We prove the following statement: Let g be a light-line-complete pseudo-Riemannian Einstein metric of indefinite signature on a connected (n>2)-dimensional manifold M. Assume that a conformally equivalent metric is also Einstein. Then, the metrics are proportional with a constant coefficient.
If in addition the manifold is closed, the assumption that the manifold is light-line-complete could be omitted.
We generalize for pseudo-Riemannian metrics a classical result of Gallot and Tanno and use it to reprove a recent result of Alekseevsky, Cortes, Galaev and Leistner that decomposable cones over complete closed pseudo-Riemannian manifolds do not exist.
We describe, by their holonomy groups, all simply connected irreducible non-locally symmetric pseudo-Riemannian SpinC manifolds which admit parallel spinors. So we generalise the Riemannian case and the pseudo-Riemannian one.
We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, and prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture.
We show that the pseudo-Riemannian geometry of submanifolds can be formulated in terms of higher order multi-linear maps. In particular, we obtain a Poisson bracket formulation of almost (para-)Kähler geometry.
We study the higher order Jacobi operator in pseudo-Riemannian geometry. We exhibit a family of manifolds so that this operator has constant Jordan normal form on the Grassmannian of subspaces of signature (r,s) for certain values of (r,s). These pseudo-Riemannian manifolds are new and non-trivial examples of higher order Osserman manifolds.
We construct a family of balanced signature pseudo-Riemannian manifolds, which arise as hypersurfaces in flat space, that are curvature homogeneous, that are modeled on a symmetric space, and that are not locally homogeneous.
We define the notion of Witt structure on the tangent bundle of a pseudo-Riemannian manifold and we introduce a connection adapted to a such structure. The notions of geodesics and symmetric spaces are revisited in this setting and applications are given in the special cases of Robinson and Fefferman manifolds.
We exhibit 3 families of complete curvature homogeneous pseudo-Riemannian manifolds which are modeled on irreducible symmetric spaces and which are not locally homogeneous. All of the manifolds have nilpotent Jacobi operators; some of the manifolds are, in addition, Jordan Osserman and Jordan Ivanov-Petrova.
I prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture stating that on a closed manifold different from the round sphere every projective (i.e., geodesic-preserving) vector field is Killing.
Spinor representations of surfaces immersed into 4-dimensional pseudo-riemannian manifolds are defined in terms of minimal left ideals and tensor decompositions of Clifford algebras. The classification of spinor fields and Dirac operators on the immersed surfaces is given. The Dirac-Hestenes spinor field on surfaces immersed into Lorentzian manifolds and on surfaces conformally immersed into Minkowski spacetime is defined.
We show that isothermic surfaces and S-Willmore surfaces are also the solutions to the corresponding Blaschke's problem for both spacelike and timelike surfaces in pseudo-Riemannian space forms. For timelike surfaces both Willmore and isothermic, we obtain a description by minimal surfaces similar to the classical results of Thomsen.
We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces and prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces
We generalize the following classical result of Fubini for pseudo-Riemannian metrics: if three essentially different metrics on $M^{n\ge 3}$ share the same unparametrized geodesics, and two of them (say, $g$ and $\bar g$) are strictly nonproportional (i.e., the minimal polynomial of $g^{iα} \bar g_{αj}$ coincides with the characteristic polynomial) at least at one point, then they have constant curvature.
We give necessary and sufficient conditions for Riemannian maps to be biharmonic. We also define pseudo umbilical Riemannian maps as a generalization of pseudo-umbilical submanifolds and show that such Riemannian maps put some restrictions on the base manifolds.
We prove an important partial case of the pseudo-Riemannian version of the projective Lichnerowicz conjecture stating that a complete manifold admitting an essential group of projective transformations is the round sphere (up to a finite cover).
Using the Lie derivative of the metric we define a class of Lie algebras of vector fields by generalising the concept of Killing vectors. As a Lie algebra they define locally a group action on the pseudo-Riemannian manifold through exponentiation. The motivation behind studying these infinitesimal group actions is the investigation of $\mathcal{I}$-degenerate pseudo-Riemannian spaces, i.e., spaces having identical polynomial curvature invariants. In particular, we show that all the known examples of $\mathcal{I}$-degenerate pseudo-Riemannian spaces possess such vector fields.
