# Obstructions to distinguished Riemannian metrics via Lorentzian geometry

Amir Babak Aazami  Clark University Worcester, MA 01610<EMAIL_ADDRESS>
###### Abstract.

We approach the problem of finding obstructions to curvature distinguished
Riemannian metrics by considering Lorentzian metrics to which they are dual in
a suitable sense. Obstructions to the latter then yield obstructions to the
former. This framework applies both locally and globally, including to compact
manifolds, and is sensitive to various aspects of curvature. Here we apply it
in two different ways. First, by embedding a Riemannian manifold into a
Lorentzian one and utilizing Penrose’s “plane wave limit,” we find necessary
local conditions, in terms of the Hessian of just one function, for large
classes of Riemannian metrics to contain within them those that have parallel
Ricci tensor, or are Ricci-flat, or are locally symmetric. Second, by
considering Riemannian metrics dual to constant curvature Lorentzian metrics
via a type of Wick rotation, we are able to rule out the existence of a family
of compact Riemannian manifolds (in all dimensions) that deviate from constant
curvature in a precise sense.

##  1\. Introduction & Overview of Results

What role may the Lorentzian geometry of general relativity play in the
service of Riemannian geometry? In this article we promote the viewpoint that
techniques and results specific to Lorentzian geometry can be used to obtain
_nonexistence_ results in the Riemannian setting. In what follows we shall
demonstrate two ways in which this can happen, one local, one global. While
these two ways are quite different in flavor from each other, they both yield
obstructions to the existence of distinguished Riemannian metrics, and they
both rest firmly on the belief that it is worthwhile to ask whether a given
Riemannian metric is somehow or other related to a distinguished Lorentzian
metric.

###  1.1. The local case.

We show in Section 2 that it is possible to “simplify” a Riemannian metric $g$
— in such a way that some features of its curvature are preserved, such as the
condition of being Ricci-flat or of being locally symmetric — by realizing $g$
as the induced metric on an appropriately chosen submanifold of a Lorentzian
manifold. There are precedents to this approach (e.g., [FG85]), but our
motivation derives from the following fact, well known to practitioners of
general relativity: Every Lorentzian manifold admits, locally, a gravitational
_plane wave_ metric via an appropriate limit, a construction known as
Penrose’s “plane wave limit” [Pen76]. As remarked by Penrose himself, his
construction is analogous to that of the tangent space to a manifold at a
given point — except that here the point is replaced by the integral curve of
a “null” vector field; i.e., one that is nonzero but orthogonal to itself. (In
Section 2.1 and the Appendix, we provide a self-contained description of
Penrose’s plane wave limit and of plane wave metrics in general.) Putting
aside the connection to gravitational physics, of interest for us here are the
following two features of this construction:

  1. i.

A plane wave metric is typically a _far simpler_ metric than the original
Lorentzian metric to which it was a limit. Indeed, when written down in so
called _Brinkmann coordinates_ (see (220) of the Appendix), only one of its
metric components is nonconstant. This has the desirable consequence that most
of the components of its curvature 4-tensor vanish.

  2. ii.

By a more general argument due to Geroch [Ger69], certain curvature properties
of the Lorentzian metric are _preserved_ in this limit — and this despite the
fact that Penrose’s limit is both local and highly coordinate-dependent.
Geroch coined this the “hereditary” properties of the limit; in Proposition 1
of Section 2.1, we provide for convenience a proof of this (well known) fact
when the metric is either Ricci-flat, or locally symmetric, or locally
conformally flat, or has parallel Ricci tensor. (In fact, if any covariant
tensor constructed from the Riemann curvature tensor and its derivatives
happens to vanish for the Lorentzian metric, then it will also vanish for its
plane wave limit.)

How may this state of affairs be of use to a Riemannian geometer? As follows:
Suppose that one has a Riemannian metric $g$, expressed locally in some choice
of coordinates, and one would like to know, e.g., if it has parallel Ricci
tensor, or if it is Ricci-flat, or if it is locally symmetric,

$\displaystyle\nabla\text{Ric}=0\hskip 14.45377pt,\hskip
14.45377pt\text{Ric}=0\hskip 14.45377pt,\hskip 14.45377pt\nabla\text{Rm}=0,$
(1)

where Ric and Rm are the Ricci tensor and curvature 4-tensor of $g$, and
$\nabla$ its Levi-Civita connection. If any of these was the case, then the
components $g_{ij}$ would satisfy a certain system of PDEs in the given
coordinates. Now, instead of examining these PDEs directly, suppose that one
realizes $g$ as a time-symmetric (i.e., totally geodesic) slice of the
Lorentzian metric

$-(dt)^{2}+g,$

and then takes the plane wave limit of this — the motivation being, of course,
precisely points i. and ii. above. The virtue of a time-symmetric embedding is
that curvature properties of $g$ either carry over directly to $-(dt)^{2}+g$
or else are only slightly altered (see Lemma 1 in Section 2.2). (While a time-
symmetric embedding is, for this reason, the most desirable choice for our
purposes, in Propositions 5 and 6 of the Appendix we show that there is a
rigid relationship between Riemannian metrics and the class of Lorentzian _pp-
wave_ metrics of which gravitational plane waves are an example.) The benefit
of applying Penrose’s plane wave limit to Riemannian geometry in this way is
that one can rule out wholesale the existence of Riemannian metrics satisfying
(1) from within large classes of Riemannian metrics — and to do so by
examining the Hessian of just one function:

######  Theorem 1.

Let $(\bar{g}_{ij}(r))_{i,j=2,\dots,n}$ be a symmetric, positive-definite
matrix of smooth functions $\bar{g}_{ij}(r)$, and let
$\mathscr{F}_{\scalebox{0.5}{$\bar{g}$}}\\!$ denote the set of all Riemannian
metrics $g_{\scalebox{0.4}{R}}$ on
$\mathbb{R}^{n}=\\{(r,x^{2},\dots,x^{n})\\}$ of the form

$\displaystyle g_{\scalebox{0.4}{R}}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=(dr)^{2}+g_{ij}(r,x^{2},\dots,x^{n})dx^{i}dx^{j}\hskip
14.45377pt,\hskip 14.45377ptg_{ij}(r,0,\dots,0)=\bar{g}_{ij}(r).$ (2)

On $\mathbb{R}^{n+1}=\\{(r,t,x^{2},\dots,x^{n})\\}$, define the Lorentzian
plane wave metric

$\displaystyle h_{\scalebox{0.4}{\emph{PW}}}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\begin{pmatrix}0&1&0&\cdots&0\\\
1&0&0&\cdots&0\\\ 0&0&\bar{g}_{22}(r)&\cdots&\bar{g}_{2n}(r)\\\
0&0&\bar{g}_{32}(r)&\cdots&\bar{g}_{3n}(r)\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\bar{g}_{n2}(r)&\cdots&\bar{g}_{nn}(r)\end{pmatrix}\cdot$

Express $h_{\scalebox{0.4}{\emph{PW}}}$ in “Brinkmann coordinates” as shown in
(220), with corresponding function $H(u,x^{2},\dots,x^{n})$ given by (228).
Let $\Delta H$ denote the _(_ Euclidean _)_ Laplacian of $H$ with respect to
$x^{2},\dots,x^{n}$. Then the following is true _:_

  1. _1._

If $\partial_{u}(\Delta H)\neq 0$, then no
$g_{\scalebox{0.4}{R}}\in\mathscr{F}_{\scalebox{0.5}{$\bar{g}$}}\\!$ can have
parallel Ricci tensor.

  2. _2._

If $\Delta H\neq 0$, then no
$g_{\scalebox{0.4}{R}}\in\mathscr{F}_{\scalebox{0.5}{$\bar{g}$}}\\!$ can be
Ricci-flat.

  3. _3._

If any $H_{iju}\neq 0$, then no
$g_{\scalebox{0.4}{R}}\in\mathscr{F}_{\scalebox{0.5}{$\bar{g}$}}\\!$ can be
locally symmetric.

Note that every Riemannian metric can be put locally in the form (2); see (40)
in Section 2.2. Finally, we mention that it is possible to extend the Ricci-
flat case to that Einstein metrics in general; see Proposition 2.

###  1.2. The global case.

We show in Section 3 that a certain one-parameter family of Riemannian metrics
$g_{\scalebox{0.5}{\emph{$\lambda$}}}$ cannot exist on any compact manifold
$M$ — not by appealing directly to a topological argument, or by constructing
vector bundles over $M$, or even by examining the conformal class of
$g_{\scalebox{0.5}{\emph{$\lambda$}}}$ — but rather by showing that if such a
$g_{\scalebox{0.5}{\emph{$\lambda$}}}$ existed, then so would a certain
Lorentzian metric $g_{\scalebox{0.4}{\emph{L}}}$ on $M$, the existence of
which is known to be impossible. The $g_{\scalebox{0.4}{\emph{L}}}$ we have in
mind is a _spherical space form_ ; i.e., a Lorentzian metric with constant
positive sectional curvature. In the noncompact setting, de Sitter spacetime
is the classic example, on $M=\mathbb{R}\times\mathbb{S}^{n}$; however, when
$M$ is compact, it is well known, by a landmark result due collectively to
[CM62] and [Kli96], that there are no such space forms in any dimension. In
order to “apply” the nonexistence of such a metric to Riemannian geometry, we
must first of all decide how to form Riemannian metrics from Lorentzian ones.
There are several ways of doing this in the literature. For example, one may
do this in the manner of the Gibbons-Hawking ansatz for producing
gravitational instantons in dimension 4 [GH78, GH79]. Or one can embed a
Riemannian manifold as a submanifold of a Lorentzian one — Proposition 6 in
Section Appendix: the geometry of plane waves and Penrose’s limit provides a
particularly interesting example of this. Or one can pursue a general form of
Wick rotation, one that is independent of coordinates, by picking a unit
timelike vector field $T$ on a Lorentzian manifold
$(M,g_{\scalebox{0.4}{\emph{L}}})$ and forming the metric

$\displaystyle g\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=g_{\scalebox{0.4}{\emph{L}}}+2T^{\flat_{\scalebox{0.3}{\emph{L}}}}\otimes
T^{\flat_{\scalebox{0.3}{\emph{L}}}},$ (3)

where $T^{\flat_{\scalebox{0.3}{\emph{L}}}}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=g_{\scalebox{0.4}{\emph{L}}}(T,\cdot)$
(“unit timelike” here means that $g_{\scalebox{0.4}{\emph{L}}}(T,T)=-1$). Note
that $g(T,T)=1$ and $T^{\flat}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=g(T,\cdot)=-g_{\scalebox{0.4}{\emph{L}}}(T,\cdot)$,
so that we may just as well have expressed (3) as
$g_{\scalebox{0.4}{\emph{L}}}=g-2T^{\flat}\otimes T^{\flat}.$ All three of
these constructions yield a Riemannian metric that is somehow “dual” to a
Lorentzian one. In this paper we center our attention on (3), and observe that
it immediately yields nonexistence results: For if
“$g_{\scalebox{0.4}{\emph{L}}}$” in (3) does not exist (e.g., if it’s a
compact spherical space form), then for any choice of $T$ neither can the
corresponding Riemannian metric $g$. The problem, however, is that for an
arbitrary choice of $T$ the curvature of $g$ can vary wildly from that of
$g_{\scalebox{0.4}{\emph{L}}}$ (see, e.g., [Ole14]); therefore, we must take
care to choose both a “nonexistent” $g_{\scalebox{0.4}{\emph{L}}}$ _and_ a
suitable choice of $T$, preferably one for which the corresponding Riemannian
metric $g$ has distinguished curvature. Here we take our cue from de Sitter
spacetime itself, which is the warped product
$(\mathbb{R}\times\mathbb{S}^{n},-dt^{2}+f(t)\mathring{g})$, where
$\mathring{g}$ is the standard Riemannian metric on $\mathbb{S}^{n}$ and where
$f(t)=r^{2}\cosh^{2}\left(\frac{t}{r}\right),$ with $r$ the radius of
$\mathbb{S}^{n}$. With respect to this metric, the vector field
$T\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\partial_{t}$
is not only unit timelike, it is also _closed_ : $dT^{\flat}=0$. The condition
of being closed plays a very important role here: For as we show in
Proposition 4 of Section 3.2, for any such $T$, the 2-tensor $\nabla
T^{\flat}$ will be symmetric, one consequence of which is that the curvature
4-tensors Rm and $\text{Rm}_{\scalebox{0.4}{\emph{L}}}$ of $g$ and
$g_{\scalebox{0.4}{\emph{L}}}$ in (3) will be related to each in a very direct
way:

