Maximally-warped metrics with harmonic curvature

TL;DR

This paper characterizes Riemannian manifolds with harmonic curvature that admit many warped-product decompositions and have Ricci tensors with simple eigenvalues, revealing a finite-dimensional moduli space of such geometries.

Contribution

It provides a detailed description of the local structure of these manifolds and establishes the existence of a finite-dimensional moduli space of their local-isometry types.

Findings

Manifolds with harmonic curvature and maximal warped-product decompositions are classified.

A finite-dimensional moduli space of such manifolds exists in each dimension > 2.

Examples include irreducible metrics that are neither Ricci-parallel nor conformally flat in higher dimensions.

Abstract

We describe the local structure of Riemannian manifolds with harmonic curvature which admit a maximum number, in a well-defined sense, of local warped-product decompositions, and at the same time their Ricci tensor has, at some point, only simple eigenvalues. We also prove that, in every given dimension greater than two, the local-isometry types of such manifolds form a finite-dimensional moduli space, and a nonempty open subset of this moduli space is realized by locally irreducible complete metrics which are neither Ricci-parallel, nor -- for dimensions greater than three -- conformally flat.

Equations

$$\mathrm{div}R=0,\mathrm{or},\mathrm{in}\mathrm{local}\mathrm{coordinates},R_{ijl}^{}{}^k{}_{,k}^{}=0,$$

$$\begin{array}[]{l}\mathrm{r}\,\mathrm{\ has\ }\hskip 0.7ptn\mathrm{\ distinct\ eigen\-values\ at\ some\ point,\ and\ an\ open\ sub\-man\-i}\hskip 0.5pt\vbox{\hbox{\vrule width=2.5pt,height=0.5pt,depth=0.0pt}\vskip 2.0pt}\hskip 0.5pt\\ \mathrm{fold\ of\ }\,\hskip 0.7pt(M\hskip-0.7pt,g)\hskip 0.4pt\,\mathrm{\ admits\ a\ nontrivial\ warp\-ed}\hskip 0.5pt\vbox{\hbox{\vrule width=2.5pt,height=0.5pt,depth=0.0pt}\vskip 2.0pt}\hskip 0.5pt\mathrm{prod\-uct\ decomposition.}\end{array}$$

$$\gamma =n-1.$$

$$\begin{array}[]{l}\mathrm{complete,\ locally\ irreducible,\ and\ neither\ con}\hskip 0.5pt\vbox{\hbox{\vrule width=2.5pt,height=0.5pt,depth=0.0pt}\vskip 2.0pt}\hskip 0.5pt\\ \mathrm{for\-mal\-ly\ flat\ (unless\ }\hskip 0.7ptn=3\mathrm{),\ nor\ Ric\-ci}\hskip 0.5pt\vbox{\hbox{\vrule width=2.5pt,height=0.5pt,depth=0.0pt}\vskip 2.0pt}\hskip 0.5pt\mathrm{par\-al\-lel.}\end{array}$$

$$\begin{array}[]{rl}\mathrm{i)}&\mathrm{div}\,R=0\,\,\mathrm{\ if\ and\ only\ if\ }\,\,\mathrm{r}\,\,\mathrm{\ is\ a\ Co\-daz\-zi\ tensor,}\\ \mathrm{ii)}&\mathrm{the\ condition\ }\,\mathrm{div}\,R=0\hskip 0.7pt\mathrm{\ implies\ constancy\ of\ }\,\mathrm{s}\hskip 0.4pt,\end{array}$$

$$\mathrm{metrics}\mathrm{with}\mathrm{div}R=0\mathrm{are}\mathrm{real}\text{vbox}\text{vrule}width=2.5pt,height=0.5pt,depth=0.0pt\text{vskip}2.0pt\mathrm{analytic}\mathrm{in}\mathrm{suitable}\mathrm{local}coordinates.$$

$$\mathrm{r}(\nabla \varphi ,\cdot )+d\Delta \varphi =\mathrm{div}[\nabla d\varphi ].$$

$$(M,g)=(\overline{M}\times \Sigma ,\overline{g}+{\varphi }^2\eta ).$$

$$\mathrm{i})\partial v{}_z^{}=dv{}_z^{}:Y\to Y,\mathrm{ii})\partial w{}_z^{}\mathrm{equals}\mathrm{the}\mathrm{restriction}\mathrm{of}\partial v{}_z^{}\mathrm{to}T{}_z^{}N.$$

$$\mathrm{if}\theta v\mathrm{is}\mathrm{also}\mathrm{a}\mathrm{Kil}\text{-}\mathrm{ling}\mathrm{field},\theta \mathrm{must}\mathrm{be}\mathrm{constant},cf.\mathrm{Remark}\text{ref}\mathrm{extkf}(\mathrm{i}).$$

$$\mathrm{the}1\text{vbox}\text{vrule}width=2.5pt,height=0.5pt,depth=0.0pt\text{vskip}2.0pt\mathrm{form}g(v,\cdot )/g(v,v),\mathrm{defined}\mathrm{wherever}v\ne 0,\mathrm{is}closed.$$

$$2\beta \xi {}_{i,j}^{}=\beta {}_{,j}^{}\xi {}_i^{}-\beta {}_{,i}^{}\xi {}_j^{},\mathrm{where}\beta =v{}^k{\xi }_k^{}=g(v,v).$$

$$\begin{array}[]{l}v\,\,\mathrm{\ is\ orthogonal\ to\ any\ geodesic\ passing\ through\ a\ zero\ of\ }\,\,v\mathrm{,\ while}\\ \mathrm{whenever\ (\ref{gvg})\ holds,the\ distribution\ }\hskip-0.7pt\,v^{\perp}\hskip-1.5pt\mathrm{\ has\ totally\ geodesic\ leaves.}\end{array}$$

$$g_{kl}^{}=0\mathrm{if}k\ne l,g_{11}^{}=1,g{}_{jj}^{}=g{}_{jj}^{}(t)\mathrm{for}t=x^1\mathrm{and}j\ge 2,$$

$$2y{}_j^{}g{}_{jj}^{}=-\dot{g}{}_{jj}^{}(\mathrm{no}\mathrm{summation}),\mathrm{with}()\dot{}=d/dt.$$

$$\ddot{\mathbf{y}}-(\mathrm{tr}\mathbf{y}+\mathbf{y})\dot{\mathbf{y}}=(\mathrm{tr}{\mathbf{y}}^2)\mathbf{y}-(\mathrm{tr}\mathbf{y}){\mathbf{y}}^2$$

$$\mathrm{s}=2\mathrm{tr}\dot{\mathbf{y}}-\mathrm{tr}{\mathbf{y}}^2-(\mathrm{tr}\mathbf{y}{)}^2\mathrm{is}\mathrm{constant}\mathrm{whenever}t\mapsto \mathbf{y}\mathrm{satisfies}(\text{ref}\mathrm{hcu}).$$

$$\dot{\mathbf{y}}=\mathbf{p},\dot{\mathbf{p}}=(\mathrm{tr}\mathbf{y}+\mathbf{y})\mathbf{p}+(\mathrm{tr}{\mathbf{y}}^2)\mathbf{y}-(\mathrm{tr}\mathbf{y}){\mathbf{y}}^2.$$

$$(\mathbf{y},\mathbf{p})\mapsto v_{(\mathbf{y},\mathbf{p})}^{}=(\mathbf{p},(\mathrm{tr}\mathbf{y}+\mathbf{y})\mathbf{p}+(\mathrm{tr}{\mathbf{y}}^2)\mathbf{y}-(\mathrm{tr}\mathbf{y}){\mathbf{y}}^2)$$

$$\mathrm{our}z,N,w\mathrm{and}\epsilon =-\mathrm{sgn}q\mathrm{satisfy}\mathrm{the}\mathrm{hypothesis}\mathrm{of}\mathrm{Lemma}\text{ref}\mathrm{trapd}.$$

$$\mp y{}_j^{}>|a|\mathrm{whenever}(y_2^{},\dots ,y_n^{},p_2^{},\dots ,p_n^{})\in U{}_{\pm }^{}\mathrm{and}j\in \{2,\dots ,n\}.$$

$$\lambda ,\mu ,\nu \in \{1,\dots ,n\},i,j,k\in \{1,\dots ,m\},a,b,c\in \{m+1,\dots ,n\}$$

$$g_{ij}^{}=\overline{g}{}_{ij}^{},g_{ia}^{}=g_{ai}^{}=0,g_{ab}^{}=e^{2\theta }{\eta }_{ab}^{}.$$

$$\begin{array}[]{l}g^{\hskip 0.7ptij}\hskip-1.5pt=\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt^{\hskip 0.4ptij},\hskip 5.0ptg^{\hskip 0.7ptia}\hskip-1.5pt=g^{\hskip 0.7ptai}\hskip-0.7pt=0\hskip 0.4pt,\hskip 5.0ptg^{\hskip 0.7ptab}\hskip-1.5pt=e^{-\hskip-0.7pt2\theta}\hskip-0.7pt\eta^{ab},\hskip 5.0pt\varGamma_{\hskip-2.1ptij}^{\hskip 0.7ptk}\hskip-0.7pt=\hskip 3.0pt\overline{\hskip-3.0pt\varGamma\hskip-0.7pt}\hskip 0.7pt_{\hskip-3.7ptij}^{\hskip 0.7ptk}\hskip 0.4pt,\hskip 5.0pt\varGamma_{\hskip-2.1ptia}^{\hskip 0.7ptk}\hskip-0.7pt=\varGamma_{\hskip-2.1ptij}^{\hskip 0.7pta}\hskip-0.7pt=0\hskip 0.4pt,\hskip 5.0pt\varGamma_{\hskip-2.1ptia}^{\hskip 0.7ptb}\hskip-0.7pt=\delta_{\hskip-0.7pta}^{b}\theta\hskip-1.5pt_{,\hskip 0.4pti}^{\phantom{i}}\hskip 0.4pt,\\ \varGamma_{\hskip-3.2ptab}^{\hskip 0.7pti}=-e^{2\theta}\hskip-1.5pt\eta_{ab}^{\phantom{i}}\hskip 0.7pt\theta^{,i},\hskip 8.0pt\varGamma_{\hskip-2.7ptab}^{\hskip 0.7ptc}=H\hskip-1.5pt_{ab}^{\hskip 0.7ptc}\hskip 0.4pt,\hskip 8.0ptR_{i\hskip-0.4ptj}^{\phantom{i}}\hskip-1.5pt=\hskip-0.7pt\hskip 3.0pt\overline{\hskip-3.0ptR\hskip-0.7pt}\hskip 0.7pt_{i\hskip-0.4ptj}^{\phantom{i}}\hskip-0.7pt-p\hskip 0.4pt[\hskip 0.7pt\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt_{\hskip-2.7pti}^{\phantom{i}}\hskip-2.2pt\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt_{\hskip-2.7ptj}^{\phantom{i}}\hskip-0.7pt\theta+\hskip-0.7pt\theta\hskip-1.5pt_{,\hskip 0.4pti}^{\phantom{i}}\hskip 0.4pt\theta\hskip-1.5pt_{,\hskip 0.4ptj}^{\phantom{i}}]\hskip 0.4pt,\end{array}$$

$$R_{ia}^{}=0,R_{ab}^{}=P{}_{ab}^{}-p^{-1}{e}^{(2-p)\theta }[\overline{\Delta }{e}^{p\theta }]{\eta }_{ab}^{}.$$

$$\begin{array}[]{l}R\hskip-0.7pt_{j\hskip-0.4ptk,\hskip 0.7pti}^{\phantom{i}}=\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt_{\hskip-2.7pti}^{\phantom{i}}\hskip-2.2pt\hskip 3.0pt\overline{\hskip-3.0ptR\hskip-0.7pt}\hskip 0.7pt\hskip-0.7pt_{j\hskip-0.4ptk}^{\phantom{i}}\hskip-0.7pt-p\hskip 0.4pt[\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt_{\hskip-2.7pti}^{\phantom{i}}\hskip-2.2pt\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt_{\hskip-2.7ptj}^{\phantom{i}}\hskip-2.2pt\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt_{\hskip-2.7ptk}^{\phantom{i}}\hskip-0.7pt\theta+\,\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt_{\hskip-2.7pti}^{\phantom{i}}\hskip-3.0pt(\hskip-0.7pt\theta\hskip-1.5pt_{,\hskip 0.4ptj}^{\phantom{i}}\hskip 0.4pt\theta\hskip-1.5pt_{,\hskip 0.4ptk}^{\phantom{i}})]\hskip 0.4pt,\hskip 28.0ptR_{i\hskip-0.4ptj,\hskip 0.7pta}^{\phantom{i}}\hskip-0.7pt=R_{a\hskip-0.4ptj,\hskip 0.7pti}^{\phantom{i}}\hskip-0.7pt=0\hskip 0.4pt,\\ R_{ib,\hskip 0.7pta}^{\phantom{i}}\hskip-0.7pt=e^{2\theta}\hskip-0.4pt(p^{-\hskip-0.7pt1}\hskip-0.7pte^{-p\theta}\hskip-0.7pt[\hskip 0.7pt\overline{\hskip-0.7pt\Delta\hskip 0.4pt}e^{p\theta}]\hskip 0.7pt\theta\hskip-1.5pt_{,\hskip 0.4pti}^{\phantom{i}}\hskip-0.7pt+\hskip-0.7pt[\hskip 3.0pt\overline{\hskip-3.0ptR\hskip-0.7pt}\hskip 0.7pt_{i\hskip-0.4ptj}^{\phantom{i}}\hskip-0.7pt-p\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt_{\hskip-2.7pti}^{\phantom{i}}\hskip-2.2pt\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt_{\hskip-2.7ptj}^{\phantom{i}}\hskip-0.7pt\theta-p\hskip 0.85pt\theta\hskip-1.5pt_{,\hskip 0.4pti}^{\phantom{i}}\hskip 0.4pt\theta\hskip-1.5pt_{,\hskip 0.4ptj}^{\phantom{i}}]\hskip 0.7pt\theta\hskip 0.4pt^{,\hskip 0.7ptj})\hskip 0.4pt\eta_{ab}^{\phantom{i}}\hskip-1.5pt-\hskip-0.7pt\theta\hskip-1.5pt_{,\hskip 0.4pti}^{\phantom{i}}\hskip 0.4ptP\hskip-2.7pt_{ab}^{\phantom{i}}\hskip 0.4pt,\\ R_{ab,\hskip 0.7pti}^{\phantom{i}}\hskip-0.7pt=-\hskip-0.7ptp^{-\hskip-0.7pt1}\hskip-0.7pte^{2\theta}\hskip-0.7pt(e^{-p\theta}\overline{\hskip-0.7pt\Delta\hskip 0.4pt}e^{p\theta})\hskip-1.5pt_{,\hskip 0.4pti}^{\phantom{i}}\hskip 0.4pt\eta_{ab}^{\phantom{i}}\hskip-1.5pt-\hskip-0.7pt2\hskip 0.4pt\theta\hskip-1.5pt_{,\hskip 0.4pti}^{\phantom{i}}\hskip 0.4ptP\hskip-2.7pt_{ab}^{\phantom{i}}\hskip 0.4pt,\hskip 16.0ptR_{ab,\hskip 0.7ptc}^{\phantom{i}}=D\hskip-1.5pt_{c}^{\phantom{i}}P\hskip-2.7pt_{ab}^{\phantom{i}}\hskip 0.4pt.\end{array}$$

