Distinguished dimensions for special Riemannian geometries
Pawel Nurowski
备注：
Section 9 has been changed, so that the relation between the 6th tensor defining SU(2)xSU(2) geometry in dimension 8 and isoparametric hypersurfaces with 6 different principal curvatures is explicit. A 5-form reducing GL(14,R) to SP(3) is added to the Appendix B
The paper is based on relations between a ternary symmetric form defining the SO(3) geometry in dimension five and Cartan's works on isoparametric hypersurfaces in spheres. As observed by Bryant such a ternary form exists only in dimensions n_k=3k+2, where k=1,2,4,8. In these dimensions it reduces the orthogonal group to the subgroups H_k\subset SO(n_k), with H_1=SO(3), H_2=SU(3), H_4=Sp(3) and H_8=F_4. This enables studies of special Riemannian geometries with structure groups H_k in dimensions n_k. The neccessary and sufficient conditions for the H_k geometries to admit the characteristic connection are given. As an illustration nontrivial examples of SU(3) geometries in dimension 8 admitting characteristic connection are provided. Among them there are examples having nonvanishing torsion and satisfying Einstein equations with respect to either the Levi-Civita or the characteristic connections.
All parabolic geometries, i.e. Cartan geometries with homogeneous model a real generalized flag manifold, admit highly interesting classes of distinguished curves. The geodesics of a projective class of connections on a manifold, conformal circles on conformal Riemannian manifolds, and Chern--Moser chains on CR--manifolds of hypersurface type are typical examples. We show that such distinguished curves are always determined by a finite jet in one point, and study the properties of such jets. We also discuss the question when distinguished curves agree up to reparametrization and discuss the di…
In this paper we construct a distinguished Riemannian geometrization on the dual 1-jet space J^{1*}(T,M) for the multi-time quadratic Hamiltonian functions. Our geometrization includes a nonlinear connection N, a generalized Cartan canonical N-linear connection (together with its local d-torsions and d-curvatures), naturally provided by a given quadratic Hamiltonian function depending on polymomenta.
In this paper, we study harmonic Riemannian submersions from 3-dimensional geometries using the ( generalized) integrability data associated to an orthonormal frame natural to a Riemannian submersion. We give complete classifications of harmonic Riemannian submersions from Thurston's 3-dimensional geometries, 3-dimensional BCV spaces and Berger sphere into a surface. We also give some explicit constructions of these harmonic Riemannian submersions.
We define a class of two dimensional surfaces conformally related to minimal surfaces in flat three dimensional geometries. By the utility of the metrics of such surfaces we give a construction of the metrics of $2 N$ dimensional Ricci flat (pseudo-) Riemannian geometries.
We investigate nonintegrable Riemannian geometries modelled after certain symmetric spaces related to the Freudenthal-Tits Magic Square. The collection of four such structures found by Nurowski is extended by further eight. A focus is given to those admitting a compatible connection with completely skew torsion.
We consider the sigma models where the base metric is proportional to the metric of the configuration space. We show that the corresponding sigma model equation admits a Lax pair. We also show that this type of sigma models in two dimensions are intimately related to the minimal surfaces in a flat pseudo Riemannian 3-space. We define two dimensional surfaces conformally related to the minimal surfaces in flat three dimensional geometries which enable us to give a construction of the metrics of some even dimensional Ricci flat pseudo Riemannian geometries.
We propose studies of special Riemannian geometries with structure groups $H_1=SO(3)\subset SO(5)$, $H_2=SU(3)\subset SO(8)$, $H_3=Sp(3)\subset SO(14)$ and $H_4=F_4\subset SO(26)$ in respective dimensions 5, 8, 14 and 26. These geometries, have torsionless models with symmetry groups $G_1=SU(3)$, $G_2=SU(3)\times SU(3)$, $G_3=SU(6)$ and $G_4=E_6$. The groups $H_k$ and $G_k$ constitute a part of the `magic square' for Lie groups. Apart from the $H_k$ geometries in dimensions $n_k$, the `magic square' Lie groups suggest studies of a finite number of other special Riemannian geometries. Among the…
Let $\left( M,J,g\right) $ be a metallic pseudo-Riemannian manifold equipped with a metallic structure $J$ and a pseudo-Riemannian metric $g$. The paper deals with interactions of Codazzi couplings formed by conjugate connections and tensor structures. The presence of Tachibana operator and Codazzi couplings presented a new characterization for locally metallic pseudo-Riemannian manifold. Also, a necessary and sufficient condition a non-integrable metallic pseudo-Riemannian manifold is a quasi metallic pseudo Riemannian manifold is derived. Finally, it is introduced metallic-like pseudo-Rieman…
Sub-Riemannian structures and non-transitive Cartan geometries via Lie groupoids
Centre Inria d’Université Côte d’Azur（阿基坦大学蔚蓝海岸Inria中心）；Julius-Maximilians-Universität Würzburg（维尔茨堡朱利叶斯-马克西米利安大学）；CMUC, University of Coimbra（科英布拉大学数学研究中心）
Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description of the structure of these curvature invariants in the cases where the background structure is one of the parabolic geometries. As an illustration, constant curvature models are discussed for certain sub-Riemannian geometries.