$\text{Rm}_{\scalebox{0.4}{\emph{L}}}=\text{Rm}+\nabla
T^{\flat}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}\nabla
T^{\flat}.$

(Here $\nabla$ is the Levi-Civita connection of $g$ and
${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$
is the Kulkarni-Nomizu product.) In such a case, our main result is the
following:

######  Theorem 2.

Let $(M,g)$ be a Riemannian $n$-manifold $(n\geq 3)$ and $T$ a unit length
closed vector field on $M$. For $\lambda\in\mathbb{R}$, the curvature 4-tensor
of $g$ takes the form

$\displaystyle\emph{\text{Rm}}=\frac{1}{2}\lambda
g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}g-2\lambda
g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}(T^{\flat}\otimes
T^{\flat})-\nabla
T^{\flat}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}\nabla
T^{\flat}$ (28)

if and only if the Lorentzian metric $g_{\scalebox{0.4}{L}}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=g-2T^{\flat}\otimes
T^{\flat}$ has constant curvature $\lambda$. If $M$ is compact, then
$\lambda=0$ and $T$ is parallel _;_ hence $g$ is flat, and its universal cover
splits isometrically as a product $\mathbb{R}\times N$.

Thus the nonexistence of compact Lorentzian spherical space forms rules out
the existence of a certain family of Riemannian metrics that are themselves
closely related to (Riemannian) space forms (as evidenced by the constant
curvature term $\frac{1}{2}\lambda
g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}g$
in (28)). For such choices of $T$ and $g$, we would argue, the relationship
(3) is valuable. With that said, we now begin in the local setting, with the
construction of a Riemannian version of Penrose’s “plane wave limit.”

##  2\. Local obstructions to Distinguished Riemannian metrics

###  2.1. Penrose’s plane wave limit

Penrose’s “plane wave limit” [Pen76] is in fact a special case of a more
general notion of “spacetime limit” due to Geroch [Ger69]. Here we give a
self-contained description of Penrose’s limit, though we point out that the
material in this Section can be found (in different notation) in [Ger69,
Pen76, Phi06]. Consider a Lorentzian metric $g$ (with Levi-Civita connection
$\nabla$) given locally in so called “null coordinates”
$(x^{0},x^{1},x^{2},\dots,x^{n})$:

$\displaystyle(g_{ij})\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\begin{pmatrix}0&1&0&0&\cdots&0\\\
1&g_{11}&g_{12}&g_{13}&\cdots&g_{1n}\\\
0&g_{21}&g_{22}&g_{23}&\cdots&g_{2n}\\\
0&g_{31}&g_{32}&g_{33}&\cdots&g_{3n}\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&g_{n1}&g_{n2}&g_{n3}&\cdots&g_{nn}\end{pmatrix}\cdot$ (29)

(In fact every Lorentzian metric $g$ can be put locally in this form, in what
is called a _null coordinate chart_ ; for the details of this construction,
consult, e.g., [Pen72, Proposition 7.14, p. 63].) Observe that
$\partial_{0}=\text{grad}\,x^{1}$ is a _null_ (or _lightlike_) vector field:
$g_{00}=0$; furthermore, because it is also a gradient, it has geodesic flow:
$\nabla_{\\!\partial_{0}}{\partial_{0}}=0$. This vector field is central to
Penrose’s construction, which proceeds as follows. First, define a new
coordinate system
$(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})$ via the
following diffeomorphism $\varphi_{\scriptscriptstyle\varepsilon}$,

$\displaystyle(x^{0},x^{1},x^{2},\dots,x^{n})\overset{\varphi_{\scriptscriptstyle\varepsilon}}{\mapsto}(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\Big{(}x^{0},\frac{x^{1}}{\varepsilon^{2}},\frac{x^{2}}{\varepsilon},\dots,\frac{x^{n}}{\varepsilon}\Big{)},$
(30)

where $\varepsilon>0$ is a constant; let

$\displaystyle g_{\scriptscriptstyle\varepsilon}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=(\varphi_{\scriptscriptstyle\varepsilon}^{-1})^{*}g$
(31)

denote the metric in these new coordinates. Since $x^{0}$ is an affine
parameter along the geodesic integral curve of $\partial_{0}$ through the
origin, observe that the limit $\varepsilon\to 0$ has the effect of “zooming
infinitesimally close to” this integral curve, pushing the remaining
coordinates $x^{1},\dots,x^{n}$ out to infinity. In fact this is precisely the
limit that Penrose will eventually take. Before doing that, define another
Lorentzian metric, $h_{\scriptscriptstyle\varepsilon}$, in the new coordinates
$(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})$, by

$\displaystyle\big{(}(h_{\scriptscriptstyle\varepsilon})_{ij}\big{)}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\underbrace{\,\begin{pmatrix}0&1&0&0&\cdots&0\\\
1&\varepsilon^{2}g_{11}&\varepsilon g_{12}&\varepsilon
g_{13}&\cdots&\varepsilon g_{1n}\\\ 0&\varepsilon
g_{21}&g_{22}&g_{23}&\cdots&g_{2n}\\\ 0&\varepsilon
g_{31}&g_{32}&g_{33}&\cdots&g_{3n}\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&\varepsilon
g_{n1}&g_{n2}&g_{n3}&\cdots&g_{nn}\end{pmatrix}\,}_{\text{defined in the
coordinates}~{}(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})},$
(32)

where each component $(h_{\scriptscriptstyle\varepsilon})_{ij}$ is
(strategically) defined as follows,

$\displaystyle(h_{\scriptscriptstyle\varepsilon})_{11}(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})\\!\\!$
$\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=$
$\displaystyle\\!\\!\varepsilon^{2}\underbrace{\,{g}_{11}(\tilde{x}^{0},\varepsilon^{2}\tilde{x}^{1},\varepsilon\,\tilde{x}^{2},\dots,\varepsilon\,\tilde{x}^{n})\,}_{=\,{g}_{11}(x^{0},x^{1},x^{2},\dots,x^{n})},$
(33)
$\displaystyle(h_{\scriptscriptstyle\varepsilon})_{22}(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})\\!\\!$
$\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=$
$\displaystyle\\!\\!\underbrace{\,g_{22}(\tilde{x}^{0},\varepsilon^{2}\tilde{x}^{1},\varepsilon\,\tilde{x}^{2},\dots,\varepsilon\,\tilde{x}^{n})\,}_{=\,{g}_{22}(x^{0},x^{1},x^{2},\dots,x^{n})},$
(34)

and similarly with the others. Note that as $\varepsilon\to 0$,

$\displaystyle\lim_{\varepsilon\to
0}(h_{\scriptscriptstyle\varepsilon})_{11}\\!\\!$
$\displaystyle\overset{\eqref{newcomp}}{=}$
$\displaystyle\\!\\!0\cdot{g}_{11}(\tilde{x}^{0},0,0,\dots,0)=0,$
$\displaystyle\lim_{\varepsilon\to
0}(h_{\scriptscriptstyle\varepsilon})_{22}\\!\\!$
$\displaystyle\overset{\eqref{newcomp2}}{=}$
$\displaystyle\\!\\!{g}_{22}(\tilde{x}^{0},0,0,\dots,0),$

and similarly with the others. Crucially, the metric
$h_{\scriptscriptstyle\varepsilon}$ is _homothetic_ to
$g_{\scriptscriptstyle\varepsilon}$; to see this, use the fact that, via (30)
and (31),

$\underbrace{\,g_{\scriptscriptstyle\varepsilon}(\partial_{\tilde{x}^{i}},\partial_{\tilde{x}^{j}})\,}_{g\left(d\varphi_{\scriptscriptstyle\varepsilon}^{-1}(\partial_{\tilde{x}^{i}}),d\varphi_{\scriptscriptstyle\varepsilon}^{-1}(\partial_{\tilde{x}^{i}})\right)}d\tilde{x}^{i}\otimes
d\tilde{x}^{j}\bigg{|}_{(\tilde{x}^{0},\tilde{x}^{1},\dots,\tilde{x}^{n})}=g(\partial_{x^{i}},\partial_{x^{j}})dx^{i}\otimes
dx^{j}\bigg{|}_{\varphi_{\scriptscriptstyle\varepsilon}^{-1}(\tilde{x}^{0},\tilde{x}^{1},\dots,\tilde{x}^{n})},$

where
$d\varphi_{\scriptscriptstyle\varepsilon}^{-1}(\partial_{\tilde{x}^{1}})=\varepsilon^{2}\,\partial_{x^{1}},d\varphi_{\scriptscriptstyle\varepsilon}^{-1}(\partial_{\tilde{x}^{2}})=\varepsilon\,\partial_{x^{2}}$,
etc., to obtain

$\displaystyle dx^{0}\otimes dx^{1}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\varepsilon^{2}\,d\tilde{x}^{0}\otimes d\tilde{x}^{1},$
$\displaystyle g_{11}(x^{0},x^{1},x^{2},\dots,x^{n})\,dx^{1}\otimes
dx^{1}\\!\\!$ $\displaystyle\overset{\eqref{newcomp}}{=}$
$\displaystyle\\!\\!\varepsilon^{2}\,(h_{\scriptscriptstyle\varepsilon})_{11}(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})\,d\tilde{x}^{1}\otimes
d\tilde{x}^{1},$ $\displaystyle
g_{12}(x^{0},x^{1},x^{2},\dots,x^{n})\,dx^{1}\otimes dx^{2}\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\varepsilon^{2}\,(h_{\scriptscriptstyle\varepsilon})_{12}(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})\,d\tilde{x}^{1}\otimes
d\tilde{x}^{2},$ $\displaystyle
g_{22}(x^{0},x^{1},x^{2},\dots,x^{n})\,dx^{2}\otimes dx^{2}\\!\\!$
$\displaystyle\overset{\eqref{newcomp2}}{=}$
$\displaystyle\\!\\!\varepsilon^{2}\,(h_{\scriptscriptstyle\varepsilon})_{22}(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})\,d\tilde{x}^{2}\otimes
d\tilde{x}^{2},$

and so on, which clearly yields the relationship

$\displaystyle
h_{\scriptscriptstyle\varepsilon}=\frac{g_{\scriptscriptstyle\varepsilon}}{\varepsilon^{2}}\cdot$
(35)