Full text

Both authors’ research was supported in part by a FAPESP- OSU 2015 Regular Research Award (FAPESP grant: 2015/50265-6)

Introduction

A Riemannian manifold is said to have harmonic curvature [1, Sect. 16.33] if

[TABLE]

$R\,$ being the curvature tensor. We consider harmonic-curvature Riemannian manifolds $\,(M\hskip-0.7pt,g)\,$ of dimensions $\,n\geq 3\,$ in which, with $\,\mathrm{r}\,$ denoting the Ricci tensor,

[TABLE]

Such warped -product decompositions have one -dimensional fibres (Corollary 1.3), and are in a one -to -one correspondence with certain one -dimensional Lie subalgebras of $\,\mathfrak{isom}(M\hskip-0.4pt^{\prime}\hskip-1.5pt,g^{\prime})$, the Lie algebra of Killing fields on the Riemannian universal covering $\,(M\hskip-0.4pt^{\prime}\hskip-1.5pt,g^{\prime})\,$ of $\,(M\hskip-0.7pt,g)\,$ (see Remark 3.2). The number $\,\gamma\,$ of these subalgebras cannot exceed $\,n\hskip-0.4pt-\hskip-0.7pt1$, cf. Corollary 4.2, and we refer to $\,g\,$ as maximally warped if

[TABLE]

Our Theorem 5.6 describes the local structure of Riemannian manifolds $\,(M\hskip-0.7pt,g)$ of dimensions $\,n\geq 3$, satisfying (0.1) – (0.3). Their local-isometry types turn out to form a $\,(2n-3)$-dimensional moduli space (Remark 5.7), and we prove (in Theorem 7.3) that some nonempty open subset of the moduli space consists of local-isometry types of such manifolds which in addition are

[TABLE]

We do not know if any compact manifold can have the properties (0.1) – (0.3). However, we observe (see Theorem 1.4) that compact Riemannian manifolds with harmonic curvature that admit global nontrivial warped-product decompositions must have fibre dimensions greater than one and, consequently, cannot be Ricci-generic in the sense of satisfying the distinct-eigenvalues clause of (0.2).

Harmonicity of the curvature always follows if the metric is Ricci-parallel, or locally reducible with harmonic-curvature factors, or conformally flat and of constant scalar curvature while, in dimension three, harmonic curvature amounts to conformal flatness plus constancy of the scalar curvature [1, Sect. 16.35 and 16.4].

Compact Riemannian manifolds with (0.1) have been studied extensively. All their known examples (aside from the three classes italicized above), listed as 2,3,4 in [1, p. 432], admit – at least locally – nontrivial warped-product decompositions with a fibre of dimension greater than one. Consequently (see Corollary 1.3 below), they are not Ricci-generic. However, for examples 2 and 4 of [1, p. 432] the warped-product structure, rather than being an Ansatz, is a consequence of geometric conditions involving multiplicities of eigenvalues of the Ricci tensor [2] or self-dual Weyl tensor [3]. The following questions about compact Riemannian manifolds with harmonic curvature, lying outside of the three italicized classes, are thus open and natural: can they be Ricci-generic? must they, locally, have a warped-product decomposition and if not, how to describe those among them which have one?

Our Theorem 7.3 yields an affirmative answer to a weaker version of the first question in which completeness replaces compactness. On the other hand, the second question provides an obvious motivation for studying condition (0.3).

The authors wish to thank the referees for constructive comments, which have resulted in a substantial improvement of the presentation.

1. Preliminaries

Manifolds (always assumed connected), mappings and tensor fields are by definition $\,C^{\infty}\hskip-1.5pt$-differentiable. By a Codazzi tensor [1, p. 435] on a Riemannian manifold one means a twice -covariant symmetric tensor field $\,S\,$ with a totally symmetric covariant derivative $\,\nabla\hskip-1.5ptS$. One then has two well-known facts [1, Sect. 16.4(ii)]:

[TABLE]

$\,\mathrm{s}\,$ being the scalar curvature. As shown by DeTurck and Goldschmidt [4],

[TABLE]

We call two (connected) real -analytic Riemannian manifolds locally isometric if they have open submanifolds that are both isometric to open submanifolds of a third such manifold. One easily sees that this is an equivalence relation. In view of the extension theorem for analytic isometries [9, Corollary 6.4 on p. 256], for two complete real -analytic Riemannian manifolds, being locally isometric to each other means the same as having isometric Riemannian universal coverings.

On a manifold with a torsion-free connection $\,\nabla\hskip-0.7pt$, the Ricci tensor $\,\mathrm{r}\,$ satisfies the Bochner identity $\,\mathrm{r}(\,\cdot\,,v)+\hskip 0.7ptd\hskip 0.7pt[\mathrm{div}\hskip 0.7ptv]=\hskip 0.7pt\mathrm{div}\hskip 1.2pt\nabla\hskip-0.7ptv$, where $\,v\,$ is any vector field. Its coordinate form $\,R\hskip-0.7pt_{jk}^{\phantom{i}}v^{\hskip 0.7ptk}\hskip-0.7pt=v^{\hskip 0.7ptk}{}_{,\hskip 0.7ptjk}^{\phantom{i}}\hskip-0.7pt-v^{\hskip 0.7ptk}{}_{,\hskip 0.7ptkj}^{\phantom{i}}$ arises via contraction from the Ricci identity $\,v^{\hskip 0.7ptl}{}_{,\hskip 0.7ptjk}^{\phantom{i}}\hskip-0.7pt-v^{\hskip 0.7ptl}{}_{,\hskip 0.7ptkj}^{\phantom{i}}\hskip-0.7pt=R\hskip-0.7pt_{jkq}^{\phantom{i}}{}^{l}\hskip 0.7ptv\hskip 0.4pt^{q}\hskip-0.7pt$. (We use the sign convention for $\,R\,$ such that $\,R\hskip-0.7pt_{jk}^{\phantom{i}}\hskip-0.7pt=R\hskip-0.7pt_{jqk}^{\phantom{i}}{}^{q}\hskip-0.7pt$.) Applied to the gradient $\,v\,$ of a function $\,\phi\,$ on a Riemannian manifold, this yields

[TABLE]

The warped product of Riemannian manifolds $\,(\hskip 3.0pt\overline{\hskip-3.0ptM\hskip-0.7pt}\hskip 0.7pt\hskip-0.7pt,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt)\,$ and $\,(\varSigma,\eta)\,$ with the warping function $\,\phi:\hskip 3.0pt\overline{\hskip-3.0ptM\hskip-0.7pt}\hskip 0.7pt\to(0,\infty)\,$ is the Riemannian manifold

[TABLE]

(The same symbols $\,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt,\eta,\phi\,$ stand here for also the pullbacks of $\,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt,\eta,\phi\,$ to the product $\,M\hskip-0.7pt=\hskip 3.0pt\overline{\hskip-3.0ptM\hskip-0.7pt}\hskip 0.7pt\times\varSigma$.) One calls $\,(\hskip 3.0pt\overline{\hskip-3.0ptM\hskip-0.7pt}\hskip 0.7pt\hskip-0.7pt,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt)\,$ and $\,(\varSigma,\eta)\,$ the base and fibre of (1.4), and refers to (1.4) as nontrivial if $\,\phi\,$ is nonconstant. From now on we assume that $\,\dim\varSigma\geq 1$.

Remark 1.1*.*

As $\,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt+\hskip-0.7pt\phi^{2}\eta=\phi^{2}\hskip 0.4pt[\hskip 0.7pt\phi^{-\hskip-0.7pt2}\hskip-0.7pt\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt+\eta\hskip 0.7pt]$, a warped product is nothing else than a Riemannian manifold conformal to a Riemannian product via multiplication by a positive function which is constant along one of the factor manifolds.

A proof of the following well-known lemma [7] is given in the Appendix.

Lemma 1.2**.**

A warped product (1.4) with a nonconstant warping function $\,\phi$ has harmonic curvature if and only if the Levi-Civita connection $\,\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\,$ of $\,(\hskip 3.0pt\overline{\hskip-3.0ptM\hskip-0.7pt}\hskip 0.7pt\hskip-0.7pt,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt)$, its Ricci tensor $\,\overline{\mathrm{r}}\hskip-0.4pt$ and the $\,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt$-gradient $\,\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-1.5pt\phi\,$ of $\,\phi\,$ satisfy three conditions :

1. (a)

$(\varSigma,\eta)\,$* is an Einstein manifold, with some Einstein constant $\,\kappa$.* 2. (b)

$\overline{\mathrm{r}}-\hskip 0.4ptp\phi^{-\hskip-1.5pt1}\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-0.7ptd\phi\,$* is a Codazzi tensor on $\,(\hskip 3.0pt\overline{\hskip-3.0ptM\hskip-0.7pt}\hskip 0.7pt\hskip-0.7pt,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt)$, where $\,p=\dim\varSigma\geq 1$.* 3. (c)

$\phi^{3}\hskip 0.4pt\overline{\mathrm{div}}\hskip 0.4pt[\phi^{-\hskip-1.5pt1}\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-0.7ptd\phi]=[(p-1)\varLambda-\kappa]\,d\phi+(1-p)\phi\,d\varLambda/2$, with $\,\varLambda=\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt(\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-1.5pt\phi\hskip-0.4pt,\hskip-1.5pt\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-1.5pt\phi)$

and the $\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt$-divergence $\hskip 0.7pt\overline{\mathrm{div}}\hskip-0.7pt$. Then, at each point of $M\hskip-1.5pt$, for the Ricci tensor $\mathrm{r}\hskip-0.7pt$ of $g$,

1. (d)

the space tangent to the fibre factor is contained in an eigenspace of $\,\mathrm{r}$.

One may rewrite (c) as a requirement involving the $\,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt$-Laplacian $\,\hskip 0.7pt\overline{\hskip-0.7pt\Delta\hskip-0.7pt}\hskip 0.7pt\phi\hskip-0.4pt$, namely,

1. (e)

$\phi^{2}[\overline{\mathrm{r}}(\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-1.5pt\phi\hskip-0.7pt,\,\cdot\,)+d\hskip 0.4pt\hskip 0.7pt\overline{\hskip-0.7pt\Delta\hskip-0.7pt}\hskip 0.7pt\phi]=[(p-1)\varLambda-\kappa]\,d\phi+(1-p/2)\phi\,d\varLambda$.

Finally, when $\,p=1$, and so $\,\kappa=0$, (c) reads $\,\overline{\mathrm{div}}\hskip 0.4pt[\phi^{-\hskip-1.5pt1}\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-0.7ptd\phi]=0$.

From (d) and, respectively, (c), we obtain two easy consequences:

Corollary 1.3**.**

In a warped-product Riemannian manifold with a nonconstant warping function, harmonic curvature, and a fibre of dimension greater than one, the Ricci tensor has, at every point, at least one multiple eigenvalue. The assumptions (0.1) – (0.2) thus imply one-dimensionality of the fibre for any warped-product decomposition in (0.2).

Theorem 1.4**.**

If a warped-product Riemannian manifold $\,(M\hskip-0.7pt,g)\,$ has a compact base $\,(\hskip 3.0pt\overline{\hskip-3.0ptM\hskip-0.7pt}\hskip 0.7pt\hskip-0.7pt,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt)$, a nonconstant warping function, and harmonic curvature, then the Einstein constant $\,\kappa$ of its fibre must be positive. Thus, the dimension of the fibre is greater than one and, in view of Corollary 1.3, $\,(M\hskip-0.7pt,g)\,$ cannot satisfy the distinct-eigenvalues clause of (0.2).

In fact, given a positive function $\,\phi\,$ on a Riemannian manifold $\,(\hskip 3.0pt\overline{\hskip-3.0ptM\hskip-0.7pt}\hskip 0.7pt\hskip-0.7pt,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt)\,$ and constants $\,\kappa,p\in\mathrm{I\!R}$, let us set $\,v=\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-1.5pt\phi\,$ and $\,w=\phi^{2}u-[(p-1)\varLambda-\kappa]\hskip 0.7ptv+(p-2)\hskip 0.4pt\phi\hskip 0.7pt\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-2.7pt_{v}^{\phantom{i}}v$, where $\,u=\hskip 0.7pt\overline{\mathrm{div}}\hskip-0.7pt\,\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-0.7ptv\,$ and $\,\varLambda=\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt(v,v)$. Then the function $\,\phi\hskip 0.7pt^{p-\hskip-0.4pt4}\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt(v,w)\,$ differs by a $\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt$-divergence from $\,-[(p-1)\varLambda-\kappa]\phi\hskip 0.7pt^{p-\hskip-0.4pt4}\hskip-0.7pt\varLambda-\phi\hskip 0.7pt^{p-\hskip-0.7pt2}\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt(\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-0.7ptv,\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-0.7ptv)$, while (c) reads $\,w=0$, as $\,2\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-2.7pt_{v}^{\phantom{i}}v=\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-1.5pt\varLambda$. (An easy exercise.)

Remark 1.5*.*

The base and fibre factor distributions of any warped product are Ricci-orthogonal to each other. (See the equality $\,R_{ia}^{\phantom{i}}\hskip-0.7pt=0\,$ in formula (A.3) of the Appendix.) Thus, if the base, or fibre, is one -dimensional, nonzero vectors tangent to it constitute eigenvectors of the Ricci tensor.

