Riemannian Signature Research Articles

Pseudo-Riemannian Geodesic Orbit Nilmanifolds of Signature (n-2,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{(n-2,2)}$$\\end{document}

The geodesic orbit property is useful and interesting in itself, and it plays a key role in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here we extend Riemannian and Lorentz results to trans-Lorentz nilmanifolds. Those are the geodesic orbit pseudo Riemannian manifolds M=G/H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M = G/H$$\\end{document} of signature (n-2,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-2,2)$$\\end{document} such that a nilpotent analytic subgroup of G is transitive on M. For that we suppose that there is a reductive decomposition g=h⊕n(vector space direct sum) with[h,n]⊂n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {g}}= {\\mathfrak {h}}\\oplus {\\mathfrak {n}}\ ext { (vector space direct sum) with } [{\\mathfrak {h}},{\\mathfrak {n}}] \\subset {\\mathfrak {n}}$$\\end{document} and n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {n}}$$\\end{document} nilpotent. When the metric is nondegenerate on [n,n]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[{\\mathfrak {n}},{\\mathfrak {n}}]$$\\end{document} we show that n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {n}}$$\\end{document} is abelian or 2-step nilpotent. That is the same result as for geodesic orbit Riemannian and Lorentz nilmanifolds. When the metric is degenerate on [n,n]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[{\\mathfrak {n}},{\\mathfrak {n}}]$$\\end{document} we show that n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {n}}$$\\end{document} is a double extension of a geodesic orbit nilmanifold of either Riemannian or Lorentz signature.

Parallel spinor flows on three-dimensional Cauchy hypersurfaces

The three-dimensional parallel spinor flow is the evolution flow defined by a parallel spinor on a globally hyperbolic Lorentzian four-manifold. We prove that, despite the fact that Lorentzian metrics admitting parallel spinors are not necessarily Ricci flat, the parallel spinor flow preserves the vacuum momentum and Hamiltonian constraints and therefore the Einstein and parallel spinor flows coincide on common initial data. Using this result, we provide an initial data characterization of parallel spinors on Ricci flat Lorentzian four-manifolds, which in turn yields the first initial data characterization of Ricci-flat pp-waves. Furthermore, we explicitly solve the left-invariant parallel spinor flow on simply connected Lie groups, obtaining along the way necessary and sufficient conditions for the flow to be immortal. These are, to the best of our knowledge, the first non-trivial examples of evolution flows of parallel spinors. Finally, we use some of these examples to construct families of -Einstein cosymplectic structures and to produce solutions to the left-invariant Ricci flow in three dimensions. This suggests the intriguing possibility of using first-order hyperbolic spinorial flows to construct special solutions of curvature flows in Riemannian signature.

The Structure of Geodesic Orbit Lorentz Nilmanifolds

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here, we carry out a major step in the structural analysis of geodesic orbit Lorentz nilmanifolds. Those are the geodesic orbit Lorentz manifolds $$M = G/H$$ such that a nilpotent analytic subgroup of G is transitive on M. Suppose that there is a reductive decomposition $${\mathfrak {g}}= {\mathfrak {h}}\oplus {\mathfrak {n}}$$ (vector space direct sum) with $${\mathfrak {n}}$$ nilpotent. When the metric is nondegenerate on $$[{\mathfrak {n}},{\mathfrak {n}}]$$ , we show that $${\mathfrak {n}}$$ is abelian or 2-step nilpotent (this is the same result as for geodesic orbit Riemannian nilmanifolds), and when the metric is degenerate on $$[{\mathfrak {n}},{\mathfrak {n}}]$$ , we show that $${\mathfrak {n}}$$ is a Lorentz double extension corresponding to a geodesic orbit Riemannian nilmanifold. In the latter case, we construct examples to show that the number of nilpotency steps is unbounded.