In particular, the Levi-Civita connections of
$h_{\scriptscriptstyle\varepsilon}$ and $g_{\scriptscriptstyle\varepsilon}$
are equal: $\nabla^{{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}}=\nabla$. The
final step in Penrose’s construction is to take the limit of
$g_{\scriptscriptstyle\varepsilon}/\varepsilon^{2}$ as $\varepsilon\to 0$,

$\lim_{\varepsilon\to
0}\frac{g_{\scriptscriptstyle\varepsilon}}{\varepsilon^{2}}=\lim_{\varepsilon\to
0}h_{\scriptscriptstyle\varepsilon},$

which limit yields a _nondegenerate_ — and _non-flat_! — Lorentzian metric:

$\displaystyle h_{\scalebox{0.4}{PW}}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\underbrace{\,\begin{pmatrix}0&1&0&0&\cdots&0\\\
1&0&0&0&\cdots&0\\\
0&0&g_{22}(\tilde{x}^{0},0,\dots,0)&g_{23}(\tilde{x}^{0},0,\dots,0)&\cdots&g_{2n}(\tilde{x}^{0},0,\dots,0)\\\
0&0&g_{32}(\tilde{x}^{0},0,\dots,0)&g_{33}(\tilde{x}^{0},0,\dots,0)&\cdots&g_{3n}(\tilde{x}^{0},0,\dots,0)\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&g_{n2}(\tilde{x}^{0},0,\dots,0)&g_{n3}(\tilde{x}^{0},0,\dots,0)&\cdots&g_{nn}(\tilde{x}^{0},0,\dots,0)\end{pmatrix}\,}_{\text{defined
in the
coordinates}~{}(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})}\cdot$
(36)

As it turns out, (36) is a distinguished and very well known metric in general
relativity, a so called _plane wave metric_ (expressed here in so called
“Rosen coordinates”), and the limit we’ve just described is now known as
Penrose’s “plane wave limit.” The most obvious feature of (36) is that _it is
a far simpler metric than $g$_; indeed, its components are functions of only
the one variable $\tilde{x}^{0}=x^{0}$, which means that its curvature
properties are given by ODEs, not PDEs. What is of fundamental importance for
us is that the limit taken was that of a one-parameter family of metrics
$h_{\scriptscriptstyle\varepsilon}$ each of which was homothetic (35) to our
original metric $g$. For this reason, certain curvature properties are
_preserved_ in this limit — in the language of [Ger69], they are “hereditary”:

######  Proposition 1.

Let $(M,g)$ be a Lorentzian manifold expressed locally in the form (29), and
let $h_{\scalebox{0.4}{\emph{PW}}}$ denote its corresponding plane wave limit
(36). Then the following is true _:_

  1. _1._

If $g$ is an Einstein metric, $\emph{\text{Ric}}=\lambda g$, then
$h_{\scalebox{0.4}{\emph{PW}}}$ is Ricci-flat.

  2. _2._

If $g$ is locally conformally flat, then so is
$h_{\scalebox{0.4}{\emph{PW}}}$.

  3. _3._

If $g$ is locally symmetric, $\nabla\emph{\text{Rm}}=0$, then so is
$h_{\scalebox{0.4}{\emph{PW}}}$.

  4. _4._

If $g$ has parallel Ricci tensor, $\nabla\emph{\text{Ric}}=0$, then so does
$h_{\scalebox{0.4}{\emph{PW}}}$.

###### Proof.

This is known (see, e.g., [Phi06, Ger69]), though our presentation differs
from these. Given the constant conformal factor in (35), the curvature
4-tensor Rm, the Ricci tensor Ric, and the Weyl tensor W transform as follows
(see, e.g., [Lee18, Theorem 7.30]):

$\displaystyle\text{Rm}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}=\varepsilon^{-2}\text{Rm}_{\scalebox{0.55}{\text{$g_{\varepsilon}$}}}\hskip
14.45377pt,\hskip
14.45377pt\text{Ric}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}=\text{Ric}_{\scalebox{0.55}{\text{$g_{\varepsilon}$}}}\hskip
14.45377pt,\hskip
14.45377pt\text{W}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}=\varepsilon^{-2}\text{W}_{\scalebox{0.55}{\text{$g_{\varepsilon}$}}}.$
(37)

Our first task is to establish that

$\displaystyle\text{Ric}_{\scalebox{0.55}{\text{$h_{\scalebox{0.4}{PW}}$}}}=\lim_{\varepsilon\to
0}\text{Ric}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}\hskip
14.45377pt,\hskip
14.45377pt\text{W}_{\scalebox{0.55}{\text{$h_{\scalebox{0.4}{PW}}$}}}=\lim_{\varepsilon\to
0}\text{W}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}\hskip 14.45377pt,\hskip
14.45377pt\nabla^{\scalebox{0.3}{PW}}\text{Rm}_{\scalebox{0.55}{\text{$h_{\scalebox{0.4}{PW}}$}}}=\lim_{\varepsilon\to
0}\nabla^{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}\text{Rm}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}.$
(38)

Indeed, an examination of (32)-(34), together with the inverse of
$h_{\scriptscriptstyle\varepsilon}$,111The inverse of
$h_{\scriptscriptstyle\varepsilon}$ is of the form
$\displaystyle\big{(}(h_{\scriptscriptstyle\varepsilon})^{ij}\big{)}=\begin{pmatrix}\varepsilon^{2}f^{00}&1&\varepsilon
f^{02}&\varepsilon f^{03}&\cdots&\varepsilon f^{0n}\\\ 1&0&0&0&\cdots&0\\\
\varepsilon f^{02}&0&h^{22}&h^{23}&\cdots&h^{2n}\\\ \varepsilon
f^{03}&0&h^{32}&h^{33}&\cdots&h^{3n}\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\ \varepsilon
f^{0n}&0&h^{n2}&h^{n3}&\cdots&h^{nn}\end{pmatrix},$ where each $f^{0i}$ is a
certain smooth function of the $g_{ij},h_{ij}$’s and where the submatrix
$(h^{ij})$ is the inverse of $h$. Every component other than the two instances
of $1$ has $\text{det}\,h$ for its denominator. yields that every Christoffel
symbol $\Gamma^{i}_{jk}$ takes the form

$\displaystyle\Gamma^{i}_{jk}=\varepsilon^{\ell}\gamma^{i}_{jk}(\tilde{x}^{0},\varepsilon^{2}\tilde{x}^{1},\varepsilon\,\tilde{x}^{2},\dots,\varepsilon\,\tilde{x}^{n})\hskip
14.45377pt,\hskip 14.45377pt0\leq\ell\leq 4,$ (39)

for some smooth function
$\gamma^{i}_{jk}(\tilde{x}^{0},\varepsilon^{2}\tilde{x}^{1},\varepsilon\,\tilde{x}^{2},\dots,\varepsilon\,\tilde{x}^{n})$;
furthermore, $\ell=0$ in (39) only for $\Gamma^{1}_{ij},\Gamma^{i}_{0j}$ with
$i,j=2,\dots,n$, and these particular Christoffel symbols involve derivatives
by $\tilde{x}^{0}$ only. Since any partial derivatives of the functions
$\gamma^{i}_{jk}(\tilde{x}^{0},\varepsilon^{2}\tilde{x}^{1},\varepsilon\,\tilde{x}^{2},\dots,\varepsilon\,\tilde{x}^{n})$
by $\tilde{x}^{1},\dots,\tilde{x}^{n}$ will vanish in the limit
$\varepsilon\to 0$,

$\displaystyle\lim_{\varepsilon\to
0}\frac{\partial\gamma^{i}_{jk}(\tilde{x}^{0},\varepsilon^{2}\tilde{x}^{1},\varepsilon\,\tilde{x}^{2},\dots,\varepsilon\,\tilde{x}^{n})}{\partial\tilde{x}^{1,2,\dots,n}}\\!\\!$
$\displaystyle=$ $\displaystyle\lim_{\varepsilon\to
0}\underbrace{\,\varepsilon^{\ell}\,}_{\ell\,=\,1,2}\frac{\partial\gamma^{i}_{jk}(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})}{\partial\tilde{x}^{1,2,\dots,n}}\Bigg{|}_{(\tilde{x}^{0},\varepsilon^{2}\tilde{x}^{1},\varepsilon\,\tilde{x}^{2},\dots,\varepsilon\,\tilde{x}^{n})}=0,$

it follows that only the Christoffel symbols of the form
$\Gamma^{1}_{ij},\Gamma^{i}_{0j}$, their products, and their partial
derivatives by $\tilde{x}^{0}$ only, will remain after the limit
$\varepsilon\to 0$ is taken. In any case, the following is clear:

  1. i.

One can take the limit $\lim_{\varepsilon\to
0}h_{\scriptscriptstyle\varepsilon}$ to obtain $h_{\scalebox{0.4}{PW}}$, and
_then_ compute
$\text{Rm}_{\scalebox{0.55}{\text{$h_{\scalebox{0.4}{PW}}$}}},\text{W}_{\scalebox{0.55}{\text{$h_{\scalebox{0.4}{PW}}$}}}$,
and $\text{Ric}_{\scalebox{0.55}{\text{$h_{\scalebox{0.4}{PW}}$}}}$ directly
from $h_{\scalebox{0.4}{PW}}$,

  2. ii.

Or one can compute
$\text{Rm}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}},\text{W}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}$,
and $\text{Ric}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}$ from the outset,
and _then_ take their limits as $\varepsilon\to 0$;

by the smooth dependence on $\varepsilon$, one obtains the same result either
way. Having established this, suppose now that
$g_{\scriptscriptstyle\varepsilon}$ is an Einstein metric, then the fact that

$\text{Ric}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}\overset{\eqref{eqn:list}}{=}\text{Ric}_{\scalebox{0.55}{\text{$g_{\varepsilon}$}}}=\lambda
g_{\scriptscriptstyle\varepsilon}\overset{\eqref{eqn:cm}}{=}\lambda(\varepsilon^{2}h_{\scriptscriptstyle\varepsilon})$

directly yields

$\text{Ric}_{\scalebox{0.55}{\text{$h_{\scalebox{0.4}{PW}}$}}}\overset{\eqref{eqn:list2}}{=}\lim_{\varepsilon\to
0}\text{Ric}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}=\lambda(0\cdot
h_{\scalebox{0.4}{PW}})=0,$

thus proving 1. above. Next, if $g_{\scriptscriptstyle\varepsilon}$ is locally
conformally flat, so that
$\text{W}_{\scalebox{0.55}{\text{$g_{\varepsilon}$}}}=0$, then

$\text{W}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}\overset{\eqref{eqn:list}}{=}\frac{\text{W}_{\scalebox{0.55}{\text{$g_{\varepsilon}$}}}}{\varepsilon^{2}}=0\hskip
14.45377pt\Rightarrow\hskip
14.45377pt\text{W}_{\scalebox{0.55}{\text{$h_{\scalebox{0.4}{PW}}$}}}\overset{\eqref{eqn:list2}}{=}\lim_{\varepsilon\to
0}\text{W}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}=0.$

If $g_{\scriptscriptstyle\varepsilon}$ is locally symmetric, so that
$\nabla\text{Rm}_{\scalebox{0.55}{\text{$g_{\varepsilon}$}}}=0$, then

$\nabla^{{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}}\text{Rm}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}\overset{\eqref{eqn:list}}{=}\frac{\nabla\text{Rm}_{\scalebox{0.55}{\text{$g_{\varepsilon}$}}}}{\varepsilon^{2}}=0\hskip
14.45377pt\Rightarrow\hskip
14.45377pt\nabla^{\scalebox{0.3}{PW}}\text{Rm}_{\scalebox{0.55}{\text{$h_{\scalebox{0.4}{PW}}$}}}\overset{\eqref{eqn:list2}}{=}\lim_{\varepsilon\to
0}\nabla^{{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}}\text{Rm}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}=0.$

Likewise if $g$ has parallel Ricci tensor. ∎

Note that the curvature properties mentioned in Proposition 1 are not the only
ones preserved in the limit; indeed, so are any other tensors that vanish and
are constructed from $\nabla,\text{Rm}$, and traces (or irreducible
components) of the latter. For a more thorough discussion of this fact,
consult [Ger69].