2. Vector fields

Lemma 2.1**.**

Let a maximal integral curve $\,(a_{-}^{\phantom{i}},a_{+}^{\phantom{i}})\ni t\mapsto x(t)\,$ of a vector field $\,v\hskip 0.7pt$ on a manifold $\,M\hskip-0.7pt$, with $\,-\infty\leq a_{-}^{\phantom{i}}\hskip-1.5pt<a_{+}^{\phantom{i}}\hskip-0.7pt\leq\infty$, some $\,t^{\prime}\hskip-0.7pt\in(a_{-}^{\phantom{i}},a_{+}^{\phantom{i}})$, and some compact set $\,C\hskip 0.4pt\subseteq M\hskip-0.7pt$, have the property that $\,x(t)\in C\hskip 0.7pt$ for all $\,t\in[\hskip 0.7ptt^{\prime}\hskip-0.7pt,a_{+}^{\phantom{i}})\,$ or, respectively, for all $\,t\in(a_{-}^{\phantom{i}},t^{\prime}]$. Then $\,a_{+}^{\phantom{i}}=\infty\,$ or, respectively, $\,a_{-}^{\phantom{i}}=-\infty$.

Consequently, a maximal integral curve of a vector field on a manifold, lying within a compact set, must be complete, that is, defined on $\,\mathrm{I\!R}$.

Proof.

For a compactly supported function $\,\chi\,$ equal to $\,1\,$ on an open set $\,\,U$ containing $\,C\hskip-0.4pt$, the curve restricted to $\,[\hskip 0.7ptt^{\prime}\hskip-0.7pt,a_{+}^{\phantom{i}})$, or to $\,(a_{-}^{\phantom{i}},t^{\prime}]$, clearly remains half-maximal (not extendible beyond $\,a_{+}^{\phantom{i}}$, or $\,a_{-}^{\phantom{i}}$) when treated as an integral curve of $\,\chi v$. On the other hand, $\,\chi v\,$ is complete due to compactness of its support. ∎

By a section of a locally trivial fibre bundle we mean, as usual, a submanifold $\,\varSigma\,$ of the total space $\,Q\,$ mapped diffeomorphically onto the base $\,M\,$ by the bundle projection $\,\mathrm{p}$. We also identify the section with the inverse $\,\psi:M\to\varSigma\,$ of the latter diffeomorphism, which makes it a mapping $\,\psi:M\to Q\,$ having $\,\mathrm{p}\circ\hskip-0.7pt\psi=\hskip 0.7pt\mathrm{Id}_{M}^{\phantom{i}}$. In the case of a vector bundle $\,Q$, a section $\,\psi$, and a zero $\,z\in M\,$ of $\,\psi$, the corresponding submanifold $\,\varSigma\,$ of $\,Q\,$ intersects the zero section $\,M\,$ at $\,z\,$ (that is, at the zero vector of the fibre $\,Q\hskip-0.7pt_{z}^{\phantom{i}}$), giving rise to the differential $\,\partial\hskip 0.4pt\psi\hskip-1.5pt_{z}^{\phantom{i}}$, defined to be the linear operator $\,T\hskip-2.7pt_{z}^{\phantom{i}}\hskip-0.7ptM\to Q\hskip-0.7pt_{z}^{\phantom{i}}$ obtained as the composite of the ordinary differential of $\,\psi:M\to Q\,$ at $\,z\,$ (the inverse of $\,d\mathrm{p}\hskip-0.7pt_{z}^{\phantom{i}}:T\hskip-2.7pt_{z}^{\phantom{i}}\hskip-0.7pt\varSigma\to T\hskip-2.7pt_{z}^{\phantom{i}}\hskip-0.7ptM$), followed by the direct-sum projection $\,T\hskip-2.7pt_{z}^{\phantom{i}}\hskip-0.7ptQ=T\hskip-2.7pt_{z}^{\phantom{i}}\hskip-0.7ptM\oplus Q\hskip-0.7pt_{z}^{\phantom{i}}\to Q\hskip-0.7pt_{z}^{\phantom{i}}$. Relative to any local coordinates at $\,z\,$ and a local trivialization of $\,Q$, the components of $\,A=\partial\hskip 0.4pt\psi\hskip-1.5pt_{z}^{\phantom{i}}$ form the matrix $\,[A_{j}^{\lambda}]=[\partial\hskip-0.7pt_{j}^{\phantom{i}}\psi^{\lambda}]$, with the partial derivatives of the components of $\,\psi\,$ evaluated at $\,z$.

Two important examples are provided by zeros $\,z\,$ of $\,\psi=v$, a vector field on $\,M\,$ (with $\,Q=T\hskip-1.5ptM$) and of $\,\psi=d\hskip-0.8ptf\hskip-1.5pt$, for a function $\,f:M\to\mathrm{I\!R}\,$ (here $\,Q=T^{*}\hskip-2.0ptM$). In the former case, $\,A=\partial v\hskip-0.7pt_{z}^{\phantom{i}}$ (in coordinates: $\,A_{j}^{k}=\partial_{j}v^{k}$), is the infinitesimal generator of the one-parameter group of linear transformations of $\,T\hskip-2.7pt_{z}^{\phantom{i}}\hskip-0.7ptM\,$ arising as the differentials, at the fixed point $\,z$, of the local diffeomorphisms forming the local flow of $\,v$. In the latter, $\,\partial d\hskip-0.8ptf\hskip-1.5pt_{z}^{\phantom{i}}=\hskip 0.7pt\mathrm{Hess}_{z}^{\phantom{i}}f\hskip-1.5pt$, the Hessian of $\,f\,$ at the critical point $\,z$.

Let $\,v\,$ be a vector field on a manifold $\,M\hskip-0.7pt$, having a zero at $\,z\in M\hskip-0.7pt$, where one assumes $\,M\,$ either to be an open submanifold of a vector space $\,Y\hskip-1.5pt$, or to have a submanifold $\,N\hskip 0.7pt$ with $\,z\in N\hskip 0.7pt$ such that $\,v\,$ is tangent to $\,N\hskip 0.7pt$ at each point of $\,N\hskip-0.7pt$. In this way $\,v$, or the restriction of $\,v\,$ to $\,N\hskip-1.5pt$, becomes a mapping $\,v:M\hskip-0.4pt\to\hskip 0.4ptY\hskip-1.5pt$, or a vector field $\,w\,$ on $\,N\hskip-1.5pt$. The equality $\,A_{j}^{k}=\partial_{j}v^{k}$ of the last paragraph, evaluated in coordinates for $\,M\,$ which are linear functionals on $\,Y\hskip-0.7pt$ or, respectively, in which $\,N$ is defined by equating some coordinate functions to $\,0$, clearly implies that

[TABLE]

Lemma 2.2**.**

Given a zero $\,z\in M\hskip 0.7pt$ of a vector field $\,v\hskip 0.7pt$ on a manifold $\,M\hskip-1.5pt$, with the differential $\,A=\partial v\hskip-0.7pt_{z}^{\phantom{i}}$, let a function $\,f:U\to\mathrm{I\!R}\,$ on a neighborhood $\,\,U$ of $\,z$ have $\,d\hskip-0.8ptf\hskip-1.5pt_{z}^{\phantom{i}}=0$. Then $\,d\sigma\hskip-1.5pt_{z}^{\phantom{i}}=0\,$ and $\,(u,u)\hskip-0.7pt_{\sigma}^{\phantom{i}}=2(Au,u)\hskip-1.5pt_{f}^{\phantom{i}}$ for the directional derivative $\,\sigma=d_{v}^{\phantom{i}}f:U\to\mathrm{I\!R}$, all $\,u\in T\hskip-2.7pt_{z}^{\phantom{i}}\hskip-0.7ptM\hskip-1.5pt$, the Hessian $\,(\hskip 3.0pt,\hskip 0.7pt)\hskip-1.5pt_{f}^{\phantom{i}}=\mathrm{Hess}_{z}^{\phantom{i}}f\hskip-1.5pt$, and $\,(\hskip 3.0pt,\hskip 0.7pt)\hskip-0.7pt_{\sigma}^{\phantom{i}}=\mathrm{Hess}_{z}^{\phantom{i}}\sigma$.

Proof.

With commas denoting, this time, partial derivatives relative to fixed local coordinates on a neighborhood of $\,z$, we have $\,\sigma=v\hskip 0.4pt^{j}\hskip-1.5ptf\hskip-1.5pt_{,j}^{\phantom{i}}$ as well as $\,\sigma\hskip-1.5pt_{,\hskip 0.7ptk}^{\phantom{i}}=v\hskip 0.4pt^{j}\hskip-1.5ptf\hskip-1.5pt_{,jk}+v\hskip 0.4pt^{j}\hskip-0.7pt{}_{,k}^{\phantom{i}}f\hskip-1.5pt_{,j}^{\phantom{i}}$ and $\,\sigma\hskip-1.5pt_{,\hskip 0.7ptkl}^{\phantom{i}}=v\hskip 0.4pt^{j}\hskip-1.5ptf\hskip-1.5pt_{,jkl}^{\phantom{i}}+v\hskip 0.4pt^{j}\hskip-0.7pt{}_{,\hskip 0.4ptl}^{\phantom{i}}f\hskip-1.5pt_{,jk}^{\phantom{i}}+v\hskip 0.4pt^{j}\hskip-0.7pt{}_{,\hskip 0.7ptk}^{\phantom{i}}f\hskip-1.5pt_{,jl}^{\phantom{i}}+v\hskip 0.4pt^{j}\hskip-0.7pt{}_{,\hskip 0.7ptkl}^{\phantom{i}}f\hskip-1.5pt_{,j}^{\phantom{i}}$. At $\,z$, both $\,v\hskip 0.4pt^{j}$ and $\,f\hskip-1.5pt_{,j}^{\phantom{i}}$ vanish, while $\,v\hskip 0.4pt^{j}\hskip-0.7pt{}_{,\hskip 0.7ptk}^{\phantom{i}}=A^{j}_{k}$. This proves our claim. ∎

Lemma 2.3**.**

Let $\,z\in N$ be a zero of a vector field $\,w\hskip 0.7pt$ on a manifold $\,N$ such that, for some $\,\varepsilon=\pm 1$, some Euclidean inner product $\,\langle\,,\rangle$ in $\,T\hskip-2.7pt_{z}^{\phantom{i}}\hskip-0.7ptN\hskip-1.5pt$, and $\,A=\varepsilon\partial w\hskip-0.7pt_{z}^{\phantom{i}}$, the bilinear form $\,\langle A\hskip 0.7pt\cdot\,,\,\cdot\,\rangle\,$ on $\,T\hskip-2.7pt_{z}^{\phantom{i}}\hskip-0.7ptN\,$ is negative definite. In this case there exist arbitrarily small neighborhoods $\,\,U\,$ of $\,z\,$ with the following property:* if a maximal integral curve $\,(a_{-}^{\phantom{i}},a_{+}^{\phantom{i}})\ni t\mapsto x(t)\,$ of $\,w\hskip 0.7pt$ and $\,t^{\prime}\hskip-0.7pt\in(a_{-}^{\phantom{i}},a_{+}^{\phantom{i}})\,$ satisfy the condition $\,x(t^{\prime})\in\,U\hskip-0.7pt$, where $\,-\infty\leq a_{-}^{\phantom{i}}\hskip-0.7pt<a_{+}^{\phantom{i}}\hskip-0.4pt\leq\infty$, then, denoting by $\,\pm\,$ the sign of $\,\varepsilon$, one has $\,a_{\pm}^{\phantom{i}}=\pm\infty$, and $\,x(t)\in\,U\,$ whenever $\,\varepsilon\hskip 0.4pt(t\hskip-0.4pt-t^{\prime})\geq 0$.*

Proof.

We fix a Riemannian metric $\,g\,$ on a neighborhood of $\,z\,$ in $\,N\hskip 0.7pt$ having $\,\langle\,,\rangle=g\hskip-0.7pt_{z}^{\phantom{i}}$. The required neighborhoods $\,\,U\,$ of $\,z\,$ are $\,g$-metric balls centered at $\,z$, small enough so as to have compact closures and be diffeomorphic images, under the $\,g$-exponential mapping at $\,z$, of the corresponding Euclidean balls around $\,0$ in $\,T\hskip-2.7pt_{z}^{\phantom{i}}\hskip-0.7ptN\hskip-1.5pt$. This gives smoothness of the function $\,f:U\to\mathrm{I\!R}\,$ such that $\,2f\,$ equals $\,\mathrm{dist}\hskip-0.4pt^{2}(z,\,\cdot\,)$, the squared $\,g$-distance from $\,z$, and using normal coordinates one obtains $\,\mathrm{Hess}_{z}^{\phantom{i}}f\hskip-0.7pt=\langle\,,\rangle$. If the $\,g$-metric ball $\,\,U\,$ is sufficiently small, Lemma 2.2 for $\,v=\varepsilon w\,$ implies negativity of $\,\sigma=\varepsilon d_{w}^{\phantom{i}}f\,$ on $\,\,U\hskip-0.7pt\smallsetminus\{z\}$, as $\,\sigma\,$ assumes at $\,z\,$ the critical value $\,0\,$ with a negative-definite Hessian. Our claim now easily follows from Lemma 2.1. ∎

Remark 2.4*.*

The same neighborhoods $\,\,U\,$ of $\,z\,$ will still satisfy the assertion of Lemma 2.3 if one replaces $\,w\,$ by $\,w/c\,$ for a constant $\,c>0\,$ and $\,(a_{-}^{\phantom{i}},a_{+}^{\phantom{i}})\ni t\mapsto x(t)$ by $\,(ca_{-}^{\phantom{i}},ca_{+}^{\phantom{i}})\ni t\mapsto x(t/c)$.

Remark 2.5*.*

For any Killing field $\,v\,$ on a Riemannian manifold $\,(M\hskip-0.7pt,g)$, the pair $\,(v,\hskip-0.7pt\nabla\hskip-0.7ptv)\,$ constitutes a parallel section of the vector bundle $\,T\hskip-1.5ptM\oplus\,\mathfrak{so}(T\hskip-1.5ptM)$ endowed with a suitable linear connection [5, Remark 17.25 on p. 547]. Therefore,

1. (i)

a Killing field on $\,M\,$ is uniquely determined by its restriction to any nonempty open subset of $\,M\hskip-0.7pt$, while 2. (ii)

assuming $\,(M\hskip-0.7pt,g)\,$ to be simply connected and real -analytic, we conclude that any Killing field $\,v\,$ on a nonempty connected open subset of $\,M\,$ has a unique extension to a Killing field on $\,M\hskip-0.7pt$.

