there are five PPN parameters instead of the three in Riemannian geometry. The classical solar system tests can readily be satisfied leaving two arbitrary parameters. These parameters could be determined from measurements of the second order gravitational red-shift and periodic perturbations in particle orbits, thus providing a consistency check on the Riemannian metric hypothesis of General Relativity. Such an experiment is possible on a satellite on an orbit with perihelion of a few solar radii.     

Twistor spinors on Lorentzian symmetric spaces

An indecomposable Riemannian symmetric space which admits non-trivial twistor spinors has constant sectional curvature. Furthermore, each homogeneous Riemannian manifold with parallel spinors is flat. In the present paper we solve the twistor equation on all indecomposable Lorentzian symmetric spaces explicitly. In particular, we show that there are — in contrast to the Riemannian case — indecomposable Lorentzian symmetric spaces with twistor spinors, which have non-constant sectional curvature and non-flat and non-Ricci flat homogeneous Lorentzian manifolds with parallel spinors.     

Growth conditions, Riemannian completeness and Lorentzian causality

The stationary-Randers correspondence (SRC) provides a deep connection between the property of standard stationary spacetimes being globally hyperbolic, and the completeness of certain Finsler metrics of Randers type defined on the fibres. In order to establish further results, we investigate pointwise conformal transformations of certain Riemannian metrics on the fibres and growth conditions on the corresponding conformal factors. In general, a conformal transformation may map a complete Riemannian metric onto a complete or incomplete metric. We prove a theorem for the growth of the conformal factor such that the conformally transformed Riemannian metric is also complete. As an application, we establish novel relations between the completeness of Riemannian metrics, growth conditions on conformal factors and the Cauchy hypersurface condition on the fibres of a standard stationary spacetime. These results also imply new conditions for the completeness of Randers-type metrics by the application of the SRC.     

Twistor forms on Riemannian products

We study twistor forms on products of compact Riemannian manifolds and show that they are defined by Killing forms on the factors. The main result of this note is a necessary step in the classification of compact Riemannian manifolds with non-generic holonomy carrying twistor forms.     

On the Flag Curvature of Invariant Randers Metrics

In the present paper, the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups is studied. We first give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and, in special case, Lie groups. We then study Randers metrics of constant positive flag curvature and complete underlying Riemannian metric on Lie groups. Finally we give some properties of those Lie groups which admit a left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field.      

Quadratic metric‐affine gravity

We consider spacetime to be a connected real 4‐manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is (purely) quadratic in curvature and study the resulting system of Euler–Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi‐Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with pp‐wave metric of parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non‐Riemannian solutions. We define the notion of a “Weyl pseudoinstanton” (metric compatible spacetime whose curvature is purely of Weyl type) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non‐Riemannian solution which is a wave of torsion in a spacetime with Minkowski metric. We discuss the possibility of using this non‐Riemannian solution as a mathematical model for the neutrino.     

On the Dirac spectrum of Riemannian foliations admitting a basic parallel 1-form

In this paper, in the special setting of a Riemannian foliation with basic, non-necessarily harmonic mean curvature we introduce a Weitzenböck-Lichnerowicz type formula which allows us to apply the classical Bochner-Lichnerowicz technique. We show that the lower bound for the eigenvalues of the basic Dirac operator can be calculated using only classical techniques. As another application, for general Riemannian foliations we calculate the above eigenvalue bound in the presence of a basic parallel 1-form, as an extension of a known result on a closed Riemannian manifold. Some results concerning the limiting case are obtained in the final part of the paper.     

The isometry groups of simply connected 3-dimensional unimodular Lie groups

For each simply connected 3-dimensional unimodular Lie group the left invariant Riemannian metrics up to automorphism are classified in Ha and Lee (2009) [2]. Using this classification, we determine on each such a Lie group the full group of isometries which depends on left invariant Riemannian metrics.     

A remark about static space times

We discuss a differential equation, whose unknowns are a function and a Riemannian metric. This equation occurs both in general relativity (static space times) and in the study of the space of Riemannian metrics on a manifold (singularities of the map from the space of metrics into the space of functions, which assigns to any metric its scalar curvature).     

Dislocations and internal length measurement in continuized crystals. I. Riemannian material space

Distributions of dislocations creating point defects are considered. These point defects are described by a metric tensor, which supplements a Burgers field responsible for dislocations. The metric tensor depends on the distribution of dislocations and defines a Riemannian geometry of the material space of a continuized crystal and thus an internal length measurement in this crystal. The dependence of the distribution of dislocations on the existence of point defects created by these dislocations is modeled by treating the Burgers field as a field defined on the Riemannian material space. Field equations, following from geometric identities, are formulated as balance equations on this Riemannian space and their source terms, responsible for interactions of dislocations and point defects, are identified.     

On the geometrical interpretation of the theory of gravitation in flat space

In the framework of the flat space-time approach to gravitational theory the equations of motion of point particles are derived from a suitable Lagrangian. Attention is paid to the distinction of the different worldline parameters in Minkowskian und Riemannian space-time. It is shown that the world-lines of point particles are geodesics in a Riemannian space-time and permit a consistent geometrical interpretation of the theory.     

A correspondence for isometric immersions into product spaces and its applications

We present a correspondence for isometric immersions that are graphs in Riemannian or semi-Riemannian warped product spaces. We use this correspondence to give several existence and non-existence theorems for hypersurfaces in Riemannian or Lorentzian spaces. In the case of surfaces, we obtain further applications regarding height estimates, harmonic representation of surfaces and the existence of holomorphic quadratic differentials in homogeneous and non-homogeneous spaces.     

On the Algebraic Index for Riemannian Étale Groupoids

In this paper, we construct an explicit quasi-isomorphism to study the cyclic cohomology of a deformation quantization over a Riemannian étale groupoid. Such a quasi-isomorphism allows us to propose a general algebraic index problem for Riemannian étale groupoids. We discuss solutions to that index problem when the groupoid is proper or defined by a constant Dirac structure on a 3-dimensional torus.     

Calibrated complex structures on the generalized tangent bundle of a Riemannian manifold

We study calibrated complex structures on the generalized tangent bundle of a Riemannian manifold M and their relationship to the Riemannian geometry of M. In particular we introduce a concept of integrability of such structures and we prove that integrability conditions are strictly related to the existence of certain Codazzi tensors on M.     

Getzler symbol calculus and deformation quantization

In this paper we give a construction of Fedosov quantization incorporating the odd variables and an analogous formula to Getzler’s pseudodifferential calculus composition formula is obtained. A Fedosov type connection is constructed on the bundle of Weyl tensor Clifford algebras over the cotangent bundle of a Riemannian manifold. The quantum algebra associated with this connection is used to define a deformation of the exterior algebra of Riemannian manifolds.