###  2.2. The plane wave limit of a Riemannian manifold

Now that we have a firm understanding of Penrose’s plane wave limit, we can
make use of Proposition 1 to arrive at results purely in Riemannian geometry.
To set the stage, let $(M,\bar{g})$ be a Riemannian $n$-manifold, and recall
the existence of _semigeodesic coordinates_ , namely, at each point of $M$,
local coordinates $(r,x^{2},\dots,x^{n})$ exist in which the integral curves
$s\mapsto(s,x^{2},\dots,x^{n})$ of $\partial_{r}$ are geodesics normal to the
level sets of $r$:

$\displaystyle\bar{g}=(dr)^{2}+\bar{g}_{ij}(r,x^{2},\dots,x^{n})dx^{i}dx^{j}.$
(40)

It follows that $\partial_{r}=\text{grad}_{\scalebox{0.5}{$\bar{g}$}}r$ and
that $r$ is a local distance function (see, e.g., [Lee18, Proposition 6.41, p.
182]). Choose such coordinates in a neighborhood $\mathscr{U}\subseteq M$, and
henceforth restrict $\bar{g}$ to $\mathscr{U}$. In order to take the “plane
wave limit of $(\mathscr{U},\bar{g}|_{\mathscr{U}})$,” we will need to
suitably embed it in a Lorentzian manifold. We do so in the simplest way
possible, by embedding $(\mathscr{U},\bar{g}|_{\mathscr{U}})$ as a “time-
symmetric” (i.e., totally geodesic) hypersurface in the $(n+1)$-dimensional
Lorentzian manifold

$\displaystyle\mathbb{R}\times\mathscr{U}\hskip 14.45377pt,\hskip
14.45377ptg\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=-(dt)^{2}+\bar{g}.$
(41)

In the coordinate basis
$\\{\partial_{t},\partial_{r},\partial_{2},\dots,\partial_{n}\\}$, this
Lorentzian metric is

$\displaystyle(g_{\alpha\beta})=\begin{pmatrix}-1&0&0&0&\cdots&0\\\
0&1&0&0&\cdots&0\\\ 0&0&\bar{g}_{22}&\bar{g}_{23}&\cdots&\bar{g}_{2n}\\\
0&0&\bar{g}_{32}&\bar{g}_{33}&\cdots&\bar{g}_{3n}\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\bar{g}_{n2}&\bar{g}_{n3}&\cdots&\bar{g}_{nn}\end{pmatrix}\cdot$

In what follows, we reserve the Latin indices $i,j,k$ for the coordinates
$x^{2},\dots,x^{n}$ and Greek indices $\alpha,\beta$ for
$t,r,x^{2},\dots,x^{n}$. The reason for this embedding is the following well
known relationship between the curvatures of $\bar{g}$ and $g$:

######  Lemma 1.

If $\bar{g}$ in (40) is locally symmetric, then so is $g$ in (41). If
$\bar{g}$ has parallel Ricci tensor, then so does $g$. Finally, if $\bar{g}$
is Einstein, $\emph{\text{Ric}}_{\scriptscriptstyle\bar{g}}=\lambda\bar{g}$,
then $\emph{\text{Ric}}_{\scriptscriptstyle g}=\lambda(g+(dt)^{2})$.

###### Proof.

Let $X,Y,Z$ be the lifts to $\mathbb{R}\times\mathscr{U}$ of vector fields on
$\mathscr{U}$, and let $\nabla^{\scriptscriptstyle g},R_{\scriptscriptstyle
g}$ and $\nabla^{\scriptscriptstyle\bar{g}},R_{\scriptscriptstyle\bar{g}}$
denote the Levi-Civita connections and curvature endomorphisms of $g$ and
$\bar{g}$, respectively. Then $\nabla^{\scriptscriptstyle
g}_{\\!X}{Y}=\nabla^{\scriptscriptstyle\bar{g}}_{\\!X}{Y}$, while
$\nabla^{\scriptscriptstyle g}_{\\!X}{\partial_{t}}=\nabla^{\scriptscriptstyle
g}_{\\!\partial_{t}}X=\nabla^{\scriptscriptstyle
g}_{\\!\partial_{t}}\partial_{t}=0$; see, e.g., [O’N83, pp. 206ff.]. Thus

$R_{\scriptscriptstyle g}(X,Y)Z=R_{\scriptscriptstyle\bar{g}}(X,Y)Z\hskip
14.45377pt,\hskip 14.45377ptR_{\scriptscriptstyle
g}(X,Y)\partial_{t}=R_{\scriptscriptstyle g}(X,\partial_{t})Y=0.$

It follows that if $\bar{g}$ is locally symmetric,
$\nabla^{\scriptscriptstyle\bar{g}}\text{Rm}_{\scriptscriptstyle\bar{g}}=0$,
then so is $g$, and if $\bar{g}$ has parallel Ricci tensor,
$\nabla^{\scriptscriptstyle\bar{g}}\text{Ric}_{\scriptscriptstyle\bar{g}}=0$,
then so does $g$. Similarly,

$\text{Ric}_{\scriptscriptstyle
g}(X,Y)=\text{Ric}_{\scriptscriptstyle\bar{g}}(X,Y)=\lambda\bar{g}\hskip
14.45377pt,\hskip 14.45377pt\text{Ric}_{\scriptscriptstyle
g}(\partial_{t},\cdot)=0,$

so that the equality $\text{Ric}_{\scriptscriptstyle g}=\lambda(g+(dt)^{2})$
holds. ∎

Our goal now is to take the plane wave limit of (41), keeping track of the
Riemannian metric $\bar{g}$ embedded within, and then apply Proposition 1. In
order to do this, we must first express the Lorentzian metric (41) in null
coordinates (29). The simplest way to do so is to change coordinates
$(t,r,x^{2},\dots,x^{n})\mapsto(x^{0},x^{1},x^{2},\dots,x^{n})$ via

$\displaystyle x^{0}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\frac{1}{\sqrt{2}}(r+t)\hskip
14.45377pt,\hskip 14.45377ptx^{1}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\frac{1}{\sqrt{2}}(r-t),$
(42)

with respect to which $g$ will take the form

$(g_{\alpha\beta})=\underbrace{\,\begin{pmatrix}0&1&0&0&\cdots&0\\\
1&0&0&0&\cdots&0\\\ 0&0&\bar{g}_{22}&\bar{g}_{23}&\cdots&\bar{g}_{2n}\\\
0&0&\bar{g}_{32}&\bar{g}_{33}&\cdots&\bar{g}_{3n}\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\bar{g}_{n2}&\bar{g}_{n3}&\cdots&\bar{g}_{nn}\end{pmatrix}\,}_{(g_{\alpha\beta})~{}\text{in
coordinates}~{}(x^{0},x^{1},x^{2},\dots,x^{n})}\hskip 7.22743pt,\hskip
7.22743pt\bar{g}_{ij}\big{(}(x^{0}+x^{1})/\sqrt{2},x^{2},\dots,x^{n}\big{)}.$

Note that the null gradient vector field $\partial_{0}$ giving rise to these
coordinates is simply

$\partial_{0}=\text{grad}\,x^{1}=\frac{1}{\sqrt{2}}(\partial_{r}+\partial_{t}).$

Now we can take the plane wave limit of $g$. Indeed, setting
$g_{\scriptscriptstyle\varepsilon}=(\varphi_{\scriptscriptstyle\varepsilon}^{-1})^{*}g$
as in (31) and bearing in mind that, with the coordinate transformation (30),
the components $\bar{g}_{ij}$ take the form

$\displaystyle\bar{g}_{ij}\big{(}(\tilde{x}^{0}+\varepsilon^{2}\tilde{x}^{1})/\sqrt{2},\varepsilon\tilde{x}^{2},\dots,\varepsilon\tilde{x}^{n}\big{)},$
(43)

it follows that the metric $h_{\scriptscriptstyle\varepsilon}$ given by (32)
is

$(({h_{\scriptscriptstyle\varepsilon}})_{\alpha\beta})=\underbrace{\,\begin{pmatrix}0&1&0&0&\cdots&0\\\
1&0&0&0&\cdots&0\\\ 0&0&\bar{g}_{22}&\bar{g}_{23}&\cdots&\bar{g}_{2n}\\\
0&0&\bar{g}_{32}&\bar{g}_{33}&\cdots&\bar{g}_{3n}\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\bar{g}_{n2}&\bar{g}_{n3}&\cdots&\bar{g}_{nn}\end{pmatrix}\,}_{\text{defined
in
coordinates}~{}(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})}\hskip
7.22743pt,\hskip 7.22743pt\text{$\bar{g}_{ij}$ given by
\eqref{eqn:bar_g_ij}}.$

Recall that this metric is conformal to $g_{\scriptscriptstyle\varepsilon}$.
Taking the limit $\varepsilon\to 0$ yields

$\displaystyle h_{\scalebox{0.4}{PW}}=\begin{pmatrix}0&1&0&\cdots&0\\\
1&0&0&\cdots&0\\\
0&0&g_{22}(\tilde{x}^{0}/\sqrt{2},0,\dots,0)&\cdots&g_{2n}(\tilde{x}^{0}/\sqrt{2},0,\dots,0)\\\
0&0&g_{32}(\tilde{x}^{0}/\sqrt{2},0,\dots,0)&\cdots&g_{3n}(\tilde{x}^{0}/\sqrt{2},0,\dots,0)\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&g_{n2}(\tilde{x}^{0}/\sqrt{2},0,\dots,0)&\cdots&g_{nn}(\tilde{x}^{0}/\sqrt{2},0,\dots,0)\end{pmatrix}\cdot$

A final, cosmetic relabeling of coordinates,

$\displaystyle(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{i})\mapsto(\sqrt{2}r,t/\sqrt{2},x^{i}),$
(44)

puts $h_{\scalebox{0.4}{PW}}$ into a form directly resembling our original
Riemannian metric $\bar{g}$:

$\displaystyle h_{\scalebox{0.4}{PW}}=2drdt+g_{ij}(r,0,\dots,0)dx^{i}dx^{j}.$
(45)

The following constitutes the fruits of our labors, connecting geometric
properties of $\bar{g}$ with those of (the much simpler metric)
$h_{\scalebox{0.4}{PW}}$:

######  Proposition 2.

Let $\bar{g}$ be a Riemannian metric expressed locally in any choice of
semgeodesic coordinates (40). Let $h_{\scalebox{0.4}{\emph{PW}}}$ in (45) be
its corresponding Lorentzian plane wave metric. Then the following is true _:_

  1. _1_.