Heisenberg-invariant self-dual Einstein manifolds

We classify all self-dual Einstein four-manifolds invariant under a principal action of the three-dimensional Heisenberg group with non-degenerate orbits. This is of interest in the study of quantum-corrected gravitational physics, since they naturally arise as scalar manifolds of particular Lorentzian and Euclidean supergravities with one-loop corrections. The metrics are explicit and we find, in particular, that the Einstein constant can take any value. Then we examine when the corresponding (Riemannian or neutral-signature) metrics are (geodesically) complete. Finally, we exhibit the solutions of non-zero Ricci-curvature as different branches of one-loop deformed universal hypermultiplets in Riemannian and neutral signature.

On Lorentzian connections with parallel skew torsion

The paper is devoted to metric connections with parallel skew-symmetric torsion in Lorentzian signature. This is motivated by recent progress in the Riemannian signature and by possible applications to supergravity theories. We provide a complete information about holonomy algebras, torsion and curvature of the considered connections up to the corresponding objects from the Riemannian signature. Various examples are constructed. It is shown how to construct all simply connected Lorentzian naturally reductive homogeneous spaces of arbitrary dimension from Riemannian naturally reductive homogeneous spaces. This leads to complete classification of Lorentzian naturally reductive homogeneous spaces in low dimensions.

The anisotropic Calderón problem on 3-dimensional conformally Stäckel manifolds

Conformally Stäckel manifolds can be characterized as the class of n -dimensional pseudo-Riemannian manifolds (M,G) on which the Hamilton–Jacobi equation G(\nabla u, \nabla u) = 0 for null geodesics and the Laplace equation -\Delta_G \, \psi = 0 are solvable by R-separation of variables. In the particular case in which the metric has Riemannian signature, they provide explicit examples of metrics admitting a set of n-1 commuting conformal symmetry operators for the Laplace–Beltrami operator \Delta_G . In this paper, we solve the anisotropic Calderón problem on compact 3 -dimensional Riemannian manifolds with boundary which are conformally Stäckel, that is we show that the metric of such manifolds is uniquely determined by the Dirichlet-to-Neumann map measured on the boundary of the manifold, up to diffeomorphims that preserve the boundary.

Einstein with skew-torsion connections on Berger spheres

The invariant metric affine connections on Berger spheres which are Einstein with skew-torsion are determined in both Riemannian and Lorentzian signature. Expressions of such connections are explicitly given. Notably, every Berger sphere with Lorentzian signature admits invariant metric affine connections which are Einstein with skew-torsion except for dimension 3. For Riemannian signature, the existence of such connections strongly depends on the dimension of the sphere and on the scale of the deformation used for the Berger metric. In particular, there are Riemannian Berger spheres, not Einstein, which admit invariant Einstein with skew-torsion affine connections.

Families of spectral triples and foliations of space(time)

We study a noncommutative analog of a spacetime foliated by spacelike hypersurfaces, in both Riemannian and Lorentzian signatures. First, in the classical commutative case, we show that the canonical Dirac operator on the total spacetime can be reconstructed from the family of Dirac operators on the hypersurfaces. Second, in the noncommutative case, the same construction continues to make sense for an abstract family of spectral triples. In the case of Riemannian signature, we prove that the construction yields in fact a spectral triple, which we call a product spectral triple. In the case of Lorentzian signature, we correspondingly obtain a “Lorentzian spectral triple,” which can also be viewed as the “reverse Wick rotation” of a product spectral triple. This construction of “Lorentzian spectral triples” fits well into the Krein space approach to noncommutative Lorentzian geometry.

Non-Regular Spacetime Geometry

PrefaceUntil recently, mathematical studies of Lorentzian causality theory assumed, explicitly or implicitly, smoothness of the metric. This changed a few years ago, after independent work by Fathi and Siconolfi, and by Chruściel and Grant. Since then many papers began to explore causality under weaker conditions such as C1,1 metrics, Lipschitz metrics, continuous metrics (and cone structures), or even topological settings. Indeed, new insights are needed when the metric is not twice-differentiable. Hence, manifolds for which the metric is not C2 might be called non-regular, though closer inspection reveals that there are not so many differences between the C2 and the C1,1 theories, so one can also refer to the non-regular theories as those whose metric has regularity below C1,1.In Riemannian signature geometers studied non-regular manifolds using comparison geometry and other approaches. Many of these techniques have yet to be exported to the Lorentzian signature since, of course, a number of difficulties have to be addressed. In spite of this, mathematical physicists study non-regular spaces with a strong motivation that stems from the desire of understanding the ultimate nature of spacetime. The idea is that at the quantum level and hence whenever the physical conditions become extreme (say gravitational collapse, or origin of the universe) the spacetime manifold should perhaps not be approximated by a smooth manifold. If this expectation is correct then an understanding of the role of differentiability conditions in general relativity might give hints to the very nature of gravity and spacetime at the quantum scale.

