We introduce a new notion of a homogeneous pair for a pseudo-Riemannian metric $g$ and a positive function $f$ on a manifold $M$ admitting a free $\mathbb{R}_{>0}$-action. There are many examples admitting this structure. For example, (a) a class of pseudo-Hessian manifolds admitting a free $\mathbb{R}_{>0}$-action and a homogeneous potential function such as the moduli space of torsion-free $G_2$-structures, (b) the space of Riemannian metrics on a compact manifold, and (c) many moduli spaces of geometric structures such as torsion-free ${\rm Spin}(7)$-structures admit this structure. Hence we provide the unified method for the study of these geometric structures.
We consider conformal transformations of the pseudo-Riemannian metric $g$ of a homogeneous pair $(g, f)$. Showing that the pseudo-Riemannian manifold $(M, (v \circ f) g)$, where $v: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ is a smooth function, has the structure of a warped product, we study the geometric structures such as the sectional curvature, geodesics and the metric completion (if $g$ is positive definite) w.r.t. $(v \circ f) g$ in terms of those on the level set of $f$. In particular, (1) we can generalize the result of Clarke and Rubinstein about the metric completion of the space of Riemannian metrics w.r.t. the conformal transformations of the Ebin metric, and (2) two canonical Riemannian metrics on the $G_2$ moduli space have different metric completions.
We classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four, and (2) between conformally-flat Riemannian manifolds of dimensions at least three.
The largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric, is the class of the conformal Riemannian P-manifolds. This class is an analogue of the class of the conformal Kähler manifolds in almost Hermitian geometry. The main aim of this work is to obtain properties of manifolds of this class with connections, whose curvature tensors have similar properties as the Kähler tensors in Hermitian geometry.
We prove conformal versions of the local decomposition theorems of de Rham and Hiepko of a Riemannian manifold as a Riemannian or a warped product of Riemannian manifolds. Namely, we give necessary and sufficient conditions for a Riemannian manifold to be locally conformal to either a Riemannian or a warped product. We also obtain other related de Rham-type decomposition theorems. As an application, we study Riemannian manifolds that admit a Codazzi tensor with two distinct eigenvalues everywhere.
Necessary and sufficient conditions for a Riemannian product to be conformally equivalent to an Einstein manifold are given. Such spaces which are complete are characterized.
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 find the index of $\widetilde{\nabla}$-quasi-conformally symmetric and $\widetilde{\nabla}$-concircularly symmetric semi-Riemannian manifolds, where $\widetilde{\nabla}$ is metric connection.
For a complete Riemannian metric, a pointwise conformal transformation may lead to a complete or incomplete transformed Riemannian metric, depending on the behavior of the conformal factor. We establish conditions on the growth of the conformal factor towards the infinity of the Riemannian metric, such that the conformally transformed Riemannian metric remains complete.
We give a classification for connected complete locally irreducible Riemannian manifolds with nonpositive curvature operator, which admit a nonzero closed or co-closed conformal Killing $L^{2}-$form. Moreover, we prove vanishing theorems for closed and co-closed conformal Killing $L^{2}-$forms on some complete Riemannian manifolds.
Conformally variational Riemannian invariants (CVIs), such as the scalar curvature, are homogeneous scalar invariants which arise as the gradient of a Riemannian functional. We establish a wide range of stability and rigidity results involving CVIs, generalizing many such results for the scalar curvature.
B. Sahin [9] introduced the notion of anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. In the present paper we extend the notion of anti-invariant and Lagrangian Riemannian submersions (a special anti-invariant Riemannian submersion) to the case of locally conformal Kaehler manifolds. We discuss the geometry of foliation and obtain some decomposition theorems for the total manifold of such submersions.
Conformally flat pseudo-Riemannian manifolds with generalized Ricci recurrent, $(GR)_n$ structure are completely classified in this short report. A conformally flat generalized Ricci recurrent pseudo-Riemannian manifold is shown to be either a de Sitter space or an anti-de Sitter space. In particular, a conformally flat generalized Ricci recurrent spacetime must be either a de Sitter spacetime or an anti-de Sitter spacetime.
Conformal Killing forms are a natural generalization of conformal vector fields on Riemannian manifolds. They are defined as sections in the kernel of a conformally invariant first order differential operator. We show the existence of conformal Killing forms on nearly Kaehler and weak G_2-manifolds. Moreover, we give a complete description of special conformal Killing forms. A further result is a sharp upper bound on the dimension of the space of conformal Killing forms.
The aim of this paper is to classify the cohomogeneity one conformal actions on the three-dimensional essential Riemannian spaces, up to orbit equivalence. Among other results, the representations of all connected Lie groups acting with cohomogeneity one or zero within the full conformal group of a given three-dimensional essential Riemannian space are determined, up to conjugacy, and the orbits are specified up to conformal equivalence.
Let $(M,g_M,\mathcal F)$ be a closed, connected Riemannian manifold with a Riemannian foliation $\mathcal F$ of nonzero constant transversal scalar curvature. When $M$ admits a transversal nonisometric conformal field, we find some generalized conditions that $\mathcal F$ is transversally isometric to the sphere.
In this paper we give bounds for the first eigenvalue of the conformal Laplacian and the Yamabe invariant of a compact Riemannian manifold, by using conditions on the Ricci curvature and the diameter and deduce certain conditions on the manifold to be conformal to a sphere.