If $\bar{g}$ is Einstein,
$\emph{\text{Ric}}_{\scriptscriptstyle\bar{g}}=\lambda\bar{g}$, then
$h_{\scalebox{0.4}{\emph{PW}}}$ satisfies
$\emph{\text{Ric}}_{\scalebox{0.55}{$h_{\scalebox{0.4}{\emph{PW}}}$}}=\lambda(dr)^{2}$.

  2. _2_.

If $\bar{g}$ is locally symmetric, then so is $h_{\scalebox{0.4}{\emph{PW}}}$.

  3. _3_.

If $\bar{g}$ has parallel Ricci tensor, then so does
$h_{\scalebox{0.4}{\emph{PW}}}$.

###### Proof.

We work in the coordinates (30), with $g_{\scriptscriptstyle\varepsilon}$ in
place of $g$ in (41). If
$\text{Ric}_{\scriptscriptstyle\bar{g}}=\lambda\bar{g}$, then by Lemma 1,
$\text{Ric}_{\scalebox{0.55}{\text{$g_{\varepsilon}$}}}=\lambda(g_{\scriptscriptstyle\varepsilon}+(dt)^{2})$.
Thus, recalling that

$t\overset{\eqref{eqn:vu2}}{=}x^{0}-x^{1}\overset{\eqref{eqn:tilde}}{=}\tilde{x}^{0}-\varepsilon^{2}\tilde{x}^{1},$

we have that

$\text{Ric}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}\overset{\eqref{eqn:list}}{=}\text{Ric}_{\scalebox{0.55}{\text{$g_{\varepsilon}$}}}=\lambda\big{(}g_{\scriptscriptstyle\varepsilon}+(dt)^{2}\big{)}\overset{\eqref{eqn:cm}}{=}\lambda\Big{(}\varepsilon^{2}h_{\scriptscriptstyle\varepsilon}+d(\tilde{x}^{0}-\varepsilon^{2}\tilde{x}^{1})^{2}\Big{)},$

so that

$\text{Ric}_{\scalebox{0.55}{\text{$h_{\scalebox{0.4}{PW}}$}}}\overset{\eqref{eqn:list2}}{=}\lim_{\varepsilon\to
0}\text{Ric}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}=\lambda\cdot 0\cdot
h_{\scalebox{0.4}{PW}}+\lambda(d\tilde{x}^{0})^{2}\overset{\eqref{eqn:lastchange}}{=}\lambda(dr)^{2}.$

For the remaining cases, we may rely directly on Proposition 1. Indeed, if $g$
is locally symmetric, then by Lemma 1 so is
$(\mathbb{R}\times\mathscr{U},g_{\scalebox{0.4}{\emph{L}}})$, so that

$\nabla^{\scalebox{0.3}{PW}}\text{Rm}_{\scalebox{0.4}{PW}}\overset{\eqref{eqn:list}}{=}\varepsilon^{-2}(\nabla^{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}\text{Rm}_{\scalebox{0.55}{\text{$h_{\varepsilon}$}}})=\varepsilon^{-2}\cdot
0=0,$

where
$\nabla^{\scalebox{0.3}{PW}}=\nabla^{\scalebox{0.55}{\text{$h_{\varepsilon}$}}}$
because the conformal factor is a constant; likewise if $g$ has parallel Ricci
tensor. Taking the limit $\varepsilon\to 0$, the proof is complete. ∎

To illustrate the Einstein case, consider on
$\mathbb{R}^{n}=\\{(r,x^{2},\dots,x^{n})\\}$ the Riemannian metric $\bar{g}$
given by

$\bar{g}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=(dr)^{2}+\sum_{i=2}^{n}e^{\sqrt{2}r}(dx^{i})^{2}$

This is an Einstein metric, with Einstein constant $\lambda=-\frac{n-1}{2}$.
Its corresponding plane wave limit (45) on
$\mathbb{R}^{n+1}=\\{(r,t,x^{2},\dots,x^{n})\\}$,

$h_{\scalebox{0.4}{PW}}=2drdt+\sum_{i=2}^{n}e^{\sqrt{2}r}(dx^{i})^{2},$

is easily verified to satisfy
$\text{Ric}_{\scalebox{0.55}{\text{$h_{\scalebox{0.4}{PW}}$}}}=-\frac{n-1}{2}(dr)^{2}$.
Moving on, as we show in Proposition 3 below, Proposition 2 takes on a more
consequential form when the Riemannian metrics (40) are thought of as defining
a type of “moduli space” over Lorentzian plane waves, for this allows one to
make statements about entire classes of Riemannian metrics:

######  Definition 1.

Let $(\bar{g}_{ij}(r))_{i,j=2,\dots,n}$ be a symmetric, positive-definite
matrix of smooth functions $\bar{g}_{ij}(r)$. Let
$\mathscr{F}_{\scalebox{0.5}{\emph{$\bar{g}$}}}$ denote the set of all
Riemannian metrics $g_{\scalebox{0.4}{R}}$ on
$\mathbb{R}^{n}=\\{(r,x^{2},\dots,x^{n})\\}$ of the form

$g_{\scalebox{0.4}{R}}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=(dr)^{2}+g_{ij}(r,x^{2},\dots,x^{n})dx^{i}dx^{j}\hskip
14.45377pt,\hskip 14.45377ptg_{ij}(r,0,\dots,0)=\bar{g}_{ij}(r).$

Any $g_{\scalebox{0.4}{R}}\in\mathscr{F}_{\scalebox{0.5}{\emph{$\bar{g}$}}}$
will be called a _Riemannian extension of $(\bar{g}_{ij}(r))$_.

######  Proposition 3.

On $\mathbb{R}^{n+1}=\\{(r,t,x^{2},\dots,x^{n})\\}$, define the plane wave

$\displaystyle h_{\scalebox{0.4}{\emph{PW}}}=\begin{pmatrix}0&1&0&\cdots&0\\\
1&0&0&\cdots&0\\\ 0&0&\bar{g}_{22}(r)&\cdots&\bar{g}_{2n}(r)\\\
0&0&\bar{g}_{32}(r)&\cdots&\bar{g}_{3n}(r)\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\bar{g}_{n2}(r)&\cdots&\bar{g}_{nn}(r)\end{pmatrix},$ (46)

with $(\bar{g}_{ij}(r))$ as in Definition 1, and let
$\mathscr{F}_{\scalebox{0.5}{\emph{$\bar{g}$}}}$ denote the set of all
Riemannian extensions $g_{\scalebox{0.4}{R}}$ of $(\bar{g}_{ij}(r))$. If
$h_{\scalebox{0.4}{\emph{PW}}}$ is not Ricci-flat, then neither is any
$g_{\scalebox{0.4}{R}}\in\mathscr{F}_{\scalebox{0.5}{\emph{$\bar{g}$}}}$. If
$h_{\scalebox{0.4}{\emph{PW}}}$ is not locally symmetric, then neither is any
$g_{\scalebox{0.4}{R}}\in\mathscr{F}_{\scalebox{0.5}{\emph{$\bar{g}$}}}$. If
$h_{\scalebox{0.4}{\emph{PW}}}$ does not have parallel Ricci tensor, then
neither will any
$g_{\scalebox{0.4}{R}}\in\mathscr{F}_{\scalebox{0.5}{\emph{$\bar{g}$}}}$.

###### Proof.

For any Riemannian extension
$g_{\scalebox{0.4}{\emph{R}}}\in\mathscr{F}_{\scalebox{0.5}{\emph{$\bar{g}$}}}$,
simply form the Lorentzian metric $g\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=-(dt)^{2}+g_{\scalebox{0.4}{\emph{R}}}$
as in (41), and use Proposition 2. ∎

This result is much more appealing when the plane wave metric (46) is
expressed in “Brinkmann coordinates” (see (220) in the Appendix), for in such
coordinates its curvature 4-tensor is effectively determined by the Hessian of
just one function:

######  Theorem 1.

Let $(\bar{g}_{ij}(r)),\mathscr{F}_{\scalebox{0.5}{\emph{$\bar{g}$}}}$, and
$h_{\scalebox{0.4}{\emph{PW}}}$ be as in Proposition 3, and express
$h_{\scalebox{0.4}{\emph{PW}}}$ in “Brinkmann coordinates” as in (220), with
corresponding function $H(u,x^{2},\dots,x^{n})$ given by (228). Let $\Delta H$
denote the _(_ Euclidean _)_ Laplacian of $H$ with respect to
$x^{2},\dots,x^{n}$. Then the following is true _:_

  1. _1._

If $\partial_{u}(\Delta H)\neq 0$, then no
$g_{\scalebox{0.4}{R}}\in\mathscr{F}_{\scalebox{0.5}{\emph{$\bar{g}$}}}$ has
parallel Ricci tensor.

  2. _2._

If $\Delta H\neq 0$, then no
$g_{\scalebox{0.4}{R}}\in\mathscr{F}_{\scalebox{0.5}{\emph{$\bar{g}$}}}$ is
Ricci-flat.

  3. _3._

If any $H_{iju}\neq 0$ _(_ $i,j=x^{2},\dots,x^{n}$_)_ , then no
$g_{\scalebox{0.4}{R}}\in\mathscr{F}_{\scalebox{0.5}{\emph{$\bar{g}$}}}$ is
locally symmetric.

###### Proof.

As shown in Proposition 6 of the Appendix, the plane wave (46) is in so called
“Rosen coordinates.” The isometry defined in (227) will put
$h_{\scalebox{0.4}{PW}}$ in the “Brinkmann coordinates”
$(v,u,x^{2},\dots,x^{n})$ shown in (220),

$\displaystyle h_{\scalebox{0.4}{PW}}=\begin{pmatrix}0&1&0&\cdots&0\\\
1&H&0&\cdots&0\\\ 0&0&1&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\cdots&1\end{pmatrix},$

with $H(u,x^{2},\dots,x^{n})$ a function quadratic in $x^{2},\dots,x^{n}$ (see
(228)). In this form, it is straightforward to verify, using the Christoffel
symbols (222), that the only nonvanishing components of Rm and Ric are

$\text{Rm}(\partial_{i},\partial_{u},\partial_{u},\partial_{j})=-\frac{1}{2}H_{ij}\hskip
14.45377pt,\hskip
14.45377pt\text{Ric}(\partial_{u},\partial_{u})=-\frac{1}{2}\Delta H.$

In particular, if $H$ is not harmonic in $x^{2},\dots,x^{n}$, then
$h_{\scalebox{0.4}{PW}}$ will not be Ricci-flat. Similarly, the only
nonvanishing components of $\nabla\text{Rm}$ and $\nabla\text{Ric}$ (when $H$
is quadratic in $x^{2},\dots,x^{n}$, which is the case here) are

$(\nabla_{\\!\partial_{u}}\text{Rm})(\partial_{i},\partial_{u},\partial_{u},\partial_{j})=-\frac{1}{2}H_{iju}\hskip
7.22743pt,\hskip
7.22743pt(\nabla_{\\!\partial_{u}}\text{Ric})(\partial_{u},\partial_{u})=-\frac{1}{2}\partial_{u}(\Delta
H).$