Embedding graphs in Lorentzian spacetime.

Geometric approaches to network analysis combine simply defined models with great descriptive power. In this work we provide a method for embedding directed acyclic graphs (DAG) into Minkowski spacetime using Multidimensional scaling (MDS). First we generalise the classical MDS algorithm, defined only for metrics with a Riemannian signature, to manifolds of any metric signature. We then use this general method to develop an algorithm which exploits the causal structure of a DAG to assign space and time coordinates in a Minkowski spacetime to each vertex. As in the causal set approach to quantum gravity, causal connections in the discrete graph correspond to timelike separation in the continuous spacetime. The method is demonstrated by calculating embeddings for simple models of causal sets and random DAGs, as well as real citation networks. We find that the citation networks we test yield significantly more accurate embeddings that random DAGs of the same size. Finally we suggest a number of applications in citation analysis such as paper recommendation, identifying missing citations and fitting citation models to data using this geometric approach.

Cosmological singularity theorems and splitting theorems for N-Bakry-Émery spacetimes

We study Lorentzian manifolds with a weight function such that the N-Bakry-Émery tensor is bounded below. Such spacetimes arise in the physics of scalar-tensor gravitation theories, including Brans-Dicke theory, theories with Kaluza-Klein dimensional reduction, and low-energy approximations to string theory. In the “pure Bakry-Émery” N = ∞ case with f uniformly bounded above and initial data suitably bounded, cosmological-type singularity theorems are known, as are splitting theorems which determine the geometry of timelike geodesically complete spacetimes for which the bound on the initial data is borderline violated. We extend these results in a number of ways. We are able to extend the singularity theorems to finite N-values N ∈ (n, ∞) and N ∈ (−∞, 1]. In the N ∈ (n, ∞) case, no bound on f is required, while for N ∈ (−∞, 1] and N = ∞, we are able to replace the boundedness of f by a weaker condition on the integral of f along future-inextendible timelike geodesics. The splitting theorems extend similarly, but when N = 1, the splitting is only that of a warped product for all cases considered. A similar limited loss of rigidity has been observed in a prior work on the N-Bakry-Émery curvature in Riemannian signature when N = 1 and appears to be a general feature.

Homogeneous cosmologies as group field theory condensates

We give a general procedure, in the group field theory (GFT) formalism for quantum gravity, for constructing states that describe macroscopic, spatially homogeneous universes. These states are close to coherent (condensate) states used in the description of Bose-Einstein condensates. The condition on such states to be (approximate) solutions to the quantum equations of motion of GFT is used to extract an effective dynamics for homogeneous cosmologies directly from the underlying quantum theory. The resulting description in general gives nonlinear and nonlocal equations for the 'condensate wavefunction' which are analogous to the Gross-Pitaevskii equation in Bose-Einstein condensates. We show the general form of the effective equations for current quantum gravity models, as well as some concrete examples. We identify conditions under which the dynamics becomes linear, admitting an interpretation as a quantum-cosmological Wheeler-DeWitt equation, and give its semiclassical (WKB) approximation in the case of a kinetic term that includes a Laplace-Beltrami operator. For isotropic states, this approximation reproduces the classical Friedmann equation in vacuum with positive spatial curvature. We show how the formalism can be consistently extended from Riemannian signature to Lorentzian signature models, and discuss the addition of matter fields, obtaining the correct coupling of a massless scalar in the Friedmann equation from the most natural extension of the GFT action. We also outline the procedure for extending our condensate states to include cosmological perturbations. Our results form the basis of a general programme for extracting effective cosmological dynamics directly from a microscopic non-perturbative theory of quantum gravity.