The curvature tensor of a pseudo-Riemannian metric, and its covariant derivatives, satisfy certain identities that hold on any manifold of dimension less or equal than $n$. In this paper, we re-elaborate recent results by Gilkey-Park-Sekigawa regarding $p$-covariant dimensional curvature identities, for $p=0,2$. To this end, we use the classical theory of natural operations, that allows us to simplify some arguments and to generalize the description of Gilkey-Park-Sekigawa. Thus, our main result describes the first space of $p$-covariant dimensional curvature identities, for any even $p$.
We show that a complete flat pseudo-Riemannian homogeneous manifold with non-abelian linear holonomy is of dimension at least 14. Due to an example constructed in a previous article by Oliver Baues and the author, this is a sharp bound. Also, we give a structure theory for the fundamental groups of complete flat pseudo-Riemannian manifolds in dimensions less than 7. Finally, we observe that every finitely generated torsion-free 2-step nilpotent group can be realized as the fundamental group of a complete flat pseudo-Riemannian manifold with abelian linear holonomy.
In this paper a thorough study of the normal form and the first integrability conditions arising from {\em bi-conformal vector fields} is presented. These new symmetry transformations were introduced in {\em Class. Quantum Grav.}\textbf{21}, 2153-2177 and some of their basic properties were addressed there. Bi-conformal vector fields are defined on a pseudo-Riemannian manifold through the differential conditions $\lie P_{ab}=ϕP_{ab}$ and $\lieΠ_{ab}=χΠ_{ab}$ where $P_{ab}$ and $Π_{ab}$ are orthogonal and complementary projectors with respect to the metric tensor $\rmg_{ab}$. One of the main re…
Riemannian first-passage percolation (FPP) is a continuum model, with a distance function arising from a random Riemannian metric in $\R^d$. Our main result is a shape theorem for this model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one.
In this paper, we derived biharmonic equations for pseudo-Riemannian submanifolds of pseudo-Riemannian manifolds which includes the biharmonic equations for submanifolds of Riemannian manifolds as a special case. As applications, we proved that a pseudo-umbilical biharmonic pseudo-Riemannian submanifold of a pseudo-Riemannian manifold has constant mean curvature, we completed the classifications of biharmonic pseudo-Riemannian hypersurfaces with at most two distinct principal curvatures, which were used to give four construction methods to produce proper biharmonic pseudo-Riemannian submanifol…
We classify pseudo-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a pseudo-Riemannian manifold. Also, we obtain the classification of the pseudo-Riemannian submersions with (para-)complex connected totally geodesic fibres from a (para-)complex pseudo-hyperbolic space onto a pseudo-Riemannian manifold.
We describe, by their holonomy groups, all complete simply connected irreducible non-locally symmetric pseudo-Riemannian SpinC manifolds which admit parallel spinors. So we generalize the Riemannian SpinC case and the pseudo-Riemannian Spin one.
In this paper we continue the study of bi-conformal vector fields started in {\em Class. Quantum Grav.} {\bf 21} 2153-2177. These are vector fields defined on a pseudo-Riemannian manifold by the differential conditions $\lie P_{ab}=ϕP_{ab}$, $\lieΠ_{ab}=χΠ_{ab}$ where $P_{ab}$, $Π_{ab}$ are orthogonal and complementary projectors with respect to the metric tensor $\rmg_{ab}$ and $\lie$ is the Lie derivative. In a previous paper we explained how the analysis of these differential conditions enabled us to derive local geometric characterizations of the most relevant cases of {\em conformally sep…
In the present paper we study pseudo-Riemannian submanifolds which have 3-planar geodesic normal sections.We consider W-curves (helices) on pseudo-Riemannian submanifolds. Finally, we give neccessary and sufficient condition for a normal section to be a W-curve on pseudo-Riemannian submanifolds.