In particular, if each $H_{iju}\neq 0$, then $h_{\scalebox{0.4}{PW}}$ will not
be locally symmetric; if $\partial_{u}(\Delta H)\neq 0$, then
$h_{\scalebox{0.4}{PW}}$ will not have parallel Ricci tensor. Applying
Proposition 3 to any Riemannian extension
$g_{\scalebox{0.4}{\emph{R}}}\in\mathscr{F}_{\scalebox{0.5}{\emph{$\bar{g}$}}}$
now completes the proof. ∎

##  3\. Global Obstructions to Distinguished Riemannian metrics

###  3.1. Introduction

We now switch our focus from the local to the global, and search for
obstructions to distinguished Riemannian metrics in the compact setting. The
obstructions we seek are to metrics that are deformations of constant
curvature metrics. Recall that a Riemannian metric $g$ has _constant _(_
sectional _)_ curvature_ if its Riemann curvature 4-tensor Rm satisfies

$\displaystyle\text{Rm}=\frac{1}{2}\lambda
g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}g$
(55)

for some $\lambda\in\mathbb{R}$. When $\lambda>0$, the Riemannian metric $g$
is called a _spherical space form_ ; such metrics have been classified in
[Wol72], and include among them the standard round spheres
$(\mathbb{S}^{n},\mathring{g})$, as well as the class of _Lens spaces_
$L(p,q)$ in dimension 3. However, there are _no_ such examples in the
Lorentzian world. Indeed, by [CM62] and [Kli96], there are no Lorentzian
spherical space forms when $M$ is compact, a foundational result in Lorentzian
geometry (see [Lun15] for a comprehensive account). They only exist in the
noncompact setting, the canonical one being _de Sitter spacetime_
$(\mathbb{S}_{1}^{n},g_{\scalebox{0.4}{dS}})$ ($n\geq 3$), which is the warped
product $(\mathbb{R}\times\mathbb{S}^{n-1},-(dt)^{2}+f(t)\mathring{g})$, where
$\mathring{g}$ is the standard Riemannian metric above and

$f(t)=r^{2}\cosh^{2}\left(\frac{t}{r}\right),$

with $r$ the radius of $\mathbb{S}^{n-1}$; consult, e.g., [Har17] and [BEE96,
p. 183ff.]. This is a complete Lorentzian manifold with constant positive
curvature ($\lambda=1/r^{2}$). The starting point of our analysis is the
observation that the vector field

$T\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!}{t}=-\partial_{t}$

is a special case of a unit length _closed_ vector field. As a consequence,
the Riemannian metric (3) formed from de Sitter spacetime by “removing the
minus sign,”

$g\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=(dt)^{2}+f(t)\mathring{g},$

is also complete (see [O’N83, Lemma 40, p. 209]) but more importantly, it has
Riemann curvature 4-tensor Rm related to that of
$g_{\scalebox{0.4}{\emph{L}}}$ in a simple and elegant way:

$\displaystyle\text{Rm}=\text{Rm}_{\scalebox{0.4}{\emph{L}}}-\nabla
T^{\flat}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}\nabla
T^{\flat}.$ (64)

We will prove, and generalize, (64) in Proposition 4 below. For now we point
out that, since de Sitter spacetime has constant curvature,
$\text{Rm}_{\scalebox{0.4}{\emph{L}}}=\frac{1}{2}g_{\scalebox{0.4}{\emph{L}}}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}g_{\scalebox{0.4}{\emph{L}}}$
(with $\lambda=1$), and since $g=g_{\scalebox{0.4}{dS}}+2(dt)^{2}$, Rm can in
fact be written directly in terms of $g$, as follows:

$\displaystyle\text{Rm}=\frac{1}{2}g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}g-2g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}(T^{\flat}\otimes
T^{\flat})-\nabla
T^{\flat}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}\nabla
T^{\flat}.$ (89)

This makes clear precisely how $g$ deviates from the constant curvature case
(55), as we describe in more detail in Section 3.4 below. In Theorem 2, we
will generalize this result by first generalizing (64), in Proposition 4.
First, let us recall some basic properties of closed vector fields.

###  3.2. Closed vector fields

Recall that a vector field $T$ on a Riemannian manifold $(M,g)$ is _closed_ if
its metric 1-form $T^{\flat}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=g(T,\cdot)$
is closed: $dT^{\flat}=0$. If $M$ is simply connected (and noncompact), then
such a $T$ is necessarily a gradient and $\nabla T^{\flat}$ is its Hessian
(with $\nabla$ the Levi-Civita connection of $g$); generally speaking, $T$ is
always at least locally a gradient. But the crucial property of such vector
fields, for our purposes, is the following:

######  Lemma 2.

Let $T$ be a unit length vector field on a Riemannian manifold $(M,g)$. Then
$T$ is closed if and only if the endomorphism

$\mathcal{D}\colon TM\longrightarrow TM\hskip 14.45377pt,\hskip
14.45377ptV\mapsto\nabla_{\\!V}{T}$

is self-adjoint with respect to $g$. Equivalently, $T$ is closed if and only
if the 2-tensor $\nabla T^{\flat}$ is symmetric.

###### Proof.

Suppose that $T$ is closed. Then it must have geodesic flow,
$\nabla_{\\!T}{T}=0$, because

$\displaystyle\underbrace{\,dT^{\flat}(T,V)\,}_{0}=2g(\nabla_{\\!T}{T},V)\hskip
14.45377pt\text{for all $V\in\mathfrak{X}(M)$}.$ (90)

Furthermore, its orthogonal complement $T^{\perp}\subseteq TM$ will be
integrable, because

$\displaystyle\underbrace{\,dT^{\flat}(X_{i},X_{j})\,}_{0}=-g(T,[X_{i},X_{j}])\hskip
14.45377pt\text{for all $X_{i},X_{j}\in\Gamma(T^{\perp})$}.$ (91)

Then, with respect to any $g$-orthonormal basis of the form
$\\{T,X_{1},\dots,X_{n-1}\\}$, observe that

$\displaystyle g(\nabla_{\\!T}{T},X_{i})\\!\\!\\!\\!\\!$
$\displaystyle\overset{\eqref{eqn:prop0}}{=}$
$\displaystyle\\!\\!\\!\\!\\!0\,=\,g(\nabla_{\\!X_{i}}{T},T),$ $\displaystyle
g(\nabla_{\\!X_{i}}{T},X_{j})\\!\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!-g(T,\nabla_{\\!X_{i}}{X_{j}})\\!\overset{\eqref{eqn:prop1}}{=}\\!-g(T,\nabla_{\\!X_{j}}{X_{i}})=g(\nabla_{\\!X_{j}}{T},X_{i}).$

It follows that $g(\nabla_{\\!V}{T},W)=g(\nabla_{\\!W}{T},V)$ for all
$V,W\in\mathfrak{X}(M)$, so that $\mathcal{D}$ is indeed self-adjoint. The
converse is also clear from these two equations. Finally, suppose that $T$ is
closed and consider the 2-tensor $\nabla T^{\flat}$; then

$\nabla
T^{\flat}(X_{i},X_{j})=-g(T,\nabla_{\\!X_{i}}{X_{j}})\overset{\eqref{eqn:prop1}}{=}-g(T,\nabla_{\\!X_{j}}{X_{i}})=\nabla
T^{\flat}(X_{j},X_{i}).$

Similarly, by (90) it will be the case that $\nabla T^{\flat}(T,X_{i})=\nabla
T^{\flat}(X_{i},T)=0$. Thus $\nabla T^{\flat}$ is symmetric; the converse is
also clear from these equations. ∎

Henceforth we will assume that $T$ is a unit length closed vector field. As
the endomorphism $\mathcal{D}$ will be self-adjoint, let
$\\{T,X_{2},\dots,X_{n}\\}$ be a local $g$-orthonormal basis of eigenvectors
of $\mathcal{D}$,

$\displaystyle\mathcal{D}(T)=\nabla_{\\!T}{T}=0\hskip 14.45377pt,\hskip
14.45377pt\mathcal{D}(X_{i})=\nabla_{\\!X_{i}}{T}=\lambda_{i}X_{i},$ (92)

for some $\lambda_{2},\dots,\lambda_{n}\in\mathbb{R}$. Now consider the
Lorentzian metric $g_{\scalebox{0.4}{\emph{L}}}$ defined by

$g_{\scalebox{0.4}{\emph{L}}}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=g-2T^{\flat}\otimes
T^{\flat}.$

Observe that $\\{T,X_{2},\dots,X_{n}\\}$ is simultaneously a
$g_{\scalebox{0.4}{\emph{L}}}$-orthonormal basis, except that $T$ is now unit
“timelike”: $g_{\scalebox{0.4}{\emph{L}}}(T,T)=-1$. The Levi-Civita connection
$\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!}{}$ of $g_{\scalebox{0.4}{\emph{L}}}$
is related to that of $g$ as follows:

######  Lemma 3.

Let $\\{T,X_{2},\dots,X_{n}\\}$ be a local $g$-orthonormal basis of
eigenvectors of $\mathcal{D}$ with eigenvalues (92). Then the Levi-Civita
connection $\nabla^{\scalebox{0.4}{L}}$ of the Lorentzian metric
$g_{\scalebox{0.4}{L}}=g-2T^{\flat}\otimes T^{\flat}$ satisfies

$\displaystyle\left\\{\begin{array}[]{ccc}\nabla^{\scalebox{0.4}{L}}_{\\!T}{T}\\!\\!\\!&=&-\nabla_{\\!T}{T}=0,\\\
\nabla^{\scalebox{0.4}{L}}_{\\!X_{i}}{T}\\!\\!\\!&=&\nabla_{\\!X_{i}}{T}=\lambda_{i}X_{i},\\\
\nabla^{\scalebox{0.4}{L}}_{\\!X_{i}}{X_{j}}\\!\\!\\!&=&\\!\\!2\lambda_{i}\delta_{ij}T+\nabla_{\\!X_{i}}{X_{j}}.\end{array}\right.$
(96)

###### Proof.

These all follow by the Koszul formula, together with (92). ∎

Lemmas 2 and 3 combine to yield an elegant relationship between the curvature
4-tensors of $g$ and $g_{\scalebox{0.4}{\emph{L}}}$; before stating it, recall
the _Kulkarni-Nomizu product_ : for any pair of symmetric 2-tensors $P,Q$, the
4-tensor
$P\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}Q$
defined by

$\displaystyle(P\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}Q)(v,w,x,y)\\!\\!$
$\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=$
$\displaystyle\\!\\!P(v,y)Q(w,x)+P(w,x)Q(v,y)$ $\displaystyle\hskip
50.58878pt-P(v,x)Q(w,y)-P(w,y)Q(v,x)$

is an algebraic curvature tensor (i.e., possessing the same symmetries as the
Riemann curvature 4-tensor; note, e.g., that
$P\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}Q=Q\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}P$
is one of these symmetries).