We treat the problem of defining, and characterising in a practical way, an appropriate class of distinguished curves for Poincaré-Einstein manifolds, and other conformally singular geometries. These "generalised geodesics" agree with geodesics away from the conformal singularity set and are shown to satisfy natural "boundary conditions" at points where they meet or cross the metric singularity set. We also characterise when they coincide with conformal circles. In the case of (Poincaré-)Einstein manifolds, we are able to provide a very general theory of first integrals for these distinguished…
In this article, we summarize the results on symmetric conformal geometries. We review the results following from the general theory of symmetric parabolic geometries and prove several new results for symmetric conformal geometries. In particular, we show that each symmetric conformal geometry is either locally flat or covered by a pseudo-Riemannian symmetric space, where the covering is a conformal map. We construct examples of locally flat symmetric conformal geometries that are not pseudo-Riemannian symmetric spaces.
In this paper, we define the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished submanifold? As a result, we prove that the solvsolitons on three-dimensional simply-connected solvable Lie groups are completely characterized by the minimality of the corresponding submanifolds.
We show that given a conformal structure whose holonomy representation fixes a totally lightlike subspace of arbitrary dimension, there is always a local metric in the conformal class off a singular set which is Ricci-isotropic and gives rise to a parallel, totally lightlike distribution on the tangent bundle. This naturally applies to parallel spin tractors resp. twistor spinors on conformal spin manifolds and clarifies which twistor spinors are locally equivalent to parallel spinors. Moreover, we study the zero set of a twistor spinor using the curved orbit decomposition for parabolic geomet…
We study 3-dimensional non-Riemannian Lorentz geometries, i.e. compact locally homogeneous Lorentz 3-manifolds with non-compact (local) isotropy group. One result is that, up to a finite cover, all such manifolds admit Lorentz metrics of (non-positive) constant sectionnal curvature. If the geometry is maximal, then the metric has constant sectionnal curvature, or is a left invariant metric on the Heisenberg group or the Sol group. These geometries, on each of the latter two groups are characterized by having a non-compact isotropy without being flat. Recall, for the need of his formulation of …
In contrast to SPD matrices, few tools exist to perform Riemannian statistics on the open elliptope of full-rank correlation matrices. The quotient-affine metric was recently built as the quotient of the affine-invariant metric by the congruence action of positive diagonal matrices. The space of SPD matrices had always been thought of as a Riemannian homogeneous space. In contrast, we view in this work SPD matrices as a Lie group and the affine-invariant metric as a left-invariant metric. This unexpected new viewpoint allows us to generalize the construction of the quotient-affine metric and t…
We present a uniform framework generalising and extending the classical theories of projective differential geometry, c-projective geometry, and almost quaternionic geometry. Such geometries, which we call \emph{projective parabolic geometries}, are abelian parabolic geometries whose flat model is an R-space $G\cdot\mathfrak{p}$ in the infinitesimal isotropy representation $\mathbb{W}$ of a larger self-dual symmetric R-space $H\cdot\mathfrak{q}$. We also give a classification of projective parabolic geometries with $H\cdot\mathfrak{q}$ irreducible which, in addition to the aforementioned class…
We study bimodule quantum Riemannian geometries over the field $\Bbb F_2$ of two elements as the extreme case of a finite-field adaptation of noncommutative-geometric methods for physics. We classify all parallelisable such geometries for coordinate algebras up to vector space dimension $n\le 3$, finding a rich moduli of examples for $n=3$ and top form degree 2, including many that are not flat. Their coordinate algebras are commutative but their differentials are not. We also study the quantum Laplacian $Δ=(\ ,\ )\nabla{\rm d}$ on our models and characterise when it has a massive eigenvector.
The Oscillator Groups,$\G_λ,$ are the only solvable, non commutative, simply connected Lie groups to admit a Lorentzian bi-invariant metric. For these groups, we give sufficient conditions for a left-invariant pseudo-Riemannian metric to be complete, we determine the group of isometries, we exhibit a left-invariant affine structure and prove that it is not Lorentzian. As an application, we provide new examples of compact pseudo-Riemannian (sometimes Lorentzian) locally symmetric, occasionally affine, manifolds, complete or incomplete.