Definition of $({\cal T}_{a},{\cal T}_{b})$-pseudosymmetric semi-Riemannian manifold is given. $({\cal T}_{a},{\cal T}_{b})$-pseudosy mmetric $(N(k),ξ)$-semi-Riemannian manifolds are classified. Some results for ${\cal T}_{a}$-pseudosymmetric $(N(k),ξ)$-semi-Riemannian manifolds are obtained. $({\cal T}_{a},{\cal T}_{b},S^{\ell})$-pseudosymmetric semi-Riemannian manifolds are defined. $({\cal T}_{a},{\cal T}_{b},S^{\ell})$-pseudosymmetric $(N(k),ξ)$-semi-Riemannian manifolds are classified. Some results for $(R,{\cal T}_{a},S^{\ell})$-pseudosymmetric $(N(k),ξ)$-semi-Riemannian manifolds are ob…
A Lie group as a 4-dimensional pseudo-Riemannian manifold is considered. This manifold is equipped with an almost product structure and a Killing metric in two ways. In the first case Riemannian almost product manifold with nonintegrable structure is obtained, and in the second case - a pseudo-Riemannian one. Each belongs to a 4-parametric family of manifolds, which are characterized geometrically.
There is a well developed theory of weakly symmetric Riemannian manifolds. Here it is shown that several results in the Riemannian case are also valid for weakly symmetric pseudo-Riemannian manifolds, but some require additional hypotheses. The topics discussed are homogeneity, geodesic completeness, the geodesic orbit property, weak symmetries, and the structure of the nilradical of the isometry group. Also, we give a number of examples of weakly symmetric pseudo-Riemannian manifolds, some mirroring the Riemannian case and some indicating the problems in extending Riemannian results to weakly…
The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields on pseudo-Riemannian quadrics. A para-Kaehler twisted anti-isometry is used to correlate harmonic vector fields on the quadrics of neutral signature.
We prove that a foliation $(M, F)$ of codimension $q$ on a $n$-dimen\-sio\-nal pseudo-Riemannian manifold is pseudo-Riemannian if and only if any geodesic that is orthogonal at one point to a leaf is orthogonal to every leaf it intersects. We show that on the graph $G = G(F)$ of a pseudo-Riemannian foliation there exists a unique pseudo-Riemannian metric such that canonical projections are pse\-u\-do-Rieman\-ni\-an submersions and the fibres of different projections are orthogonal at common points. Relatively this metric the induced foliation $(G,\mathbb{F})$ on the graph is pseudo-Riemannian …
We define a class of Riemannian and pseudo-Riemannian 2-step nilpotent Lie groups with nondegenerate centers that generalize the H-type groups of Kaplan. Examples are given and geometric properties are investigated.
We study isometries in the contact sub-pseudo-Riemannian geometry. In particular we give an upper bound on the dimension of the isometry group of a general sub-pseudo-Riemannian manifold and prove that the maximal dimension is attained for the left invariant structures on the Heisenberg group.
We classify the space-like biharmonic surfaces in 3-dimension pseudo-Riemannian space form, and construct explicit examples of proper biharmonic hypersurfaces in general ADS space.
We describe the possible holonomy groups of simply connected irreducible non-locally symmetric pseudo-Riemannian spin manifolds which admit parallel spinors.
We consider four dimensional conformally flat homogeneous pseudo Riemannian manifolds. According to forms (Seger types) of the Ricci operator, we provide a full classification of four dimensional pseudo Riemannian conformally flat homogeneous Ricci solitons.
We exhibit several families of Jacobi-Videv pseudo-Riemannian manifolds which are not Einstein. We also exhibit Jacobi-Videv algebraic curvature tensors where the Ricci operator defines an almost complex structure.
We exhibit pseudo Riemannian manifolds which are Szabó nilpotent of arbitrary order, or which are Osserman nilpotent of arbitrary order, or which are Ivanov-Petrova nilpotent of order 3.
We show that pseudo-Riemannian almost quaternionic homogeneous spaces with index 4 and an H-irreducible isotropy group are locally isometric to a pseudo-Riemannian quaternionic Kähler symmetric space if the dimension is at least 16. In dimension 12 we give a non-symmetric example.