Given a nontrivial Killing vector field $\,v\,$ on a Riemannian manifold and a function $\,\theta\hskip-0.7pt$, the obvious equality $\,\pounds\hskip-1.0pt_{\theta\hskip-0.7ptv}^{\phantom{i}}g\hskip 0.7pt\,=\,\hskip 0.7pt\theta\hskip-0.7pt\pounds\hskip-1.0pt_{v}^{\phantom{i}}g\hskip 0.7pt+2\hskip 0.7ptd\theta\odot g(v,\,\cdot\,)\,$ clearly implies that

[TABLE]

3. Integrable-complement Killing fields

This section presents a well-known correspondence – see, for instance, the Appendix in [10] – between warped -product decompositions with a one -dimensional fibre and certain special Killing fields.

Let $\,v\,$ a nontrivial Killing field on a Riemannian manifold $\,(M\hskip-0.7pt,g)\,$ such that, on the dense (by Remark 2.5(i)) complement of its zero set, the distribution $\,v^{\perp}$ is integrable. In other words, locally, at points with $\,v\neq 0$, multiplying $\,v\,$ by a suitable positive function one obtains a gradient vector field. Equivalently,

[TABLE]

Namely, (3.1) is necessary: for $\,\xi=\,g(v,\,\cdot\,)$, due to skew-symmetry of $\,\nabla\hskip-0.7pt\xi$, the integrability condition $\,\xi\wedge\hskip 0.7ptd\hskip 0.4pt\xi=0\,$ has the local-coordinate expression $\,\xi\hskip 0.4pt_{i,j}^{\phantom{i}}\xi_{k}^{\phantom{i}}+\xi\hskip-0.7pt_{j,k}^{\phantom{i}}\xi\hskip 0.4pt_{i}^{\phantom{i}}+\xi_{k,i}^{\phantom{i}}\xi_{j}^{\phantom{i}}=0$, which transvected with $\,v\hskip 0.4pt^{k}$ yields $\,v\hskip 0.4pt^{k}\xi_{k}^{\phantom{i}}\xi\hskip 0.4pt_{i,j}^{\phantom{i}}=v\hskip 0.4pt^{k}\xi_{k,j}^{\phantom{i}}\xi\hskip 0.4pt_{i}^{\phantom{i}}-v\hskip 0.4pt^{k}\xi_{k,i}^{\phantom{i}}\xi\hskip-0.7pt_{j}^{\phantom{i}}$, or

[TABLE]

Closedness of $\,\xi/\hskip-0.7pt\beta\,$ amounts to symmetry of $\,\nabla(\xi/\hskip-0.7pt\beta)$, and so it now follows since (3.2) with $\,\xi\hskip 0.4pt_{i,j}^{\phantom{i}}\hskip-0.7pt=-\hskip 0.4pt\xi\hskip-0.7pt_{j,i}^{\phantom{i}}$ implies symmetry of $\,\beta^{2}(\xi\hskip 0.4pt_{i}^{\phantom{i}}/\hskip-0.7pt\beta)\hskip-1.5pt_{,\hskip 0.4ptj}^{\phantom{i}}=\beta\hskip 0.4pt\xi\hskip 0.4pt_{i,j}^{\phantom{i}}-\beta\hskip-1.5pt_{,\hskip 0.4ptj}^{\phantom{i}}\xi\hskip 0.4pt_{i}^{\phantom{i}}$ in $\,i,j$.

If $\,v\,$ is a Killing field, $\,g(v,\hskip 0.7pt\dot{\hskip-0.7ptx\hskip 0.7pt}\hskip-0.7pt)\,$ is constant along any geodesic $\,t\mapsto x=x(t)$, as $\,d\hskip 0.7pt[g(v,\hskip 0.7pt\dot{\hskip-0.7ptx\hskip 0.7pt}\hskip-0.7pt)]/dt=g(\nabla\hskip-2.7pt_{\hskip 0.4pt\dot{\hskip-0.4ptx\hskip 0.4pt}\hskip-0.4pt}^{\phantom{i}}v,\hskip 0.7pt\dot{\hskip-0.7ptx\hskip 0.7pt}\hskip-0.7pt)=0$. Then, with the orthogonal complement $\,v^{\perp}$ only defined away from the zero set of $\,v$, one easily sees that

[TABLE]

Remark 3.1*.*

Local Killing fields $\,v\,$ satisfying (3.1), outside of their zero sets, if treated as defined only up to multiplication by nonzero constants, stand in a natural one -to -one correspondence with local warped -product decompositions of $\,g$ that have a one -dimensional fibre. Here $\,v\,$ is tangent to the fibre direction.

Namely, such a local decomposition is uniquely determined by the base and fibre factor distributions. Just one of them suffices, the other being its (necessarily integrable) orthogonal complement. That $\,v\,$ locally spans the fibre factor distribution of a warped product follows from Remark 1.1 and the local version of de Rham’s decomposition theorem: in view of (3.2), rewritten as $\,2\beta v\hskip 0.4pt^{i}\hskip-1.5pt{}_{,\hskip 0.4ptj}^{\phantom{i}}\hskip-0.7pt=v\hskip 0.4pt^{i}\beta\hskip-1.5pt_{,\hskip 0.4ptj}^{\phantom{i}}\hskip-0.7pt-\beta\hskip 0.4pt^{,i}\xi\hskip-0.7pt_{j}^{\phantom{i}}$, where $\,\beta=v\hskip 0.4pt^{k}\xi_{k}^{\phantom{i}}=g(v,v)$, and [1, Theorem 1.159], $\,v\,$ is $\,\hat{g\hskip 2.0pt}\hskip-1.3pt$-parallel for the conformally related metric $\,\hat{g\hskip 2.0pt}\hskip-1.3pt=g/\hskip-0.7pt\beta$, with $\,d_{v}^{\phantom{i}}\beta=2g(\nabla\hskip-2.7pt_{v}^{\phantom{i}}v,v)=0\,$ due to skew-adjointness of $\,\nabla\hskip-0.7ptv$. Conversely, for a warped product with a one -dimensional fibre, the required Killing field $\,v\,$ comes from a local flow of local isometries of the fibre (cf. formula (A.2) in the Appendix), (2.2) implying uniqueness of $\,v\,$ up to a constant factor.

Remark 3.2*.*

From Remarks 2.5 and 3.1 it follows that, in the case of a real -analytic Riemannian manifold $\,(M\hskip-0.7pt,g)$, denoting by $\,\mathfrak{isom}(M\hskip-0.4pt^{\prime}\hskip-1.5pt,g^{\prime})\,$ the Lie algebra of Killing fields on the Riemannian universal covering $\,(M\hskip-0.4pt^{\prime}\hskip-1.5pt,g^{\prime})\,$ of $\,(M\hskip-0.7pt,g)$, one has a natural bijective correspondence between the one -dimensional Lie subalgebras of $\,\mathfrak{isom}(M\hskip-0.4pt^{\prime}\hskip-1.5pt,g^{\prime})\,$ spanned by Killing fields $\,v\,$ satisfying (3.1), and the local warped -product decompositions, with one -dimensional fibres, of $\,g\,$ restricted to the dense open set where $\,v\neq 0$. As before, $\,v\,$ is tangent to the fibre direction.

Lemma 3.3**.**

Let an open ball $\,B$ around $\,0$ in a Euclidean $\,n$-space, $\,n\geq 2$, admit a connection $\,\nabla$ such that all line segments through $\,0$ in $\,B$ are $\,\nabla\hskip-1.5pt$-totally geodesic and tangent at all points $\,x\in B\smallsetminus\{0\}\hskip 0.7pt$ to some codimension-one foliation $\,\mathcal{F}\hskip-0.7pt$ on $\,B\smallsetminus\{0\}\hskip 0.7pt$ having $\,\nabla\hskip-1.5pt$-totally geodesic leaves. Then $\,n=2$.

Proof.

Fix a leaf $\,L\,$ of $\,\mathcal{F}\hskip-0.7pt$ and $\,x\in L\,$ such that the $\,\nabla\hskip-1.5pt$-exponential mapping $\,\exp_{x}^{\phantom{i}}$ sends a Euclidean open ball $\,B^{\prime}$ centered at $\,0\,$ in $\,T\hskip-2.7pt_{x}^{\phantom{i}}\hskip-0.4ptB$, diffeomorphically, onto a neighborhood $\,\exp_{x}^{\phantom{i}}(B^{\prime})\,$ of $\,0\,$ in $\,B$. Thus, $\,\exp_{x}^{\phantom{i}}(B^{\prime}\hskip-0.7pt\cap T\hskip-2.7pt_{x}^{\phantom{i}}\hskip-0.4ptL)\smallsetminus J\hskip 0.7pt\subseteq\hskip-0.7ptL$, for $\,J=B\cap\{qx:q\in(-\infty,0\hskip 0.7pt]\}$, and so $\,J\hskip 0.7pt\subseteq\hskip-0.7ptL$. (The leaves of $\,\mathcal{F}\hskip 0.7pt$ are locally closed, being, locally, the level sets of a submersion.) Hence $\,L\hskip 0.4pt\cup\hskip-0.4pt\{0\}\,$ is a smooth $\,\nabla\hskip-1.5pt$-totally geodesic submanifold of $\,B$, with some tangent space $\,V\hskip-0.4pt$ at $\,0$, meaning in turn that $\,L\hskip 0.4pt\cup\hskip-0.4pt\{0\}=B\hskip 0.4pt\cap\hskip 0.4ptV\hskip-0.7pt$. Consequently, $\,n=2$, for otherwise any two such codimension-one subspaces $\,V$ of our Euclidean $\,n$-space would have a nontrivial intersection. ∎

When $\,\mathcal{F}\hskip 0.7pt$ is real -analytic, we can also obtain the above assertion by applying, to a sphere $\,\varSigma\,$ around $\,0\,$ in $\,B$, Haefliger’s theorem [6] which states that a transversally orientable real -analytic codimension-one foliation may exist on a compact manifold $\varSigma\,$ only if the fundamental group of $\,\varSigma\,$ has an element of infinite order.

Remark 3.4*.*

Kobayashi [8] showed that the zero set of any Killing vector field on a Riemannian manifold $\,(M\hskip-0.7pt,g)\,$ is either empty, or its connected components are mutually isolated totally geodesic submanifolds of even codimensions.

For a nontrivial Killing field $\,v\,$ with (3.1), the above codimensions must all equal $\,2$. This is immediate if one fixes a zero $\,z\,$ of $\,v\,$ and applies Lemma 3.3 to a small ball $\,B\hskip 0.7pt$ in the normal space at $\,z\,$ of the connected component through $\hskip 0.7ptz\hskip 0.7pt$ such that $\,\exp_{z}^{\phantom{i}}$ maps $\,B\hskip 0.7pt$ diffeomorphically onto a submanifold $\,N$ of $\,M\hskip-0.7pt$, with $\,\nabla$ and $\,\mathcal{F}\hskip 0.7pt$ denoting the $\,\exp_{z}^{\phantom{i}}$-pullback of the Levi-Civita connection of the submanifold metric $\,h\,$ on $\,N$ and, respectively, of the foliation on $\,N\hskip-0.7pt\smallsetminus\{z\}\,$ the leaves of which are intersections of $\,N\hskip 0.7pt$ and the leaves of $\,v^{\perp}\hskip-1.5pt$, the latter defined wherever $\,v\neq 0$. (The local flow of $\,v\,$ preserves $\,N\hskip 0.7pt$ and $\,h$, and so $\,v\,$ is tangent to $\,N\hskip-0.7pt$.) The restriction of $\,v\,$ to $\,N\hskip 0.7pt$ now constitutes an $\,h$-Killing field $\,w\,$ having just one zero, at $\,z$, and satisfying (3.1) (for $\,h,w\,$ rather than $\,g,v$), so that (3.3) allows us to use Lemma 3.3.

4. Multiply-warped metrics with $\,\mathrm{div}\hskip 0.7ptR=0$

Lemma 4.1**.**

Suppose that the Ricci tensor of a real-analytic Riemannian $\,n\hskip-0.4pt$-manifold $\,(M\hskip-0.7pt,g)\,$ has $\,n\hskip 0.7pt$ distinct eigenvalues at some point and, with the notation of Remark 3.2, $\,\mathfrak{a}_{2}^{\phantom{i}},\dots,\mathfrak{a}_{m}^{\phantom{i}}$ are distinct one -dimensional Lie subalgebras of $\,\mathfrak{isom}(M\hskip-0.4pt^{\prime}\hskip-1.5pt,g^{\prime})\,$ spanned by Killing fields $\,v_{2}^{\phantom{i}},\dots,v_{m}^{\phantom{i}}$ such that each $\,v=v\hskip-1.5pt_{j}^{\phantom{i}}$ satisfies (3.1). Then $\,m\hskip 0.7pt\leq n$, and $\,g(v\hskip-1.5pt_{j}^{\phantom{i}},v_{k}^{\phantom{i}})=0\,$ as well as $\,[v\hskip-1.5pt_{j}^{\phantom{i}},v_{k}^{\phantom{i}}]=0\,$ if $\,j\neq k$. Finally, $\,g(\nabla\hskip-2.7pt_{u}^{\phantom{i}}v\hskip-1.5pt_{j}^{\phantom{i}},v_{k}^{\phantom{i}})=0\,$ whenever $\,j,k,l\in\{2,\dots,m\}\,$ and $\,u=v_{l}^{\phantom{i}}$.

Proof.