We give a new characterisation of the unparametrised geodesics, or distinguished curves, for affine, pseudo-Riemannian, conformal, and projective geometry. This is a type of moving incidence relation. The characterisation is used to provide a very general theory and construction of quantities that are necessarily conserved along the curves. The formalism immediately yields explicit formulae for these curve first integrals. The usual role of Killing tensors and conformal Killing tensors is recovered as a special case, but the construction shows that a significantly larger class of equation solu…
This paper is devoted to a coordinate-free approach to several classic geometries such as hyperbolic (real, complex, quaternionic), elliptic (spherical, Fubini-Study), and lorentzian (de Sitter, anti de Sitter) ones. These geometries carry a certain simple structure that is in some sense stronger than the riemannian structure. Their basic geometrical objects have linear nature and provide natural compactifications of classic spaces. The usual riemannian concepts are easily derivable from the strong structure and thus gain their coordinate-free form. Many examples illustrate fruitful features o…
We show how to compute tensor derivatives and curvature tensors using affine connections. This allows for all computations to be obtained without using coordinate systems, in a way that parallels the computations appearing in classical Riemannian Geometry. In particular, we obtain Bianchi identities for the curvature tensor of any anisotropic connection, we compare the curvature tensors of any two anisotropic connections, and we find a family of anisotropic connections which are well suited to study the geometry of Finsler metrics.
In this paper we develop the distinguished (d-) Riemannian differential geometry (in the sense of d-connections, d-torsions, d-curvatures and some geometrical Maxwell-like and Einstein-like equations) for the polymomentum Hamiltonian which governs the multi-time electrodynamics.
We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L^2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi's metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geometry in detail. Unlike the Ebin metric, the geodesic equation involves non-local terms, and we solve it explicitly by using a constant of the motion. We then determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is …
A classic problem with intriguing implications at the level of both applied differential geometry and theoretical physics is dealt with in this short work: Is there any criterion in order to decide whether a pseudo-Riemannian space can be locally described using curvature scalars solely? Surprisingly enough, this question is susceptible of a very simple and elegant answer. In order to avoid unnecessary complexity, the analysis is restricted to local rather than global considerations, without any loss of not only the generality but also the insights to the initial problem.
Motivated by the rich geometry of conformal Riemannian manifolds and by the recent development of geometries modeled on homogeneous spaces $G/P$ with $G$ semisimple and $P$ parabolic, Weyl structures and preferred connections are introduced in this general framework. In particular, we extend the notions of scales, closed and exact Weyl connections, and Rho--tensors, we characterize the classes of such objects, and we use the results to give a new description of the Cartan bundles and connections for all parabolic geometries.
A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly Kähler or nearly parallel $\mathrm{G}_2$-manifolds, Sasakian and $3$-Sasakian manifolds, or twistor spaces over quaternion-Kähler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds c…
We study several linear connections (the first canonical, the Chern, the well adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those with totally skew-symmetric torsion) which can be defined on the four geometric types of $(J^2=\pm1)$-metric manifolds. We characterize when such a connection is adapted to the structure, and obtain a lot of results about coincidence among connections. We prove that the first canonical and the well adapted connections define a one-parameter family of adapted connections, named canonical connections, thus extending to almost Norden and almo…
We study the types of non-integrable $\mathrm{G}$-structures on Riemannian manifolds. In particular, geometric types admitting a connection with totally skew-symmetric torsion are characterized. 8-dimensional manifolds equipped with a $\Spin(7)$-structure play a special role. Any geometry of that type admits a unique connection with totally skew-symmetric torsion. Under weak conditions on the structure group we prove that this geometry is the only one with this property. Finally, we discuss the automorphism group of a Riemannian manifold with a fixed non-integrable $\mathrm{G}$-structure.
We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from $S^1$ to the plane modulo the group of diffeomorphisms of $S^1$, acting as reparameterizations. In particular we investigate the metric for a constant $A> 0$: $$ G^A_c(h,k) := \int_{S^1}(1+A\ka_c(þ)^2)< h(þ),k(þ) > |c'(þ)| dþ$$ where $\ka_c$ is the curvature of the curve $c$ and $h,k$ are normal vector fields to $c$. The term $A\ka^2$ is a sort of geometric Tikhonov regularization because, for A=0, the geodesic distance between any 2 distinct curves is 0, while for $A>0$ …
We report on some aspects and recent progress in certain problems in the sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the corresponding Yamabe problems on the round spheres, the Lichnerowicz-Obata first eigenvalue estimates, and the relation between these two problems. A motivation from the Riemannian case highlights new and old ideas which are then developed in the settings of Iwasawa sub-Riemannian geometries.
Chevalley's theorem and it's converse, the Sheppard-Todd theorem, assert that finite reflection groups are distinguished by the fact that the ring of invariant polynomials is freely generated. We show that in the Euclidean case, a weaker condition suffices to characterize finite reflection groups, namely that a freely-generated polynomial subring is closed with respect to the gradient product.