Self-Dual Conformal Gravity

We find necessary and sufficient conditions for a Riemannian four-dimensional manifold (M, g) with anti-self-dual Weyl tensor to be locally conformal to a Ricci-flat manifold. These conditions are expressed as the vanishing of scalar and tensor conformal invariants. The invariants obstruct the existence of parallel sections of a certain connection on a complex rank-four vector bundle over M. They provide a natural generalisation of the Bach tensor which vanishes identically for anti-self-dual conformal structures. We use the obstructions to demonstrate that LeBrun’s anti-self-dual metrics on connected sums of $${\mathbb{CP}^2}$$ s are not conformally Ricci-flat on any open set. We analyze both Riemannian and neutral signature metrics. In the latter case we find all anti-self-dual metrics with a parallel real spinor which are locally conformal to Einstein metrics with non-zero cosmological constant. These metrics admit a hyper-surface orthogonal null Killing vector and thus give rise to projective structures on the space of β-surfaces.

Einstein-like geometric structures on surfaces

An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenerate immersed hypersurface in flat affine space. The author has defined for AH structures Einstein equations, which specialize on the one hand to the usual Einstein Weyl equations and, on the other hand, to the equations for affine hyperspheres. Here these equations are solved for Riemannian signature AH structures on compact orientable surfaces, the deformation spaces of solutions are described, and some aspects of the geometry of these structures are related. Every such structure is either Einstein Weyl (in the sense defined for surfaces by Calderbank) or is determined by a pair comprising a conformal structure and a cubic holomorphic differential, and so by a convex flat real projective structure. In the latter case it can be identified with a solution of the Abelian vortex equations on an appropriate power of the canonical bundle. On the cone over a surface of genus at least two carrying an Einstein AH structure there are Monge-Amp\`ere metrics of Lorentzian and Riemannian signature and a Riemannian Einstein K\"ahler affine metric. A mean curvature zero spacelike immersed Lagrangian submanifold of a para-K\"ahler four-manifold with constant para-holomorphic sectional curvature inherits an Einstein AH structure, and this is used to deduce some restrictions on such immersions.

Almost Einstein and Poincaré–Einstein manifolds in Riemannian signature

An almost Einstein manifold satisfies equations which are a slight weakening of the Einstein equations; Einstein metrics, Poincaré–Einstein metrics, and compactifications of certain Ricci-flat asymptotically locally Euclidean structures are special cases. The governing equation is a conformally invariant overdetermined PDE on a function. Away from the zeros of this function the almost Einstein structure is Einstein, while the zero set gives a scale singularity set which may be viewed as a conformal infinity for the Einstein metric. In this article there are two main results: we give a simple classification of the possible scale singularity spaces of almost Einstein manifolds; we derive geometric results which explicitly relate the intrinsic (conformal) geometry of the conformal infinity to the conformal structure of the ambient almost Einstein manifold. The latter includes new results for Poincaré–Einstein manifolds. Classes of examples are constructed. A compatible generalisation of the constant scalar curvature condition is also developed. This includes almost Einstein as a special case, and when its curvature is suitably negative, is closely linked to the notion of an asymptotically hyperbolic structure.

Signature-change events in emergent spacetimes with anisotropic scaling

We investigate the behaviour of quantum fields coupled to a spacetime geometry exhibiting finite regions of Euclidean (Riemannian) signature. Although from a gravity perspective this situation might seem somewhat far fetched, we will demonstrate its direct physical relevance for an explicitly realizable condensed matter system whose linearized perturbations experience an effective emergent spacetime geometry with externally controllable signature. This effective geometry is intrinsically quantum in origin, and its signature is determined by the details of the microscopic structure. At the level of the effective field theory arising from our condensed matter system we encounter explicit anisotropic scaling in time and space. Here Lorentz symmetry is an emergent symmetry in the infrared. This anisotropic scaling of time and space cures some of the technical problems that arise when working within a canonical quantisation scheme obeying strict Lorentz invariance at all scales, and so is helpful in permitting signature change events to take place.