Remarks 3.2 and 1.5 imply that all $\,v\hskip-1.5pt_{j}^{\phantom{i}}$, wherever nonzero, are mutually nonproportional eigenvectors of the Ricci tensor, which makes them pointwise orthogonal to one another, as well as invariant, up to constant factors – by (2.2) – under each other’s local flows. Thus, $\,m\leq n+1\,$ and, as $\,[v,w]=\pounds\hskip-1.0pt_{v}^{\phantom{i}}\hskip-0.4ptw\,$ for $\,v=v\hskip-1.5pt_{j}^{\phantom{i}}$ and $\,u=v_{k}^{\phantom{i}}$, one gets $\,[v\hskip-1.5pt_{j}^{\phantom{i}},v_{k}^{\phantom{i}}]=cv_{k}^{\phantom{i}}$ with some constant $\,c\,$ depending on $\,j\,$ and $\,k$. Switching $\,j\,$ and $\,k$, we see that $\,c=0$. Now let $\,u=v_{l}^{\phantom{i}}$, $\,v=v\hskip-1.5pt_{j}^{\phantom{i}}$ and $\,u=v_{k}^{\phantom{i}}$, where $\,j,k,l\in\{2,\dots,m\}$. We have $\,g(\nabla\hskip-2.7pt_{u}^{\phantom{i}}v,w)=0\,$ if $\,u=w\,$ (due to the Killing property of $\,v$) and, therefore, also when $\,v=w\,$ (since $\,u,v\,$ commute). Also, $\,g(\nabla\hskip-2.7pt_{u}^{\phantom{i}}v,w)=0$ in the remaining case, with $\,u,v\,$ different from $\,w\,$ (and hence orthogonal to $\,w$): as a consequence of (3.3), outside of the zero set of $\,w\,$ the distribution $\,w^{\perp}$ has totally geodesic leaves. This proves the final claim of the lemma, implying in turn that, if one had $\,m=n+1$, all $\,v\hskip-1.5pt_{j}$ would be parallel, leading to flatness of $\,g$, and contradicting the Ricci-eigenvalues assumption. ∎

Due to DeTurck and Goldschmidt’s real -analyticity theorem (1.2), we may combine Lemma 4.1 with Remark 3.2 and Corollary 1.3, obtaining

Corollary 4.2**.**

Under the assumptions (0.1) – (0.2), the integer $\,\gamma\,$ defined in the Introduction does not exceed $\,n\hskip-0.4pt-\hskip-0.7pt1$.

5. The local structure

Given an open interval $\,I\subseteq\mathrm{I\!R}$, we introduce a Riemannian metric $\,g\,$ on the open set $\,I\hskip-1.5pt\times\hskip-1.5pt\mathrm{I\!R}\hskip-0.7pt^{n\hskip-0.4pt-\hskip-0.7pt1}\hskip-0.7pt\subseteq\mathrm{I\!R}\hskip-0.7pt^{n}\hskip-1.5pt$, $\,n\geq 2$, by declaring its component functions in the Cartesian coordinates $\,x^{1}\hskip-0.7pt,x\hskip 0.4pt^{2}\hskip-0.7pt,\dots,x\hskip 0.4pt^{n}$ to be

[TABLE]

where $\,I\ni t\mapsto(g_{22}^{\phantom{i}}(t),\dots,g_{nn}^{\phantom{i}}(t))\in(0,\infty)^{n\hskip-0.4pt-\hskip-0.7pt1}$ is any prescribed smooth curve. We also define the functions $\,y_{2}^{\phantom{i}},\dots,y_{n}^{\phantom{i}}$ and $\,\mathbf{y}\,=\hskip 0.7pt\mathrm{diag}\hskip 0.7pt(\hskip-0.7pty_{2}^{\phantom{i}},\dots,y_{n}^{\phantom{i}})$ of the variable $\,t\in I\hskip-0.7pt$, valued in $\,\mathrm{I\!R}\,$ and, respectively, in the real vector space $\,\mathrm{I\!E}\cong\mathrm{I\!R}\hskip-0.7pt^{n\hskip-0.4pt-\hskip-0.7pt1}$ of all diagonal $\,(n\hskip-0.4pt-\hskip-0.7pt1)\times(n\hskip-0.4pt-\hskip-0.7pt1)\,$ matrices, by

[TABLE]

Remark 5.1*.*

If $\,I\hskip-0.7pt=\mathrm{I\!R}\,$ while $\,\mp y\hskip-0.7pt_{j}(t)\geq\delta\,$ whenever $\,\pm\hskip 0.4ptt\,$ is sufficiently large and positive, for both signs $\,\pm\hskip 0.7pt$, some constant $\,\delta>0$, and all $\,j\geq 2$, then the above metric $\,g\,$ is complete. In fact, (5.2) gives $\,\log g\hskip-0.7pt_{j\hskip-0.4ptj}^{\phantom{i}}(t)\to\infty\,$ as $\,|\hskip 0.4ptt|\to\infty$, so that $\,g\hskip-0.7pt_{j\hskip-0.4ptj}^{\phantom{i}}(t)\geq a\,$ with some constant $\,a\in(0,1]\,$ and all $\,t\in\mathrm{I\!R}$, which in turn gives $\,g\geq ag^{\prime}$ (positive semidefiniteness of $\,g-ag^{\prime}$) for the standard Euclidean metric $\,g^{\prime}\hskip-0.7pt$. Completeness of $\,g^{\prime}$ now implies that of $\,g$, as $\,g\hskip 0.4pt$-bounded sets have compact closures due to the resulting inequality $\,\mathrm{dist}\hskip 0.7pt\geq\hskip 0.7pta\,\mathrm{dist}\hskip-0.4pt^{\prime}$ between distance functions.

Let us consider the second-order autonomous ordinary differential equation

[TABLE]

imposed on a $\,C^{2}\hskip-0.7pt$ curve $\,I\ni t\mapsto\mathbf{y}\in\mathrm{I\!E}$, in which $\,\mathbf{y}\hskip-0.2pt\dot{\hskip 0.0pt\mathbf{y}\hskip 0.0pt}\,$ and $\,\mathbf{y}^{2}\hskip-0.7pt=\mathbf{y}\hskip-0.2pt\mathbf{y}\,$ represent diagonal-matrix products, while $\,\mathrm{tr}\hskip 1.7pt\mathbf{y}\,$ also denotes $\,\mathrm{tr}\hskip 1.7pt\mathbf{y}\,$ times the identity.

Lemma 5.2**.**

For a metric $\,g\,$ on $\,I\hskip-0.7pt\times\mathrm{I\!R}\hskip-0.7pt^{n\hskip-0.4pt-\hskip-0.7pt1}\hskip-0.7pt$ defined by (5.1) and the corresponding curve $\,I\ni t\mapsto\mathbf{y}\,=\hskip 0.7pt\mathrm{diag}\hskip 0.7pt(\hskip-0.7pty_{2}^{\phantom{i}},\dots,y_{n}^{\phantom{i}})\,$ with (5.2), at every point $\,(t,\mathbf{x})\in I\hskip-0.7pt\times\mathrm{I\!R}\hskip-0.7pt^{n\hskip-0.4pt-\hskip-0.7pt1}\hskip-1.5pt$, each of the coordinate vectors $\,\partial\hskip-0.7pt_{k}^{\phantom{i}}$, $\,k=1,\dots,n$, is an eigenvector of the Ricci tensor of $\,g\,$ with an eigenvalue $\,\mu_{k}^{\phantom{i}}$ depending on $\,t$.

1. (a)

Specifically, $\,\mu_{1}^{\phantom{i}}=\mathrm{tr}\hskip 1.7pt\dot{\hskip 0.0pt\mathbf{y}\hskip 0.0pt}-\mathrm{tr}\hskip 1.7pt\mathbf{y}^{2}$ and $\,\mu_{j}^{\phantom{i}}=\dot{y}\hskip-0.7pt_{j}^{\phantom{i}}-y\hskip-0.7pt_{j}^{\phantom{i}}\hskip 0.7pt\mathrm{tr}\hskip 1.7pt\mathbf{y}\,$ if $\,j\geq 2$. 2. (b)

The scalar curvature $\,\mathrm{s}\hskip 0.7pt$ of $\,g\,$ equals $\,2\hskip 1.7pt\mathrm{tr}\hskip 1.7pt\dot{\hskip 0.0pt\mathbf{y}\hskip 0.0pt}-\mathrm{tr}\hskip 1.7pt\mathbf{y}^{2}\hskip-0.7pt-(\mathrm{tr}\hskip 1.7pt\mathbf{y})^{2}\hskip-0.7pt$. 3. (c)

$\partial\hskip-0.7pt_{2}^{\phantom{i}},\dots,\partial\hskip-0.7pt_{n}^{\phantom{i}}$* are $\,g\hskip-0.4pt$-Killing fields with integrable orthogonal complements.* 4. (d)

Given any fixed $\,\mathbf{x}\in\mathrm{I\!R}\hskip-0.7pt^{n\hskip-0.4pt-\hskip-0.7pt1}\hskip-1.5pt$, the curve $\,I\hskip-0.7pt\ni t\mapsto(t,\mathbf{x})\,$ is a $\,g\hskip-0.4pt$-geodesic. 5. (e)

$g\,$* has harmonic curvature if and only if (5.3) holds.*

Proof.

We assume $\,j,k,l\,$ to range over $\,\{2,\dots,n\}\,$ and be mutually distinct. Repeated indices are not summed over. First, (c) is obvious as $\,g_{11}^{\phantom{i}},g_{1\hskip-0.4ptj}^{\phantom{i}},g\hskip-0.7pt_{j\hskip-0.4ptj}^{\phantom{i}},g\hskip-0.7pt_{jk}^{\phantom{i}}$ only depend on $\,t=x^{1}\hskip-0.7pt$. Also, $\,\varGamma_{\hskip-2.7pt11}^{\hskip 0.7pt1}=\varGamma_{\hskip-2.7pt11}^{\hskip 0.7ptj}=0$, proving (d), while $\,\varGamma_{\hskip-2.7pt1j}^{\hskip 0.7pt1}=\varGamma_{\hskip-2.7pt1j}^{\hskip 0.7ptk}=\varGamma_{\hskip-2.7ptj\hskip-0.4ptj}^{\hskip 0.7ptj}=\varGamma_{\hskip-2.7ptj\hskip-0.4ptj}^{\hskip 0.7ptk}=\varGamma_{\hskip-2.7ptjk}^{\hskip 0.7ptj}=\varGamma_{\hskip-2.7ptjk}^{\hskip 0.7ptl}=0\,$ and $\,g^{j\hskip-0.4ptj}\varGamma_{\hskip-2.7ptj\hskip-0.4ptj}^{\hskip 0.7pt1}=-\varGamma_{\hskip-2.7pt1j}^{\hskip 0.7ptj}=y\hskip-0.7pt_{j}^{\phantom{i}}$. Hence $\,R_{11}^{\phantom{i}}=\mu_{1}^{\phantom{i}}$ and $\,g^{j\hskip-0.4ptj}R\hskip-0.4pt_{j\hskip-0.4ptj}^{\phantom{i}}=\mu_{j}^{\phantom{i}}$ for $\,\mu_{1}^{\phantom{i}},\mu_{j}^{\phantom{i}}$ as in (a). This yields (a), and hence (b). (Each $\,\partial\hskip-0.7pt_{k}^{\phantom{i}}$ spans the fibre direction of a warped -product decomposition, and we may use Remark 1.5.) Next, $\,R_{11,\hskip 0.7ptj}^{\phantom{i}}=R_{1j,\hskip 0.7pt1}^{\phantom{i}}=R_{1j,\hskip 0.7ptk}^{\phantom{i}}=R\hskip-0.4pt_{jk,\hskip 0.7pt1}^{\phantom{i}}=R\hskip-0.4pt_{jk,\hskip 0.7ptj}^{\phantom{i}}=R\hskip-0.4pt_{j\hskip-0.4ptj,\hskip 0.7ptk}^{\phantom{i}}=R\hskip-0.4pt_{jk,\hskip 0.7ptl}^{\phantom{i}}=0$. Finally, $\,g^{j\hskip-0.4ptj}R\hskip-0.4pt_{j1,\hskip 0.7ptj}^{\phantom{i}}=y\hskip-0.7pt_{j}^{\phantom{i}}(\mu_{j}^{\phantom{i}}\hskip-0.7pt-\hskip-0.4pt\mu_{1}^{\phantom{i}})\,$ and $\,g^{j\hskip-0.4ptj}R\hskip-0.4pt_{j\hskip-0.4ptj,\hskip 0.7pt1}^{\phantom{i}}=\dot{\mu}_{j}^{\phantom{i}}$, so that (1.1.i) implies (e), ∎

We refer to a solution $\,I\ni t\mapsto\mathbf{y}\in\mathrm{I\!E}\,$ of (5.3) as maximal if it cannot be extended to a larger open interval, and call it Ricci-generic whenever the $\,n\,$ values $\,\mu_{k}^{\phantom{i}}=\mu_{k}^{\phantom{i}}(t)\,$ of Lemma 5.2(a) are all distinct at some $\,t\in I\hskip 0.7pt$ (or, equivalently, no two among the functions $\,\mu_{1}^{\phantom{i}},\dots,\mu_{n}^{\phantom{i}}$ coincide everywhere in $\,I$).

Example 5.3**.**

Two non-Ricci-generic maximal solutions of (5.3) are defined by $\,\mathbf{y}=-\hskip-0.7pt2\hskip 0.4pt\tanh\hskip 0.7ptnt\,$ and $\,\mathbf{y}=2\hskip 0.4pt\tan\hskip 0.7ptnt\,$ (times the identity $\,\mathbf{1}$), with $\,I\hskip-0.7pt=\mathrm{I\!R}\,$ or $\,I\hskip-0.7pt=(-\pi/(2n),\pi/(2n))$. In fact, $\,2\dot{\hskip 0.0pt\mathbf{y}\hskip 0.0pt}=n\hskip 0.4pt(\mathbf{y}^{2}\hskip-1.5pt\mp 4)\,$ and so $\,\ddot{\hskip 0.0pt\mathbf{y}\hskip 0.0pt}=n\mathbf{y}\dot{\hskip 0.0pt\mathbf{y}\hskip 0.0pt}$, while for multiples $\,\mathbf{y}$ of $\,\mathbf{1}\,$ the right-hand side of (5.3) vanishes and $\,\mathrm{tr}\hskip 1.7pt\mathbf{y}+\mathbf{y}=n\mathbf{y}$.

Example 5.4**.**

Any solution $\,\mathbf{y}\,=\hskip 0.7pt\mathrm{diag}\hskip 0.7pt(\hskip-0.7pty_{2}^{\phantom{i}},\dots,y_{n}^{\phantom{i}})\,$ of (5.3), where $\,n\geq 2$, can be trivially extended to the solution $\,\mathrm{diag}\hskip 0.7pt(\hskip-0.7pty_{2}^{\phantom{i}},\dots,y_{n}^{\phantom{i}},0,\dots,0)\,$ with a number $\,m>0$ of additional zero components. The new metric defined using (5.1) – (5.2) is isometric to the Riemannian product of the original $\,g\,$ and a flat metric on $\,\mathrm{I\!R}\hskip-0.7pt^{m}\hskip-0.7pt$.