######  Proposition 4.

Let $(M,g)$ be a Riemannian $n$-manifold equipped with a unit length closed
vector field $T$. Define the Lorentzian metric

$g_{\scalebox{0.4}{L}}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=g-2T^{\flat}\otimes
T^{\flat}.$

Then $g_{\scalebox{0.4}{L}}$ has Riemann curvature 4-tensor
$\emph{\text{Rm}}_{\scalebox{0.4}{L}}$ given by

$\emph{\text{Rm}}_{\scalebox{0.4}{L}}=\emph{\text{Rm}}+\nabla
T^{\flat}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}\nabla
T^{\flat},$

where $\nabla$ and Rm are, respectively, the Levi-Civita connection and
Riemann curvature 4-tensor of $g$.

###### Proof.

For Rm and $\text{Rm}_{\scalebox{0.4}{\emph{L}}}$, we adopt the sign
convention

$\text{Rm}(v,w,x,y)=g(\nabla_{\\!v}{\\!\nabla_{\\!w}{\,x}},y)-g(\nabla_{\\!w}{\nabla_{\\!v}{\,x}},y)-g(\nabla_{\\![v,w]}{x},d).$

We now compute and compare all the components of Rm and
$\text{Rm}_{\scalebox{0.4}{\emph{L}}}$ with respect to the $g$\- and
$g_{\scalebox{0.4}{\emph{L}}}$-orthonormal basis (92):

  1. 1.

We start with the components
$\text{Rm}_{\scalebox{0.4}{\emph{L}}}(X_{i},T,T,X_{j})$:

$\displaystyle\text{Rm}_{\scalebox{0.4}{\emph{L}}}(X_{i},T,T,X_{j})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{i}}{\cancelto{0}{\\!\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!T}{T}}},X_{j})-\\!\\!\\!\\!\\!\underbrace{\,g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!T}{\overbrace{\,\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{i}}{T}\,}^{\lambda_{i}X_{i}}},X_{j})\,}_{\text{$T(\lambda_{i})\delta_{ij}+\lambda_{i}g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!T}{X_{i}},\,X_{j})$}}\\!\\!\\!\\!-\underbrace{\,g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\![X_{i},T]}{T},X_{j})\,}_{\text{$g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{j}}{\\!T},[X_{i},T])$}}$
(106) $\displaystyle=$
$\displaystyle-T(\lambda_{i})\delta_{ij}-\lambda_{i}g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!T}{X_{i}},X_{j})-\lambda_{j}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\underbrace{\,g_{\scalebox{0.4}{\emph{L}}}(X_{j},[X_{i},T])\,}_{\text{$g_{\scalebox{0.4}{\emph{L}}}(X_{j},\\!\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{i}}{T})-g_{\scalebox{0.4}{\emph{L}}}(X_{j},\\!\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!T}{X_{i}})$}}$
$\displaystyle=$
$\displaystyle-\big{(}T(\lambda_{i})+\lambda_{j}\lambda_{i}\big{)}\delta_{ij}+\underbrace{\,g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!T}{X_{i}},X_{j})\,}_{g(\nabla_{\\!T}{X_{i}},X_{j})}(\lambda_{j}-\lambda_{i})$
$\displaystyle=$ $\displaystyle\text{Rm}(X_{i},T,T,X_{j}),$ (107)

where the last equality is due to the fact that the same answer would have
been reached starting with

$\text{Rm}(X_{i},T,T,X_{j})=g(\nabla_{\\!X_{i}}{\\!\nabla_{\\!T}{T}},X_{j})-g(\nabla_{\\!T}{\nabla_{\\!X_{i}}{T}},X_{j})-g(\nabla_{\\![X_{i},T]}{T},X_{j}).$

  2. 2.

Next, the components
$\text{Rm}_{\scalebox{0.4}{\emph{L}}}(X_{i},X_{k},T,X_{j})$, with $k\neq i$:

$\displaystyle\text{Rm}_{\scalebox{0.4}{\emph{L}}}(X_{i},X_{k},T,X_{j})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\underbrace{\,g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{i}}{{\overbrace{\,\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{k}}{T}\,}^{\lambda_{k}X_{k}}}},X_{j})\,}_{g(\nabla_{\\!X_{i}}{\\!\nabla_{\\!X_{k}}{T}},\,X_{j})}-\underbrace{\,g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{k}}{\overbrace{\,\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{i}}{T}\,}^{\lambda_{i}X_{i}}},X_{j})\,}_{g(\nabla_{\\!X_{k}}{\\!\nabla_{\\!X_{i}}{T}},\,X_{j})}$
(108) $\displaystyle\hskip
126.47249pt-\underbrace{\,g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\![X_{i},X_{k}]}{T},X_{j})\,}_{\text{$g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{j}}{\\!T},\,[X_{i},X_{k}])$}}$
$\displaystyle=$
$\displaystyle\\!\\!g(\nabla_{\\!X_{i}}{\\!\nabla_{\\!X_{k}}{T}},X_{j})-g(\nabla_{\\!X_{k}}{\\!\nabla_{\\!X_{i}}{T}},X_{j})-\underbrace{\,g(\nabla_{\\!X_{j}}{T},[X_{i},X_{k}])\,}_{g(\nabla_{\\![X_{i},X_{k}]}{T},\,X_{j})}$
$\displaystyle=$ $\displaystyle\\!\\!\text{Rm}(X_{i},X_{k},T,X_{j}),$

where in the first and second equalities we’ve used the fact that
$g_{\scalebox{0.4}{\emph{L}}}(\cdot,X)=g(\cdot,X)$ for any $X\in T^{\perp}$,
which includes $[X_{i},X_{k}]$ by integrability of $T^{\perp}$.

  3. 3.

Finally, the components
$\text{Rm}_{\scalebox{0.4}{\emph{L}}}(X_{k},X_{i},X_{j},X_{l})$, with $k\neq
i$, $l\neq j$:

$\displaystyle\text{Rm}_{\scalebox{0.4}{\emph{L}}}(X_{k},X_{i},X_{j},X_{l})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\underbrace{\,g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{k}}{\\!\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{i}}{X_{j}}},X_{l})\,}_{(a)}-\underbrace{\,g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{i}}{\\!\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{k}}{X_{j}}},X_{l})\,}_{(b)}$
$\displaystyle\hskip
126.47249pt-\underbrace{\,g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\![X_{k},X_{i}]}{X_{j}},X_{l})\,}_{(c)},$

will be computed term-by-term, using the Koszul formula, beginning with $(a)$:

$\displaystyle
g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{k}}{\\!\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{i}}{X_{j}}},X_{l})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{k}}{\underbrace{\nabla_{\\!X_{i}}{X_{j}}}_{``Z\,"}},X_{l})+2\lambda_{i}\delta_{ij}\,g_{\scalebox{0.4}{\emph{L}}}(\underbrace{\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{k}}{T}}_{\lambda_{k}X_{k}},X_{l})$
$\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2}\Big{[}X_{k}(g_{\scalebox{0.4}{\emph{L}}}(Z,X_{l}))+Z(g_{\scalebox{0.4}{\emph{L}}}(X_{l},X_{k}))-X_{l}(g_{\scalebox{0.4}{\emph{L}}}(X_{k},Z))$
$\displaystyle\hskip 14.45377pt-
g_{\scalebox{0.4}{\emph{L}}}(Z,[X_{k},X_{l}])-g_{\scalebox{0.4}{\emph{L}}}(X_{l},[Z,X_{k}])+g_{\scalebox{0.4}{\emph{L}}}(X_{k},[X_{l},Z])\Big{]}$
$\displaystyle\hskip
144.54pt+\,2\lambda_{i}\lambda_{k}\delta_{ij}\delta_{kl}.$

Every “$g_{\scalebox{0.4}{\emph{L}}}(\cdot,\cdot)$” term here contains at
least one $X\in T^{\perp}$, so that it equals $g(\cdot,\cdot)$. We thus have
that

$\displaystyle
g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{k}}{\\!\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{i}}{X_{j}}},X_{l})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2}\Big{[}X_{k}(g(Z,X_{l}))+Z(g(X_{l},X_{k}))-X_{l}(g(X_{k},Z))$
(109) $\displaystyle\hskip
14.45377pt-g(Z,[X_{k},X_{l}])-g(X_{l},[Z,X_{k}])+g(X_{k},[X_{l},Z])\Big{]}$
$\displaystyle\hskip 144.54pt+\,2\lambda_{i}\lambda_{k}\delta_{ij}\delta_{kl}$
$\displaystyle=$
$\displaystyle\\!\\!g(\nabla_{\\!X_{k}}{Z},X_{l})+2\lambda_{i}\lambda_{k}\delta_{ij}\delta_{kl}$
$\displaystyle=$
$\displaystyle\\!\\!g(\nabla_{\\!X_{k}}{\\!\nabla_{\\!X_{i}}{X_{j}}},X_{l})+2\lambda_{i}\lambda_{k}\delta_{ij}\delta_{kl}.$

Similarly with $(b)$,

$\displaystyle
g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{i}}{\\!\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{k}}{X_{j}}},X_{l})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{i}}{\underbrace{\nabla_{\\!X_{k}}{X_{j}}}_{``Z\,"}},X_{l})+2\lambda_{k}\delta_{kj}\,g_{\scalebox{0.4}{\emph{L}}}(\underbrace{\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!X_{i}}{T}}_{\lambda_{i}X_{i}},X_{l})$
(110) $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2}\Big{[}X_{i}(g_{\scalebox{0.4}{\emph{L}}}(Z,X_{l}))+Z(g_{\scalebox{0.4}{\emph{L}}}(X_{l},X_{i}))-X_{l}(g_{\scalebox{0.4}{\emph{L}}}(X_{i},Z))$
$\displaystyle\hskip 14.45377pt-
g_{\scalebox{0.4}{\emph{L}}}(Z,[X_{i},X_{l}])-g_{\scalebox{0.4}{\emph{L}}}(X_{l},[Z,X_{i}])+g_{\scalebox{0.4}{\emph{L}}}(X_{i},[X_{l},Z])\Big{]}$
$\displaystyle\hskip 144.54pt+\,2\lambda_{k}\lambda_{i}\delta_{kj}\delta_{il}$
$\displaystyle=$
$\displaystyle\\!\\!g(\nabla_{\\!X_{i}}{\\!\nabla_{\\!X_{k}}{X_{j}}},X_{l})+2\lambda_{i}\lambda_{k}\delta_{kj}\delta_{il}.$

For $(c)$, observe that because $[X_{k},X_{i}]\in T^{\perp}$, it is of the
form $[X_{k},X_{i}]=\sum_{r=1}^{n-1}a_{r}X_{r}$ with the $a_{r}$’s smooth
locally defined functions, which in turn implies that
$\nabla^{\scalebox{0.4}{\emph{L}}}_{\\![X_{k},X_{i}]}{X_{j}}=\nabla_{\\![X_{k},X_{i}]}{X_{j}}+2a_{j}\lambda_{j}T$.
Hence

$\displaystyle
g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\![X_{k},X_{i}]}{X_{j}},X_{l})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!g(\nabla_{\\![X_{k},X_{i}]}{X_{j}},X_{l}).$ (111)

Taken together, (109), (110), and (111) yield

$\displaystyle\text{Rm}_{\scalebox{0.4}{\emph{L}}}(X_{k},X_{i},X_{j},X_{l})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\text{Rm}(X_{k},X_{i},X_{j},X_{l})+2\lambda_{i}\lambda_{k}(\delta_{ij}\delta_{kl}-\delta_{kj}\delta_{il})$
(120) $\displaystyle=$
$\displaystyle\\!\\!\text{Rm}(X_{k},X_{i},X_{j},X_{l})+(\nabla
T^{\flat}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}\nabla
T^{\flat})(X_{k},X_{i},X_{j},X_{l}).$