We provide an intrinsic notion of curved cosets for arbitrary Cartan geometries, simplifying the existing construction of curved orbits for a given holonomy reduction. To do this, we define an intrinsic holonomy group, which is shown to coincide precisely with the standard definition of the holonomy group for Cartan geometries in terms of an associated principal connection. These curved cosets retain many characteristics of their homogeneous counterparts, and they behave well under the action of automorphisms. We conclude the paper by using the machinery developed to generalize the de Rham dec…
On a smooth $n$-manifold $M$ with $n \geq 3$, we study pairs $(g,T)$ consisting of a Riemannian metric $g$ and a unit length closed vector field $T$. Motivated by how Ricci solitons generalize Einstein metrics via a distinguished vector field, we propose to generalize space forms by considering those pairs $(g,T)$ whose corresponding Lorentzian metric $g_{\scriptscriptstyle L} = g - 2T^{\flat} \otimes T^{\flat}$ has constant curvature. We show by examples that such pairs exist when $M$ is noncompact, and that complete metrics exist among them. When $M$ is compact, however, the situation is mor…
This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature.
This note is about the interplay between the almost-hermitian and Riemannian geometries of a manifold. These geometries can be seen to interact through curvature. The main result is an obstruction equation to the integrability of almost-complex structures orthogonal with respect to Riemannian metrics with constrained sectional curvature. Several geometric consequences ensue, such as a formula for the norm of the Levi-Civita covariant derivative of a hypothetical orthogonal complex structure. Our results lead to a partial recovery of the well-known fact that the round $6$-sphere $S^6$ is not he…
Geometries with parallel, skew-symmetric and closed torsion
Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西蒙·斯托伊洛数学研究所）；Universidade Estadual de Campinas（坎皮纳斯州立大学）
It is well-known that the Einstein condition on warpedgeometries requires the fibres to be necessarily Einstein. However, exact warped solutions have often been obtained using one- and two-dimensional bases. In this paper, keeping the dimensions and signatures of the base and the fibre independently arbitrary, we obtain all Ricci-flat warped metrics with flat base in closed form and show that the number of free parameters is one less than the dimension of the base. Without any assumptions on the base and fibre geometry, we then show that a warped geometry is flat, i.e, has vanishing Riemann cu…
Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the invariant Ricci flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish geome…
The seven and nine dimensional geometries associated with certain classes of supersymmetric $AdS_3$ and $AdS_2$ solutions of type IIB and D=11 supergravity, respectively, have many similarities with Sasaki-Einstein geometry. We further elucidate their properties and also generalise them to higher odd dimensions by introducing a new class of complex geometries in $2n+2$ dimensions, specified by a Riemannian metric, a scalar field and a closed three-form, which admit a particular kind of Killing spinor. In particular, for $n\ge 3$, we show that when the geometry in $2n+2$ dimensions is a cone we…
For a smooth (locally trivial) principal bundle in Ehresmann's sense, the relation between the commuting vertical and horizontal actions of the structural Lie group and the structural Lie groupoid (isomorphisms between vertical fibers) is regarded as a special case of a symmetrical concept of conjugation between "principal" Lie groupoid actions, allowing possibly non-locally trivial bundles. A diagrammatic description of this concept via a symmetric "butterfly diagram" allows its "internalization" in a wide class of categories (used by "working mathematicians") whenever they are endowed with t…
We derive the equations of chains for path geometries on surfaces by solving the equivalence problem of a related structure: sub-Riemannian geometry of signature $(1,1)$ on a contact 3-manifold. This approach is significantly simpler than the standard method of solving the full equivalence problem for path geometry. We then use these equations to give a characterization of projective path geometries in terms of their chains (the chains projected to the surface coincide with the paths) and study the chains of four examples of homogeneous path geometries. In one of these examples (horocycles in …
We show that the degenerate special Lagrangian equation, recently introduced by Rubinstein-Solomon, induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, geodesics in the space of positive Lagrangian sections of a smooth Calabi-Yau torus fibration are governed by the Riemannian DSL on the base manifold. We then develop their analytic techniques, specifically modifications of the Dirichlet duality theory of Harvey-Lawson, in the Ri…
This paper analyses the parabolic geometries generated by a free $n$-distribution in the tangent space of a manifold. It shows that certain holonomy reductions of the associated normal Tractor connections, imply preferred connections with special properties, along with Riemannian or sub-Riemannian structures on the manifold. It constructs examples of these holonomy reductions in the simplest cases. The main results, however, lie in the free 3-distributions. In these cases, there are normal Fefferman constructions over CR and Lagrangian contact structures corresponding to holonomy reductions to…