The set of maximal solutions of (5.3) is obviously preserved by the group $\,K$ acting on it via replacement of $\,\mathbf{y}\hskip 0.7pt$ with $\,t\mapsto\pm\mathbf{y}(b\pm t)$, where $\,b\in\mathrm{I\!R}\,$ and $\,\pm$ is either sign, combined with permutations of the components $\,y_{2}^{\phantom{i}},\dots,y_{n}^{\phantom{i}}$. We will use the term $\,K\hskip-2.0pt$-equivalence when two maximal solutions lie in the same $\,K\hskip-1.0pt$- orbit.

Remark 5.5*.*

Nonzero real numbers $\,a\hskip 0.7pt$ act on maximal solutions $\,t\mapsto\mathbf{y}(t)\,$ of (5.3) by sending them to $\,t\mapsto a\mathbf{y}(at)$. (The new metric arising via (5.1) – (5.2) is isometric to $\,g/a^{2}\hskip-0.7pt$.) The group $\,K\hskip 0.7pt$ defined above, obviously isomorphic to the direct product of the isometry group of $\,\mathrm{I\!R}\,$ and the symmetric group $\,S\hskip-0.7pt_{n\hskip-0.4pt-\hskip-0.7pt1}^{\phantom{i}}$, along with the multiplicative group $\,\mathrm{I\!R}\smallsetminus\{0\}\,$ acting as described here, together generate an action of a semidirect product of $\,K\hskip 0.7pt$ and $\,(0,\infty)$.

Theorem 5.6**.**

For any $\,n\geq 3$, the construction summarized by (5.1) – (5.2) provides a bijective correspondence between two sets consisting, respectively, of

1. (i)

all $\,K\hskip-2.0pt$-equivalence classes of maximal Ricci-generic solutions to (5.3), and 2. (ii)

all local-isometry types of Riemannian $\,n\hskip-0.4pt$-manifolds with (0.1) – (0.3).

For the meaning of local-isometry types, see (1.2) and the paragraph following it.

Proof.

We need to show that the mapping from (i) to (ii) is: (A) well-defined, (B) injective, and (C) surjective.

Part (A) easily follows from Lemma 5.2 combined with the comment on $\,g/a^{2}$ in Remark 5.5, the latter applied to $\,a=\pm 1$. To obtain (B), note that the local-isometry type of a metric $\,g\,$ arising from (5.1) – (5.3) determines the $\,K\hskip-1.5pt$-equivalence class of the maximal Ricci-generic solution $\,t\mapsto\mathbf{y}\,$ of (5.3). Namely, the $\,g\hskip 0.4pt$-Killing fields $\,\partial\hskip-0.7pt_{2}^{\phantom{i}},\dots,\partial\hskip-0.7pt_{n}^{\phantom{i}}$, valued in eigenvectors of the Ricci tensor of $\,g\,$ (see Lemma 5.2), are – due to the Ricci-generic condition and (2.2) – unique up to permutations and multiplication by nonzero constants, which makes $\,y_{2}^{\phantom{i}},\dots,y_{n}^{\phantom{i}}$, defined by (5.1) with $\,g\hskip-0.7pt_{j\hskip-0.4ptj}^{\phantom{i}}\hskip-0.7pt=g(\partial\hskip-0.7pt_{j}^{\phantom{i}},\partial\hskip-0.7pt_{j}^{\phantom{i}})$, also unique up to permutations. The variable $\,t$, being an arc-length parameter of $\,g\hskip 0.4pt$-geodesics orthogonal to $\,\partial\hskip-0.7pt_{2}^{\phantom{i}},\dots,\partial\hskip-0.7pt_{n}^{\phantom{i}}$, cf. Lemma 5.2(d) and (5.1), is in turn unique up to substitutions by $\,b\pm t$, for constants $\,b$, as required.

Finally, to prove (C), we fix $\,(M\hskip-0.7pt,g)\,$ of dimension $\,n\geq 3\,$ satisfying (0.1) – (0.3). Corollary 1.3 and (1.2), along with Remarks 3.2 and 2.5(ii), allow us to choose $\,\mathfrak{a}_{2}^{\phantom{i}},\dots,\mathfrak{a}_{n}^{\phantom{i}}$ and $\,v_{2}^{\phantom{i}},\dots,v_{n}^{\phantom{i}}$ as in Lemma 4.1 for $\,m=n$, and a point $\,x\in M$ at which all $\,v\hskip-1.5pt_{j}^{\phantom{i}}$ are nonzero. (From now on $\,j\,$ ranges over $\,\{2,\dots,n\}$.) By the Lie -bracket assertion of Lemma 4.1, the local flow of each $\,v\hskip-1.5pt_{j}^{\phantom{i}}$ preserves all $\,v\hskip-1.5pt_{j}^{\phantom{i}}$ and, consequently, also a unit vector field $\,v_{1}^{\phantom{i}}$ on a neighborhood of $\,x$, orthogonal to all $\,v\hskip-1.5pt_{j}^{\phantom{i}}$. Since $\,v_{1}^{\phantom{i}}$ and all $\,v\hskip-1.5pt_{j}^{\phantom{i}}$ commute with one another, they constitute the coordinate vector fields of a local coordinate system $\,x^{1}\hskip-1.5pt=t,x\hskip 0.4pt^{2}\hskip-0.7pt,\dots,x\hskip 0.4pt^{n}$ on a neighborhood of $\,x$, in which the metric $\,g\,$ has the form (5.1) as a consequence of the last two lines of Lemma 4.1, with $\,m=n$. (In particular, the assertion $\,g(\nabla\hskip-2.7pt_{u}^{\phantom{i}}v\hskip-1.5pt_{j}^{\phantom{i}},v_{k}^{\phantom{i}})=0$, for $\,u=v_{l}^{\phantom{i}}$ and $\,j,k,l\in\{2,\dots,n\}$, applied to $\,j=k$, shows that $\,g\hskip-0.7pt_{j\hskip-0.4ptj}^{\phantom{i}}\hskip-0.7pt=g(v\hskip-1.5pt_{j}^{\phantom{i}},v\hskip-1.5pt_{j}^{\phantom{i}})\,$ only depend on the variable $\,t=x^{1}\hskip-0.7pt$.) Now Lemma 5.2(e) yields (C). ∎

Remark 5.7*.*

The component version of (5.3) states that $\,\ddot{y}\hskip-0.7pt_{j}^{\phantom{i}}\hskip-0.7pt-(\mathrm{tr}\hskip 1.7pt\mathbf{y}+y\hskip-0.7pt_{j}^{\phantom{i}})\hskip 0.7pt\dot{y}\hskip-0.7pt_{j}^{\phantom{i}}$ equals $\,y\hskip-0.7pt_{j}^{\phantom{i}}[\mathrm{tr}\hskip 1.7pt\mathbf{y}^{2}\hskip-0.7pt-(\mathrm{tr}\hskip 1.7pt\mathbf{y})\hskip 0.4pty\hskip-0.7pt_{j}^{\phantom{i}}]$. A solution $\,t\mapsto\mathbf{y}\,$ of (5.3) for $\,n\geq 3$, with any prescribed value at $\,t=0$, may be chosen so as to make the values $\,\mu_{1}^{\phantom{i}}(0),\dots,\mu_{n}^{\phantom{i}}(0)\,$ mutually distinct. (By Lemma 5.2(a), this amounts to using $\,\dot{\hskip 0.0pt\mathbf{y}\hskip 0.0pt}(0)\,$ that realizes $\hskip 0.7pt(\mu_{2}^{\phantom{i}}(0),\dots,\mu_{n}^{\phantom{i}}(0))$ lying outside a finite union of specific hyperplanes in $\,\mathrm{I\!E}$.) Consequently, the local-isometry types in Theorem 5.6(ii) form a moduli space of dimension $\,2n-3$.

6. The scalar-curvature integral

Not surprisingly, in the light of (1.1.ii) and parts (b), (e) of Lemma 5.2,

[TABLE]

Lemma 6.1**.**

For any solution $\,I\ni t\mapsto\mathbf{y}\in\mathrm{I\!E}\,$ of (5.3) defined on $\,\mathrm{I\!R}$, and not identically equal to zero, one must have $\,\mathrm{s}\hskip 0.7pt<0\,$ in (6.1).

Proof.

Under the assumption that $\,\mathrm{s}\geq 0$, (6.1) gives $\,2\hskip 1.7pt\mathrm{tr}\hskip 1.7pt\dot{\hskip 0.0pt\mathbf{y}\hskip 0.0pt}\geq\hskip 0.7pt\mathrm{tr}\hskip 1.7pt\mathbf{y}^{2}\hskip-0.7pt+(\mathrm{tr}\hskip 1.7pt\mathbf{y})^{2}$ for our solution $\,\mathrm{I\!R}\ni t\mapsto\mathbf{y}\in\mathrm{I\!E}$, and so $\,\mathrm{tr}\hskip 1.7pt\mathbf{y}\,$ is nondecreasing and nonconstant. Fixing $\,t^{\prime}\hskip-0.7pt\in\mathrm{I\!R}\,$ such that $\,\mathrm{tr}\hskip 1.7pt\mathbf{y}(t^{\prime})\neq 0$, we define a constant $\,c>0$ by $\,(n\hskip-0.4pt-\hskip-0.7pt1)\hskip 0.7ptc^{2}\hskip-0.7pt=[\mathrm{tr}\hskip 1.7pt\mathbf{y}(t^{\prime})]^{2}\hskip-0.7pt$. Depending on whether $\,\mathrm{tr}\hskip 1.7pt\mathbf{y}(t^{\prime})\,$ is positive or negative, monotonicity of $\,\mathrm{tr}\hskip 1.7pt\mathbf{y}\,$ gives $\,(\mathrm{tr}\hskip 1.7pt\mathbf{y})^{2}\hskip-0.7pt\geq(n\hskip-0.4pt-\hskip-0.7pt1)\hskip 0.7ptc^{2}$ on $\,[\hskip 0.7ptt^{\prime},\infty)\,$ or, respectively, on $\,(-\infty,t^{\prime}]$. The Schwarz inequality $\,(\mathrm{tr}\hskip 1.7pt\mathbf{x})^{2}\hskip-0.7pt\leq(n\hskip-0.4pt-\hskip-0.7pt1)\hskip 1.7pt\mathrm{tr}\hskip 1.7pt\mathbf{x}^{2}$ now shows that $\,\mathrm{tr}\hskip 1.7pt\mathbf{y}^{2}\hskip-0.7pt\geq\hskip 0.7ptc^{2}$ on $\,[\hskip 0.7ptt^{\prime},\infty)$, or on $\,(-\infty,t^{\prime}]$. The relation $\,2\hskip 1.7pt\mathrm{tr}\hskip 1.7pt\dot{\hskip 0.0pt\mathbf{y}\hskip 0.0pt}\geq\hskip 0.7pt\mathrm{tr}\hskip 1.7pt\mathbf{y}^{2}\hskip-0.7pt+(\mathrm{tr}\hskip 1.7pt\mathbf{y})^{2}$ (see above) thus yields $\,2\hskip 1.7pt\mathrm{tr}\hskip 1.7pt\dot{\hskip 0.0pt\mathbf{y}\hskip 0.0pt}\geq\hskip 0.7ptc^{2}\hskip-0.7pt+(\mathrm{tr}\hskip 1.7pt\mathbf{y})^{2}\hskip-0.7pt$, that is, $\,\hskip 0.4pt\dot{\hskip-0.4pt\alpha\hskip 0.4pt}\hskip-0.4pt\geq\hskip 0.7ptc^{2}$ on $\,[\hskip 0.7ptt^{\prime},\infty)\,$ or $\,(-\infty,t^{\prime}]$, where $\,\alpha=2\tan^{-\hskip-0.7pt1}(\mathrm{tr}\hskip 1.7pt\mathbf{y}\hskip-0.7pt/\hskip-0.7ptc)$. Consequently, $\,\alpha\to\pm\infty$ as $\,t\to\pm\infty\,$ for some sign $\,\pm\hskip 0.7pt$, contrary to boundedness of $\,\alpha$. ∎

Remark 6.2*.*

A Riemannian manifold $\,(I\hskip-0.7pt\times\mathrm{I\!R}\hskip-0.7pt^{n\hskip-0.4pt-\hskip-0.7pt1}\hskip-1.5pt,g)\,$ arising from (5.1) – (5.3), which makes it real -analytic, may be locally isometric to a compact (and hence complete) real -analytic Riemannian manifold, in the sense of the paragraph following (1.2), even if the solution $\,I\ni t\mapsto\mathbf{y}\in\mathrm{I\!E}\,$ of (5.3) has no extension to one defined on $\,\mathrm{I\!R}$. This is illustrated by the trivial extension (Example 5.4), with $\hskip 0.7ptm>0\hskip 0.7pt$ additional zeros, of the solution $\,y_{2}^{\phantom{i}}(t)=2\hskip 0.4pt\tan\hskip 0.7pt2t\,$ of Example 5.3, for $\,n=2$, further modified using $\,a=1/2\,$ in Remark 5.5, so as to become $\,t\mapsto(\tan\hskip 0.7ptt,0,\dots,0)$. Since the latter realizes (5.2) with $\,g_{22}^{\phantom{i}}=\cos^{2}\hskip-1.5ptt$, it represents, locally, a product of the standard sphere $\,\mathrm{S}^{2}$ with a flat torus $\,\mathrm{T}^{m}\hskip-0.7pt$.