In this paper a conformal classification of three dimensional left-invariant sub-Riemannian contact structures is carried out; in particular we will prove the following dichotomy: either a structure is locally conformal to the Heisenberg group $\mathbb H_3$, or its conformal classification coincides with the metric one. If a structure is locally conformally flat, then its conformal group is locally isomorphic to $SU(2,1)$.
Let M be a smooth compact manifold without boundary. We consider two smooth Sub-Semi-Riemannian metrics on M. Under suitable conditions, we show that they are almost conformally isometric in an Lp sense. Assume also that M carries a Riemannian metric with parallel Ricci curvature. Then an equation of Ricci type, is in some sense solvable, without assuming any closeness near a special metric.
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 consider a 3-dimensional Riemannian manifold V with a metric g and an affinor structure q. The local coordinates of these tensors are circulant matrices. In V we define an almost conformal transformation. Using that definition we construct an infinite series of circulant metrics which are successively almost conformaly related. In this case we get some properties.
The class W_1 of conformal Riemannian P-manifolds is the largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric. This class is an analogue of the class of conformal Kaehler manifolds in almost Hermitian geometry. In the present work we study the natural connections on the manifolds (M, P, g) from the class W_1, i.e. the linear connections preserving the almost product structure P and the Riemannian metric g. We find necessary and sufficient conditions the curvature tensor of such a connection to …
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.
The goal of this paper is to study conformal Yamabe soliton and $*$-Yamabe soliton, whose potential vector field is torse forming. Here, we have characterized conformal Yamabe soliton admitting potential vector field as torse forming with respect to Riemannian connection, semi-symmetric metric connection and projective semi-symmetric connection on Riemannian manifold. We have also shown the nature of $*$-Yamabe soliton with torse forming vector field on Riemannian manifold admitting Riemannian connection. Lastly we have developed an example to corroborate some theorems regarding Riemannian con…
We prove the existence of smooth closed hypersurfaces of prescribed mean curvature homeomorphic to $S^n$ for small $n, n\le6$, provided there are barriers.
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.
Let $(M, \bar g)$ be an $n$-dimensional complete Riemannian manifold. In this paper, we considers the following conformal scalar curvature rigidity problem: Given a compact smooth domain $Ω$ with $\partial Ω$, can one find a conformal metric $g$ whose scalar curvature $R[g]\ge R[\bar g]$ on $Ω$ and the mean curvature $H[g] \ge H[ \bar g]$ on $\partial Ω$ with $\bar g = g$ on $\partial Ω$? We prove that $\bar g = g$ on some smooth domains in a general Riemannian manifold, which is an extension of the previous results given by Qing and Yuan, and Hang and Wang.
In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T. K. Pan. Then we explain why it is so difficult to find examples, other than trivial ones, of spaces having at least two closed, conformal and homothetic vector fields. We then focus on isometric immersions, firstly generalizing a theorem of J. Simons on cones with parallel mean curvature to …
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…
We introduce a class of null hypersurfaces of a semi-Riemannian manifold, namely, screen quasi-conformal hypersurfaces, whose geometry may be studied through the geometry of its screen distribution. In particular, this notion allows us to extend some results of previous works to the case in which the sectional curvature of the ambient space is different from zero. As applications, we study umbilical, isoparametric and Einstein null hypersurfaces in Lorentzian space forms and provide several classification results.
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 consider the conformal class of the Riemannian product $g_0 + g$, where $g_0$ is the constant curvature metric on $S^m$ and $g$ is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of constant scalar curvature in the conformal class grows at least linearly with respect to the square root of the scalar curvature of $g$. This is obtained by studying radial solutions of the equation $Δu -λu + λu^p =0$ on $S^m$, and the number of solutions in terms of $λ$.
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.
This is the fourth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. The present paper lays out the second half of this entire work: The second half proves certain purely algebraic statements regarding local Riemannian invariants; these we…
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 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.
Let $(M^n,g)$ be an $n$-dimensional compact connected Riemannian manifold with smooth boundary. We show that the presence of a nontrivial conformal gradient vector field on $M$, with an appropriate control on the Ricci curvature makes $M$ to be isometric to a hemisphere of $\mathbb{S}^{n}$. We also prove that if an Einstein manifold admits nonzero conformal gradient vector field, then its scalar curvature is positive and it is isometric to a hemisphere of $\mathbb{S}^{n}$. Furthermore, we prove that if $ M $ admits a nontrivial conformal vector field and has constant scalar curvature, then the…
Given a simple Lie group G of rank 1, we consider compact pseudo-Riemannian manifolds (M,g) of signature (p,q) on which G can act conformally. Precisely, we determine the smallest possible value for the index min(p,q) of the metric. When the index is optimal and G non-exceptional, we prove that the metric must be conformally flat, confirming the idea that in a "good" dynamical context, a geometry is determined by its automorphisms group. This completes anterior investigations on pseudo-Riemannian conformal actions of semi-simple Lie groups of maximal real-rank. Combined with these results, we …
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.
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 …
This is the last in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. The present paper, jointly with [6,7] gives a proof of an algebraic Proposition regarding local Riemannian invariants, which lies at the heart of our resolution of the Dese…
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.