Conformal Operators on Forms and Detour Complexes on Einstein Manifolds

For even dimensional conformal manifolds several new conformally invariant objects were found recently: invariant differential complexes related to, but distinct from, the de Rham complex (these are elliptic in the case of Riemannian signature); the cohomology spaces of these; conformally stable form spaces that we may view as spaces of conformal harmonics; operators that generalise Branson’s Q-curvature; global pairings between differential form bundles that descend to cohomology pairings. Here we show that these operators, spaces, and the theory underlying them, simplify significantly on conformally Einstein manifolds. We give explicit formulae for all the operators concerned. The null spaces for these, the conformal harmonics, and the cohomology spaces are expressed explicitly in terms of direct sums of subspaces of eigenspaces of the form Laplacian. For the case of non-Ricci flat spaces this applies in all signatures and without topological restrictions. In the case of Riemannian signature and compact manifolds, this leads to new results on the global invariant pairings, including for the integral of Q-curvature against the null space of the dimensional order conformal Laplacian of Graham et al.

Yang-Mills Detour Complexes and Conformal Geometry

Working over a pseudo-Riemannian manifold, for each vector bundle with connection we construct a sequence of three differential operators which is a complex (termed a Yang-Mills detour complex) if and only if the connection satisfies the full Yang-Mills equations. A special case is a complex controlling the deformation theory of Yang-Mills connections. In the case of Riemannian signature the complex is elliptic. If the connection respects a metric on the bundle then the complex is formally self-adjoint. In dimension 4 the complex is conformally invariant and generalises, to the full Yang-Mills setting, the composition of (two operator) Yang-Mills complexes for (anti-)self-dual Yang-Mills connections. Via a prolonged system and tractor connection a diagram of differential operators is constructed which, when commutative, generates differential complexes of natural operators from the Yang-Mills detour complex. In dimension 4 this construction is conformally invariant and is used to yield two new sequences of conformal operators which are complexes if and only if the Bach tensor vanishes everywhere. In Riemannian signature these complexes are elliptic. In one case the first operator is the twistor operator and in the other sequence it is the operator for Einstein scales. The sequences are detour sequences associated to certain Bernstein-Gelfand-Gelfand sequences.

The conformal deformation detour complex for the obstruction tensor

On pseudo-Riemannian manifolds of even dimension n ≥ 4 n\geq 4 , with everywhere vanishing (Fefferman-Graham) obstruction tensor, we construct a complex of conformally invariant differential operators. The complex controls the infinitesimal deformations of obstruction-flat structures, and, in the case of Riemannian signature the complex is elliptic.

Supersymmetric wrapped membranes,AdS2spaces, and bubbling geometries

We perform a systematic study, in eleven dimensional supergravity, of the geometry of wrapped brane configurations admitting $AdS_2$ limits. Membranes wrapping holomorphic curves in Calabi-Yau manifolds are found to exhibit some novel features; in particular, for fourfolds or threefolds, the gravitational effect of the branes on the overall transverse space is only weakly restricted by the kinematics of the Killing spinor equation. We also study the $AdS_2$ limits of the wrapped brane supergravity descriptions. For membranes wrapped in a two-fold, we derive a set of $AdS_2$ supersymmetry conditions which upon analytic continuation coincide precisely with those for the half-BPS bubbling geometries of LLM. From membranes wrapped in a three-fold, we obtain a set of $AdS_2$ supersymmetry conditions which upon analytic continuation describe a class of spacetimes which we identify as quarter-BPS bubbling geometries in M-theory, with $SO(4)\times SO(3)\times U(1)$ isometry in Riemannian signature. We also study fivebranes wrapping a special lagrangian five-cycle in a fivefold, in the presence of membranes wrapping holomorphic curves, and employ the wrapped brane supersymmetry conditions to derive a classification of the general minimally supersymmetric $AdS_2$ geometry in M-theory.