7. Completeness

Let $\,n\geq 3$. In the usual fashion, (5.3) is equivalent to the first-order system

[TABLE]

Solutions $\,t\mapsto\mathbf{y}\,$ of (5.3) thus correspond to integral curves $\,t\mapsto(\mathbf{y},\mathbf{p})\,$ of the vector field $\,v\,$ on $\,\mathrm{I\!E}\hskip-0.4pt\times\hskip-0.7pt\mathrm{I\!E}\,$ represented by (7.1), and expressed as

[TABLE]

when identified with a mapping $\,\mathrm{I\!E}\hskip-0.4pt\times\hskip-0.7pt\mathrm{I\!E}\to\mathrm{I\!E}\hskip-0.4pt\times\hskip-0.7pt\mathrm{I\!E}$. This $\,v\,$ has an obvious curve $\,\mathrm{I\!R}\ni q\mapsto q(\mathbf{1},\mathbf{0})\,$ of zeros, where $\,\mathbf{1}\in\mathrm{I\!E}\,$ is the identity. Evaluating the differentials of $\,v:\mathrm{I\!E}\hskip-0.4pt\times\hskip-0.7pt\mathrm{I\!E}\to\mathrm{I\!E}\hskip-0.4pt\times\hskip-0.7pt\mathrm{I\!E}\,$ at $\,q(\mathbf{1},\mathbf{0})$, and of the function $\,\mathrm{I\!E}\hskip-0.4pt\times\hskip-0.7pt\mathrm{I\!E}\ni(\mathbf{y},\mathbf{p})\mapsto\mathrm{s}=2\hskip 1.7pt\mathrm{tr}\hskip 1.7pt\mathbf{p}-\hskip 0.7pt\mathrm{tr}\hskip 1.7pt\mathbf{y}^{2}\hskip-0.7pt-(\mathrm{tr}\hskip 1.7pt\mathbf{y})^{2}\hskip-0.7pt\in\mathrm{I\!R}$, cf. (6.1), at any $\,(\mathbf{y},\mathbf{p})\in\mathrm{I\!E}\hskip-0.4pt\times\hskip-0.7pt\mathrm{I\!E}$, we obtain $\,dv\hskip-0.7pt_{q(\mathbf{1}\hskip-0.4pt,\hskip 0.4pt\mathbf{0})}^{\phantom{i}}(\hat{\mathbf{y}},\hat{\mathbf{p}})=(\hat{\mathbf{p}},\,nq\hskip 0.7pt\hat{\mathbf{p}}+q^{2}\hskip 0.7pt\mathrm{tr}\hskip 1.7pt\hat{\mathbf{y}}-(n\hskip-0.4pt-\hskip-0.7pt1)\hskip 0.4ptq^{2}\hat{\mathbf{y}})\,$ and $\,d\hskip 0.4pt\mathrm{s}_{(\mathbf{y}\hskip-1.5pt,\hskip 0.4pt\mathbf{p})}^{\phantom{i}}(\hat{\mathbf{y}},\hat{\mathbf{p}})=2\hskip 0.4pt[\hskip 0.4pt\mathrm{tr}\hskip 1.7pt\hat{\mathbf{p}}-\hskip 0.7pt\mathrm{tr}\hskip 1.7pt\mathbf{y}\hat{\mathbf{y}}-(\mathrm{tr}\hskip 1.7pt\mathbf{y})\mathrm{tr}\hskip 1.7pt\hat{\mathbf{y}}]$. When $\,q\neq 0$, the linear endomorphism $\,dv\hskip-0.7pt_{q(\mathbf{1}\hskip-0.4pt,\hskip 0.4pt\mathbf{0})}^{\phantom{i}}$ of $\,\mathrm{I\!E}\hskip-0.4pt\times\hskip-0.7pt\mathrm{I\!E}\,$ is diagonalizable, with the eigenvalues $\,0,n\hskip 0.4ptq,(n\hskip-0.4pt-\hskip-0.7pt1)\hskip 0.4ptq,q\,$ of multiplicities $\,1,1,n\hskip-0.4pt-\hskip-0.7pt2,n\hskip-0.4pt-\hskip-0.7pt2$, the eigenspace for each of the four eigenvalues $\,\lambda\,$ consisting of all $\,(\hat{\mathbf{y}},\hat{\mathbf{p}})\,$ such that $\,\hat{\mathbf{p}}=\lambda\hat{\mathbf{y}}\,$ and either $\,\hat{\mathbf{y}}$ equals a multiple of the identity (for $\,\lambda\in\{0,n\hskip 0.4ptq\}$), or $\,\mathrm{tr}\hskip 1.7pt\hat{\mathbf{y}}=0\,$ (if $\,\lambda\in\{(n\hskip-0.4pt-\hskip-0.7pt1)\hskip 0.4ptq,q\}$).

On the other hand, $\,\mathrm{s}\,$ has no critical points in $\,\mathrm{I\!E}\times\hskip-0.7pt\mathrm{I\!E}$, and $\,v\,$ is tangent to the level sets of $\,\mathrm{s}$. The latter sets are codimension-one real -analytic submanifolds of $\,\mathrm{I\!E}\hskip-0.4pt\times\hskip-0.7pt\mathrm{I\!E}$, and those among them intersecting the curve $\,\mathrm{I\!R}\ni q\mapsto q(\mathbf{1},\mathbf{0})$ correspond, by (6.1), to $\,\mathrm{s}=-n(n\hskip-0.4pt-\hskip-0.7pt1)\hskip 0.4ptq^{2}\hskip-0.7pt$, that is, to all nonpositive values of $\,\mathrm{s}$. If we fix $\,q\neq 0$, the tangent space at $\,z=q(\mathbf{1},\mathbf{0})\,$ of the hypersurface $\,N\hskip 0.7pt$ given by $\,\mathrm{s}=-n(n\hskip-0.4pt-\hskip-0.7pt1)\hskip 0.4ptq^{2}\hskip-0.7pt$, equal to the kernel of $\,d\hskip 0.4pt\mathrm{s}_{q(\mathbf{1}\hskip-0.4pt,\hskip 0.4pt\mathbf{0})}^{\phantom{i}}$, coincides, due to dimensional reasons, with the span of the eigenspaces of $\,dv\hskip-0.7pt_{q(\mathbf{1}\hskip-0.4pt,\hskip 0.4pt\mathbf{0})}^{\phantom{i}}$ for the three nonzero eigenvalues $\,n\hskip 0.4ptq,(n\hskip-0.4pt-\hskip-0.7pt1)\hskip 0.4ptq,q$. (See the preceding paragraph and the above formula for $\,d\hskip 0.4pt\mathrm{s}_{(\mathbf{y}\hskip-1.5pt,\hskip 0.4pt\mathbf{p})}^{\phantom{i}}(\hat{\mathbf{y}},\hat{\mathbf{p}})$.) From (2.1) it now follows that $\,\partial w\hskip-0.7pt_{z}^{\phantom{i}}$, for the vector field $\,w\,$ on $\,N\hskip 0.7pt$ arising as the restriction of $\,v$, is diagonalizable, with positive (or, negative) eigenvalues. Thus, as $\,z=q(\mathbf{1},\mathbf{0})$,

[TABLE]

Remark 7.1*.*

Whenever $\,c\in\mathrm{I\!R}\smallsetminus\{0\}$, the assignment $\,(\mathbf{y},\mathbf{p})\mapsto(c\mathbf{y},c^{2}\hskip-0.7pt\mathbf{p})\,$ is a diffeomorphism $\,F\hskip-3.0pt_{c}^{\phantom{i}}:\mathrm{I\!E}\hskip-0.4pt\times\hskip-0.7pt\mathrm{I\!E}\to\mathrm{I\!E}\hskip-0.4pt\times\hskip-0.7pt\mathrm{I\!E}$, sending our vector field $\,v\,$ to $\,v/c$, and pulling the function $\,\mathrm{s}\,$ back to $\,c^{2}\mathrm{s}$. Using our $\,N\hskip 0.7pt$ given by $\,\mathrm{s}=-n(n\hskip-0.4pt-\hskip-0.7pt1)\hskip 0.4ptq^{2}$ we obtain a diffeomorphism $\,(0,\infty)\times\hskip-0.7ptN\ni(c,x)\mapsto F(c,x)=F\hskip-3.0pt_{c}^{\phantom{i}}(x)\,$ onto the open set in $\,\mathrm{I\!E}\hskip-0.4pt\times\hskip-0.7pt\mathrm{I\!E}$ on which $\,\mathrm{s}<0$, as one sees defining its inverse by $\,F\hskip 0.4pt^{-\hskip-1.5pt1}\hskip-0.7pt(x^{\prime})=(c,x)$, if $\,\mathrm{s}(x^{\prime})<0$, with $\,\hskip 0.7ptc,x\,$ such that $\,n(n\hskip-0.4pt-\hskip-0.7pt1)\hskip 0.4pt(cq)^{2}\hskip-0.7pt=-\mathrm{s}(x^{\prime})\,$ and $\,x=F(1/c,x^{\prime})$.

In the next theorem, we fix an integer $\,n\geq 3$, again denoting by $\,\mathrm{I\!E}\,$ the space of all diagonal $\,(n\hskip-0.4pt-\hskip-0.7pt1)\times(n\hskip-0.4pt-\hskip-0.7pt1)\,$ matrices, and by $\,\mathbf{1}\in\mathrm{I\!E}\,$ the identity.

Theorem 7.2**.**

For any $\,(\xi,\zeta)\in\mathrm{I\!R}\times(0,\infty)$, every maximal solution $\,t\mapsto\mathbf{y}\,$ of (5.3) with $\,(\mathbf{y}(0),\dot{\hskip 0.0pt\mathbf{y}\hskip 0.0pt}(0))\,$ sufficiently close to $\,(\xi\mathbf{1},-\zeta\mathbf{1})\,$ in $\,\mathrm{I\!E}\hskip-0.4pt\times\hskip-0.7pt\mathrm{I\!E}\,$ has the domain $\,\mathrm{I\!R}$, and the metric $\,g\,$ on $\,\mathrm{I\!R}\hskip-0.7pt^{n}$ defined by (5.1) – (5.2) is complete.

Proof.

The solution $\,\mathrm{I\!R}\ni t\mapsto\mathbf{y}\hskip-1.5pt_{1,0}^{\phantom{i}}(t)=-\hskip-0.7pt2\hskip 0.4pt\tanh\hskip 0.7ptnt\,$ (times the identity $\,\mathbf{1}$) of Example 5.3 leads, via Remark 5.5, to further solutions $\,t\mapsto\mathbf{y}\hskip-1.5pt_{a,b}^{\phantom{i}}(t)=a\mathbf{y}\hskip-1.5pt_{1,0}^{\phantom{i}}(at+b)$, where $\,a,b\in\mathrm{I\!R}\,$ and $\,a\neq 0$. Suitably chosen and fixed such $\,a,b\,$ clearly realize, at $\,t=0$, any prescribed initial data $\,(\xi\mathbf{1},-\zeta\mathbf{1})=(\mathbf{y}\hskip-1.5pt_{a,b}^{\phantom{i}}(0),\dot{\hskip 0.0pt\mathbf{y}\hskip 0.0pt}\hskip-1.5pt_{a,b}^{\phantom{i}}(0))\in\mathrm{I\!R}\times(0,\infty)$. Setting $\,x\hskip-0.7pt_{a,b}^{\phantom{i}}(t)=(\mathbf{y}\hskip-1.5pt_{a,b}^{\phantom{i}}(t),\dot{\hskip 0.0pt\mathbf{y}\hskip 0.0pt}\hskip-1.5pt_{a,b}^{\phantom{i}}(t))\,$ and $\,z_{\pm}=\mp 2|a|(\mathbf{1},\mathbf{0})\,$ we get $\,x\hskip-0.7pt_{a,b}^{\phantom{i}}(t)\to z_{\pm}$ as $\,t\to\pm\infty$. In the discussion preceding (7.3), applied to $\,q=\mp 2|a|$, both choices of the sign $\,\pm\,$ lead to the same $\,N\hskip-0.7pt$, given by $\,\mathrm{s}=-n(n\hskip-0.4pt-\hskip-0.7pt1)\hskip 0.4ptq^{2}\hskip-0.7pt$, and the same $\,w$, while $\,z_{+}^{\phantom{i}},z_{-}^{\phantom{i}}\in N\hskip 0.7pt$ are two different zeros of $\,w$. Using (7.3) we now choose neighborhoods $\,\,U\hskip-2.3pt_{\pm}^{\phantom{i}}$ of $\,z_{\pm}^{\phantom{i}}$ in $\,N\hskip 0.7pt$ satisfying the assertion of Lemma 2.3 for $\,x(t)\,$ equal to our $\,x\hskip-0.7pt_{a,b}^{\phantom{i}}(t)$, and $\,t^{\prime}_{\pm}\in\mathrm{I\!R}\,$ with $\,x\hskip-0.7pt_{a,b}^{\phantom{i}}(t^{\prime}_{\pm})\in\,U\hskip-2.3pt_{\pm}^{\phantom{i}}$. Since $\,z_{\pm}=\mp 2|a|(\mathbf{1},\mathbf{0})$, we may also require that

[TABLE]

By continuity, $\,x(t^{\prime}_{\pm})\in\,U\hskip-2.3pt_{\pm}^{\phantom{i}}$ for some neighborhood $\,\,U\hskip-1.7pt_{0}^{\phantom{i}}$ of $\,x\hskip-0.7pt_{a,b}^{\phantom{i}}(0)\,$ in $\,N\hskip 0.7pt$ and all integral curves $\,t\mapsto x(t)\in N\hskip 0.7pt$ of $\,w\,$ with $\,x(0)\in\,U\hskip-1.7pt_{0}^{\phantom{i}}$. The image of $\,(0,\infty)\times U\hskip-1.7pt_{0}^{\phantom{i}}$ under the diffeomorphism $\,F\hskip 0.7pt$ of Remark 7.1 is now a neighborhood of $\,x\hskip-0.7pt_{a,b}^{\phantom{i}}(0)=(\xi\mathbf{1},-\zeta\mathbf{1})$ in $\,\mathrm{I\!E}\hskip-0.4pt\times\hskip-0.7pt\mathrm{I\!E}$, the existence of which constitutes our assertion: according to Remark 7.1, this $\,F\hskip-0.7pt$-image equals the union of $\,F\hskip-3.0pt_{c}^{\phantom{i}}(U\hskip-1.7pt_{0}^{\phantom{i}})\,$ over $\,c>0$, and each $\,F\hskip-3.0pt_{c}^{\phantom{i}}$ maps $\,N\hskip 0.7pt$ diffeomorphically onto the $\,\mathrm{s}\hskip 0.4pt$-preimage of the value $\,-n(n\hskip-0.4pt-\hskip-0.7pt1)\hskip 0.4pt(cq)^{2}\hskip-0.7pt$, while the push-forward, under $\,F\hskip-3.0pt_{c}^{\phantom{i}}:N\hskip-0.7pt\to F\hskip-3.0pt_{c}^{\phantom{i}}(N)$, of $\,w\,$ obtained by restricting $\,v\,$ to $\,N\hskip-0.7pt$, is the restriction of $\,v/c\,$ to $\,F\hskip-3.0pt_{c}^{\phantom{i}}(N)$. However, the discussion preceding (7.3), and (7.3) itself, apply to every $\,q\neq 0$, and the use of $\,v/c\,$ rather than $\,v\,$ makes no difference (Remark 2.4). Now (7.4) combined with Remark 5.1 yields completeness of $\,g$. ∎

Our next result shows that the examples arising from Lemma 5.2(e) are not generally Ricci-parallel, or locally reducible, or (when $\,n\geq 4$) conformally flat.