We consider four-dimensional homogeneous pseudo-Riemannian manifolds with non-trivial isotropy and completely classify the cases giving rise to non-trivial homogeneous Ricci solitons. In particular, we show the existence of non-compact homogeneous (and also invariant) pseudo-Riemannian Ricci solitons which are not isometric to solvmanifolds, and of conformally flat homogeneous pseudo-Riemannian Ricci solitons which are not symmetric.
We classify the semi-Riemannian submersions from a pseudo-hyperbolic space onto a Riemannian manifold under the assumption that the fibres are connected and totally geodesic. Also we obtain the classification of the semi-Riemannian submersions from a complex pseudo-hyperbolic space onto a Riemannian manifold under the assumption that the fibres are complex, connected and totally geodesic submanifolds.
We construct the first known examples of compact pseudo-Riemannian manifolds having an essential group of conformal transformations, and which are not conformally flat. Our examples cover all types $(p,q)$, with $2 \leq p \leq q$.
In this paper, we formulate a procedure to obtain a generalization of Milnor frames for left-invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on left-invariant Riemannian metrics, and is based on the moduli space of left-invariant pseudo-Riemannian metrics. As one of applications, we show that any left-invariant pseudo-Riemannian metrics of arbitrary signature on the Lie groups of real hyperbolic spaces have constant sectional curvatures.
The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold $(M,g)$ is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of $(M,g)$. We characterize the following simply connected pseudo-Riemannian manifolds admitting such subbundles in terms of their holonomy algebras: Riemannian manifolds; Lorentzian manifolds; pseudo-Riemannian manifolds with irreducible holonomy algebras; pseudo-Riemannian manifolds of neutral signature admitting two complementary parallel isotropic distributions.
We give a concise proof that large classes of optimal (constant curvature or Einstein) pseudo-Riemannian metrics are maximally symmetric within their conformal class.
In this paper, we discuss the heat flow of a pseudo-harmonic map from a closed pseudo-Hermitian manifold to a Riemannian manifold with non-positive sectional curvature, and prove the existence of the pseudo-harmonic map which is a generalization of Eells-Sampson's existence theorem. We also discuss the uniqueness of the pseudo-harmonic representative of its homotopy class which is a generalization of Hartman theorem, provided that the target manifold has negative sectional curvature.
Harvey-Lawson and Anciaux introduced the notion of austere submanifolds in pseudo-Riemannian geometry. We give an equivalent condition for an orbit of the isotropy representations for semisimple pseudo-Riemannian symmetric space to be an austere submanifold in a pseudo-sphere in terms of restricted root system theory with respect to Cartan subspaces. By using the condition we give examples of austere orbits.
Notions of compatible and almost compatible pseudo-Riemannian metrics, which are motivated by the theory of compatible (local and nonlocal) Poisson structures of hydrodynamic type and generalize the notion of flat pencil of metrics, are introduced and studied.
This paper provides a study of algebraic Ricci solitons in the pseudo-Riemannian case. In the Riemannian case, all nontrivial homogeneous algebraic Ricci solitons are expanding algebraic Ricci solitons. In this paper, we obtain a steady algebraic Ricci soliton and a shrinking algebraic Ricci soliton in the Lorentzian setting.
We provide examples of naturally reductive pseudo-Riemannian spaces, in particular an example of a naturally reductive pseudo-Riemannian 2-step nilpotent Lie group $(N, < \,,\,>_N)$, such that $< \,,\,>_N$ is invariant under a left action and for which the center is degenerate. The metric does not correspond to a bi-invariant one.
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.
For an arbitrary subalgebra $\mathfrak{h}\subset\mathfrak{so}(r,s)$, a polynomial pseudo-Riemannian metric of signature $(r+2,s+2)$ is constructed, the holonomy algebra of this metric contains $\mathfrak{h}$ as a subalgebra. This result shows the essential distinction of the holonomy algebras of pseudo-Riemannian manifolds of index bigger or equal to 2 from the holonomy algebras of Riemannian and Lorentzian manifolds.