Theorem 7.3**.**

The local-isometry types of Riemannian $\,n$-manifolds satisfying (0.1) – (0.4) form a set with a nonempty interior in the $\,(2n-3)$-dimensional moduli space of Remark 5.7.

Proof.

According to Theorem 5.6, the local-isometry types of all $\,n$-dimensional $\,(M\hskip-0.7pt,g)\,$ with (0.1) – (0.3) arise from (5.1) when one chooses a maximal Ricci-generic solution $\,I\ni t\mapsto\mathbf{y}\in\mathrm{I\!E}\,$ of (5.3), and then fixes a smooth curve $\,I\ni t\mapsto(g_{22}^{\phantom{i}}(t),\dots,g_{nn}^{\phantom{i}}(t))\in(0,\infty)^{n\hskip-0.4pt-\hskip-0.7pt1}\,$ satisfying (5.2). Restricting our discussion to the case where $\,0\in I\hskip-0.7pt$, and then parametrizing such solutions (allowed, this time, not to be Ricci-generic) by their initial data at $\,t=0$, we identify them with points of a specific Euclidean space, and completeness of $\,g\,$ is guaranteed by Theorem 7.2 once one assumes (as we do from now on) that the initial data range over a certain nonempty open subset of the latter space. Now, as in Remark 5.7, if $\,n\geq 3$, we can make the Ricci eigenvalue functions $\,\mu_{1}^{\phantom{i}}(0),\dots,\mu_{n}^{\phantom{i}}(0)\,$ of Lemma 5.2 mutually distinct (which leads to Ricci-genericity) just by ensuring that $\,(\mu_{2}^{\phantom{i}}(0),\dots,\mu_{n}^{\phantom{i}}(0))$ does not lie within a specific finite union of hyperplanes in $\,\mathrm{I\!E}$. However, rather than using any prescribed $\,\mathbf{y}(0)$, cf. Remark 5.7, let us require $\,y_{1}^{\phantom{i}}(0),\dots,y_{n}^{\phantom{i}}(0)\,$ to be all nonzero. This amounts to imposing on the solution $\,t\mapsto\mathbf{y}\,$ of (5.3) a further open condition implying (see the proof of Lemma 5.2) that $\,R\hskip-0.4pt_{j1,\hskip 0.7ptj}^{\phantom{i}}(0)\neq 0$, and so $\,g$ is not Ricci-parallel. In the proof of Lemma 5.2 we also saw that $\,\varGamma_{\hskip-2.7ptj\hskip-0.4ptj}^{\hskip 0.7pt1}(0)\neq 0$ and, consequently, $\,g\,$ cannot be locally reducible. (If it were, the Ricci eigenvector fields $\,\partial\hskip-0.7pt_{1}^{\phantom{i}},\dots,\partial\hskip-0.7pt_{n}^{\phantom{i}}$ of Lemma 5.2, with distinct eigenvalue functions $\,\mu_{1}^{\phantom{i}},\dots,\mu_{n}^{\phantom{i}}$, would each be tangent to one or the other parallel factor distribution, giving $\varGamma_{\hskip-2.7ptj\hskip-0.4ptj}^{\hskip 0.7pt1}\hskip-0.7pt=0\hskip 0.7pt$ with some $\,j=2,\dots,n$.) For $\,k\neq j$, one easily verifies that $\,g^{j\hskip-0.4ptj}g^{kk}\hskip-1.5ptR\hskip-0.4pt_{jkjk}^{\phantom{i}}=-y\hskip-0.7pt_{j}^{\phantom{i}}y\hskip-0.7pt_{k}^{\phantom{i}}$. Therefore, if $\,W\hskip-1.5pt$ denotes the Weyl tensor, $\,(n\hskip-0.4pt-\hskip-0.7pt1)(n\hskip-0.4pt-\hskip-0.7pt2)g^{j\hskip-0.4ptj}g^{kk}W\hskip-3.4pt_{jkjk}^{\phantom{i}}\hskip 0.7pt=\,\,2\hskip 1.7pt\mathrm{tr}\hskip 1.7pt\dot{\hskip 0.0pt\mathbf{y}\hskip 0.0pt}\,-\,\mathrm{tr}\hskip 1.7pt\mathbf{y}^{2}\hskip 0.7pt-\,(\mathrm{tr}\hskip 1.7pt\mathbf{y})^{2}+\,(n\hskip-0.4pt-\hskip-0.7pt1)[(y\hskip-0.7pt_{j}^{\phantom{i}}+y\hskip-0.7pt_{k}^{\phantom{i}})\hskip 0.7pt\mathrm{tr}\hskip 1.7pt\mathbf{y}-(n\hskip-0.4pt-\hskip-0.7pt2)y\hskip-0.7pt_{j}^{\phantom{i}}y\hskip-0.7pt_{k}^{\phantom{i}}-\dot{y}\hskip-0.7pt_{j}^{\phantom{i}}-\dot{y}\hskip-0.7pt_{k}^{\phantom{i}}]$, where $\,\dot{y}\hskip-0.7pt_{j}^{\phantom{i}}$ appears with the coefficient $\,3-n$. An enhanced version of the last open condition thus precludes conformal flatness of our examples when $\,n\geq 4$. ∎

Appendix: Warped products with harmonic curvature

For the reader’s convenience, we gather here some facts that are well known [7] and easily verified. The repeated indices are always summed over. In (1.4) we set $\,m=\dim\hskip 3.0pt\overline{\hskip-3.0ptM\hskip-0.7pt}\hskip 0.7pt\,$ and $\,p=\dim\varSigma$, assuming that $\,mp\geq 1\,$ and $\,\phi:\hskip 3.0pt\overline{\hskip-3.0ptM\hskip-0.7pt}\hskip 0.7pt\to(0,\infty)\,$ is nonconstant. Thus, $\,\dim M=n\,$ with $\,n=m+p\,\geq\hskip 0.7pt2$. We use product coordinates $\,x^{\lambda}$ in $\,M\hskip-0.7pt$, consisting of local coordinates $\,x^{i}$ for $\,\hskip 3.0pt\overline{\hskip-3.0ptM\hskip-0.7pt}\hskip 0.7pt\,$ and $\,x^{a}$ for $\,\varSigma$, declaring

[TABLE]

to be our index ranges. Therefore, $\,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt_{i\hskip-0.4ptj}^{\phantom{i}}$ as well as $\,\theta\hskip-0.7pt=\hskip-0.7pt\log\phi\,$ depend only on the variables $\,x^{k}\hskip-0.7pt$,$\,\eta_{ab}^{\phantom{i}}$ only on $\,x^{c}\hskip-0.7pt$, that is, $\,\partial\hskip-0.4pt_{a}^{\phantom{i}}\hskip 0.7pt\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt\hskip-0.4pt_{i\hskip-0.4ptj}^{\phantom{i}}\hskip 0.7pt=\,\partial\hskip-0.4pt_{a}^{\phantom{i}}\theta\hskip 0.7pt=\,\partial_{i}^{\phantom{i}}\eta_{ab}^{\phantom{i}}\hskip 0.7pt=\,0$. Furthermore,

[TABLE]

For the Christoffel symbols $\,\varGamma_{\hskip-2.7pt\lambda\mu}^{\hskip 0.7pt\nu},\,\hskip 3.0pt\overline{\hskip-3.0pt\varGamma\hskip-0.7pt}\hskip 0.7pt_{\hskip-2.7ptij}^{\hskip 0.7ptk},\,H\hskip-1.5pt_{ab}^{\hskip 0.7ptc}$ of $\,g,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt,\eta$, their Ricci-tensor components $\,R_{\lambda\mu}^{\phantom{i}},\,\hskip 3.0pt\overline{\hskip-3.0ptR\hskip-0.7pt}\hskip 0.7pt_{i\hskip-0.4ptj}^{\phantom{i}},\,P\hskip-2.7pt_{ab}^{\phantom{i}}$, and the components $\,\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt_{\hskip-2.7pti}^{\phantom{i}}\hskip-2.2pt\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt_{\hskip-2.7ptj}^{\phantom{i}}\hskip-0.7pt\theta\,$ of the $\,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt\hskip 0.4pt$-Hessian of $\,\theta$, one has

[TABLE]

while, in terms of the $\,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt$-Laplacian $\,\hskip 0.7pt\overline{\hskip-0.7pt\Delta\hskip-0.7pt}\hskip 0.4pt$,

[TABLE]

The components $\,R\hskip-0.7pt_{\lambda\mu\hskip-0.4pt,\hskip 0.7pt\nu}^{\phantom{i}},\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt_{\hskip-2.7pti}^{\phantom{i}}\hskip-2.2pt\hskip 3.0pt\overline{\hskip-3.0ptR\hskip-0.7pt}\hskip 0.7pt\hskip-0.7pt_{jk}^{\phantom{i}},\hskip 0.7ptD\hskip-1.5pt_{c}^{\phantom{i}}P\hskip-2.7pt_{ab}^{\phantom{i}}$ of the covariant derivatives of the Ricci tensors of $\,g,\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt,\eta\,$ satisfy, with the usual conventions $\,\theta\hskip-1.5pt_{,\hskip 0.4pti}^{\phantom{i}}\hskip-0.7pt=\partial\hskip-0.7pt_{i}^{\phantom{i}}\theta\,$ and $\,\theta\hskip 0.4pt^{,\hskip 0.4pti}\hskip-0.7pt=\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt^{i\hskip-0.4ptj}\partial\hskip-0.7pt_{j}^{\phantom{i}}\theta$, the relations

[TABLE]

Let (a) – (e) refer to parts of Lemma 1.2, which we now proceed to prove. First,

1. (f)

$R_{ab,\hskip 0.7pti}^{\phantom{i}}\hskip-0.7pt=R_{ib,\hskip 0.7pta}^{\phantom{i}}\,\hskip 0.7pt$ for all $\,\,i,a,b\,\,$ as in (A.1) if and only if one has (a) and (e).

In fact, it suffices to verify (f) on the dense set $\,(U\hskip-1.0pt\cup U^{\prime})\times\hskip-0.4pt\varSigma\,\hskip 0.7pt\subseteq\hskip 0.7ptM\hskip-0.7pt$, for the interior $\,\,U$ of the zero set of $\,d\hskip 0.4pt\theta\,$ in $\,\hskip 3.0pt\overline{\hskip-3.0ptM\hskip-0.7pt}\hskip 0.7pt\,$ and the subset $\,\,U^{\prime}$ on which $\,d\hskip 0.4pt\theta\neq 0$. On $\,\,U\hskip-1.5pt$, according to (A.4), $\,R_{ab,\hskip 0.7pti}^{\phantom{i}}\hskip-0.7pt=0=R_{ib,\hskip 0.7pta}^{\phantom{i}}$ since $\,\overline{\hskip-0.7pt\Delta\hskip 0.4pt}e^{p\theta}\hskip-0.7pt=0$. Similarly, on $\,\,U^{\prime}\hskip-0.7pt$, the equality $\,R_{ab,\hskip 0.7pti}^{\phantom{i}}\hskip-0.7pt=R_{ib,\hskip 0.7pta}^{\phantom{i}}$ amounts, by (A.4), to the condition $\,P\hskip-2.7pt_{ab}^{\phantom{i}}\hskip-0.7pt=\kappa\hskip 0.4pt\eta_{ab}^{\phantom{i}}$, for a function $\,\kappa\,$ on $\,\varSigma$ which must be constant, as it depends only on the variables $\,x^{\hskip 0.7ptj}$ that are local coordinates in $\,\hskip 3.0pt\overline{\hskip-3.0ptM\hskip-0.7pt}\hskip 0.7pt\hskip-0.7pt$. Formulae (A.4) also show that $\,\kappa\,$ is characterized by the relation $\,-\kappa\hskip 0.4pte^{-\hskip-0.7pt2\theta}d\hskip 0.4pt\theta=p^{-\hskip-0.7pt1}(\hskip-0.4ptd\hskip 0.4pt[e^{-p\theta}\overline{\hskip-0.7pt\Delta\hskip 0.4pt}e^{p\theta}]+e^{-p\theta}\hskip-0.4pt[\hskip 0.7pt\overline{\hskip-0.7pt\Delta\hskip 0.4pt}e^{p\theta}]\,d\hskip 0.4pt\theta)+\overline{\mathrm{r}}(\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-0.7pt\theta,\,\cdot\,)-p\hskip 0.4pt\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt(\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-0.7pt\theta,\hskip-1.5pt\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-0.7pt\theta)\hskip 0.4ptd\hskip 0.4pt\theta-p\hskip 0.7ptd[\hskip 1.2pt\overline{\hskip-1.2ptg\hskip-0.3pt}\hskip 0.3pt(\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-0.7pt\theta,\hskip-1.5pt\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-0.7pt\theta)]/2\,$ which, rewritten in terms of $\,\phi=e^{\theta}\hskip-0.7pt$, becomes (e).

The equivalence of (e) and (c) is in turn obvious from (1.3). Next, by (A.4),

1. (g)

$R\hskip-0.7pt_{j\hskip-0.4ptk,\hskip 0.7pti}^{\phantom{i}}\hskip-0.7pt=R\hskip-0.7pt_{ik,\hskip 0.7ptj}^{\phantom{i}}\,\hskip 0.7pt$ for all $\,\,i,j,k\,\,$ with (A.1) if and only if (b) holds,

since $\,\phi^{-\hskip-1.5pt1}\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-0.7ptd\phi=\hskip 0.7pt\overline{\hskip-0.7pt\nabla\hskip-0.7pt}\hskip 0.7pt\hskip-0.7ptd\theta+d\theta\otimes\hskip 0.7ptd\theta$. The main claim of Lemma 1.2 is thus immediate: harmonicity of the curvature amounts to the Codazzi equation for the Ricci tensor, cf. (1.1.i), while (A.4) clearly reduces the latter to the cases (f) – (g).

Finally, (d) follows from (a) and (A.3).