Given (107), (108), and (120), and the fact that $(\nabla
T^{\flat}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}\nabla
T^{\flat})(T,\cdot,\cdot,\cdot)=0$, the proof is complete. ∎

Note that the components (107) and (108) can also be found in [Ole14];
however, the most important curvature component in our Proposition 4, namely
(120), has not (to the best of our knowledge) appeared explicitly in the
literature. Be that as it may, here are two examples illustrating Proposition
4, both in the noncompact setting, beginning with the canonical one:

Example 1. For de Sitter spacetime
$(\mathbb{R}\times\mathbb{S}^{n-1},-dt^{2}+f(t)\mathring{g})$,

$T\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!}{t}=-\partial_{t}$

is a unit length closed vector field. It will also be so for the corresponding
Riemannian metric $g\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=g_{\scalebox{0.4}{\emph{L}}}+2T^{\flat_{\scalebox{0.4}{\emph{L}}}}\otimes
T^{\flat_{\scalebox{0.4}{\emph{L}}}}=dt^{2}+f(t)\mathring{g}.$

Example 2. On $\mathbb{R}^{3}=\\{(x^{1},x^{2},x^{3})\\}$, consider a
Lorentzian metric $g_{\scalebox{0.4}{\emph{L}}}$ whose components
$({g_{\scalebox{0.4}{\emph{L}}}}_{ij})$ with respect to the coordinate basis
$\\{\partial_{1},\partial_{2},\partial_{3}\\}$ take the form

$({g_{\scalebox{0.4}{\emph{L}}}}_{ij})=\begin{pmatrix}0&1&0\\\ 1&H(x^{1})&0\\\
0&0&H(x^{1})/2\end{pmatrix},$

where $H(x^{1})$ is a smooth function, to be determined below, such that
$g_{\scalebox{0.4}{\emph{L}}}$ will have constant positive curvature on an
open subset of $\mathbb{R}^{3}$. In dimension 3, this is equivalent to being
an Einstein metric with positive Einstein constant, which we can take to be
$1$: $\text{Ric}_{\scalebox{0.4}{\emph{L}}}=g_{\scalebox{0.4}{\emph{L}}}.$
Though we forego the computations here, it is straightforward to show that
this will hold if and only if $H(x^{1})=(x^{1}+a)^{2}/2,a\in\mathbb{R}.$ For
example, if we take $a=2$, then $g_{\scalebox{0.4}{\emph{L}}}$ will be a
Lorentzian metric io the open subset $\\{(x^{1},x^{2},x^{3}):x^{1}>-2\\}$.
Finally, let $f(x^{1})$ be a smooth function and consider its
$g_{\scalebox{0.4}{\emph{L}}}$-gradient:

$\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!}{f}=-\frac{f^{\prime}}{2}(x^{1}+2)^{2}\,\partial_{1}+f^{\prime}\,\partial_{2}.$

Another computation shows that
$g_{\scalebox{0.4}{\emph{L}}}(\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!}{f},\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!}{f})=-1$
if and only if

$(f^{\prime})^{2}=\frac{2}{(x^{1}+2)^{2}}\cdot$

Taking the smooth solution
$f(x^{1})=\sqrt{2}\,\text{ln}\big{(}\frac{x^{1}+2}{2}\big{)}$, we thus have
that the pair
$(g_{\scalebox{0.4}{\emph{L}}},\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!}{f})$,
when restricted to $\\{(x^{1},x^{2},x^{3}):x^{1}>-2\\}$, yields a Lorentzian
manifold with constant positive curvature and a closed, unit timelike vector
field. The corresponding Riemannian metric $g$ is then $g\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=g_{\scalebox{0.4}{\emph{L}}}+2df\otimes
df.$ With respect to $g$, $\nabla^{\scalebox{0.4}{\emph{L}}}_{\\!}{f}$ will
also be a unit length closed vector field.

###  3.3. Obstructions to Riemannian metrics in the compact setting

We are now ready to prove our main Theorem of this section:

######  Theorem 2.

Let $(M,g)$ be a Riemannian $n$-manifold $(n\geq 3)$ and $T$ a unit length
closed vector field on $M$. For $\lambda\in\mathbb{R}$, the curvature 4-tensor
of $g$ takes the form

$\displaystyle\emph{\text{Rm}}=\frac{1}{2}\lambda
g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}g-2\lambda
g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}(T^{\flat}\otimes
T^{\flat})-\nabla
T^{\flat}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}\nabla
T^{\flat}$ (146)

if and only if the Lorentzian metric $g_{\scalebox{0.4}{L}}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=g-2T^{\flat}\otimes
T^{\flat}$ has constant curvature $\lambda$. If $M$ is compact, then
$\lambda=0$ and $T$ is parallel _;_ hence $g$ is flat, and its universal cover
splits isometrically as a product $\mathbb{R}\times N$.

###### Proof.

Suppose a pair $(g,T)$ satisfying (146) exists on a compact manifold $M$ of
dimension $\geq 3$, and such that $g_{\scalebox{0.4}{\emph{L}}}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=g-2T^{\flat}\otimes
T^{\flat}$ has constant curvature $\lambda$. We first consider the case
$\lambda\leq 0$; it turns out that the obstruction in this case occurs at the
level of the Ricci tensor, as follows. Set

$g_{\scalebox{0.4}{\emph{L}}}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=g-2T^{\flat}\otimes
T^{\flat}.$

Then

$\displaystyle\text{Ric}(T,T)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\text{Rm}(X_{i},T,T,X_{i})$ (147)
$\displaystyle\overset{\eqref{eqn:Ric1}}{=}$
$\displaystyle\sum_{i=1}^{n-1}\text{Rm}_{\scalebox{0.4}{\emph{L}}}(X_{i},T,T,X_{i})=\text{Ric}_{\scalebox{0.4}{\emph{L}}}(T,T),$

so that if $(M,g_{\scalebox{0.4}{\emph{L}}})$ has constant curvature
$\lambda\leq 0$ — more generally, if $(M,g_{\scalebox{0.4}{\emph{L}}})$ is
Einstein with nonpositive Einstein constant — then

$\displaystyle\text{Ric}(T,T)=\text{Ric}_{\scalebox{0.4}{\emph{L}}}(T,T)=(n-1)\lambda\underbrace{\,g_{\scalebox{0.4}{\emph{L}}}(T,T)\,}_{-1}\geq
0.$ (148)

Next, setting $i=j$ in (106) and summing over $i=1,\dots,n-1$ yields the
following Bochner-type equation,

$\displaystyle
T(\text{div}\,T)=-\text{Ric}(T,T)-\sum_{i=1}^{n-1}\lambda_{i}^{2},$ (149)

where we’ve used the fact that

$\text{div}\,T=\sum_{i=1}^{n-1}\lambda_{i}=\text{tr}_{g}\nabla T^{\flat}.$

Now, via the Schwarz inequality

$\sum_{i=1}^{n-1}\lambda_{i}^{2}\geq\frac{1}{n-1}(\lambda_{1}+\cdots+\lambda_{n-1})^{2},$

(149) reduces to

$\displaystyle T(\text{div}\,T)\ \leq\
-\text{Ric}(T,T)-\frac{(\text{div}\,T)^{2}}{n-1},$ (150)

which permits, in turn, the following well known Riccati analysis: since $T$
is complete ($M$ being compact),

$\text{Ric}(T,T)\geq 0\overset{\eqref{eqn:Bochner3}}{\hskip
14.45377pt\Rightarrow\hskip
14.45377pt}\text{div}\,T=0\overset{\eqref{eqn:Bochner1}}{\hskip
14.45377pt\Rightarrow\hskip 14.45377pt}\text{Ric}(T,T)=\lambda_{i}=0.$

It follows that that the case $\text{Ric}(T,T)>0$ cannot occur, and if
$\text{Ric}(T,T)=0$, then $T$ must be parallel. The former implies that
$\lambda<0$ cannot occur; the latter, that if $\lambda=0$, then $\nabla
T^{\flat}$ and (146) both vanish, in which case $(M,g)$ is flat. Furthermore,
its universal covering splits isometrically as a product $\mathbb{R}\times N$,
by the de Rham Decomposition Theorem (see [Pet16, p. 384]). This settles the
case $\lambda\leq 0$. For the case $\lambda>0$, we will employ a different
strategy; indeed, since $\text{Ric}(T,T)<0$ when $\lambda>0$ (via (148)), the
Riccati analysis we applied to (150) is unavailable here. Instead, we
substitute $g=g_{\scalebox{0.4}{\emph{L}}}+2T^{\flat}\otimes T^{\flat}$ into
Proposition 4 (note that $T^{\flat_{\scalebox{0.4}{\emph{L}}}}\otimes
T^{\flat_{\scalebox{0.4}{\emph{L}}}}=T^{\flat}\otimes T^{\flat}$), to obtain

$\displaystyle\text{Rm}_{\scalebox{0.4}{\emph{L}}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\text{Rm}+\nabla
T^{\flat}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}\nabla
T^{\flat}$ (159) $\displaystyle\overset{\eqref{eqn:1}}{=}$
$\displaystyle\\!\\!\frac{1}{2}\lambda
g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}g-2\lambda
g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}(T^{\flat}\otimes
T^{\flat})$ (176) $\displaystyle=$ $\displaystyle\\!\\!\frac{1}{2}\lambda
g_{\scalebox{0.4}{\emph{L}}}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}g_{\scalebox{0.4}{\emph{L}}}+2\lambda
g_{\scalebox{0.4}{\emph{L}}}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}(T^{\flat}\otimes
T^{\flat})-2\lambda
g_{\scalebox{0.4}{\emph{L}}}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}(T^{\flat}\otimes
T^{\flat})$ (201) $\displaystyle=$ $\displaystyle\\!\\!\frac{1}{2}\lambda
g_{\scalebox{0.4}{\emph{L}}}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}g_{\scalebox{0.4}{\emph{L}}},$
(210)

where we’ve used the fact that $(T^{\flat}\otimes
T^{\flat})\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}(T^{\flat}\otimes
T^{\flat})=0$. But as mentioned in the Introduction, such a (compact)
Lorentzian manifold is impossible when $\lambda>0$, by [CM62] and [Kli96]. ∎

###  3.4. Concluding remarks

We close with four remarks on Theorem 2.

  1. i.

In the proof of Theorem 2, it was only the case $\lambda>0$ that was truly
nontrivial, relying in an essential way on the nonexistence of compact
Lorentzian spherical space forms [CM62, Kli96]. The case $\lambda\leq 0$, on
the other hand, relied instead on the well known Bochner technique (149).

  2. ii.

Observe that compact Lorentzian manifolds with constant _negative_ curvature
certainly do exist; examples can be found, e.g., in [KR85] and [Gol85]. So do
flat ones: any $(\mathbb{S}^{1}\\!\times\\!N,-dt^{2}\oplus h)$, with $(N,h)$ a
compact flat Riemannian manifold, yields an example, with
$\nabla^{\scalebox{0.4}{\emph{L}}}t$ serving the role of $T$ above.

  3. iii.

The sectional curvature of (146) has the following behavior: The term
$g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}(T^{\flat}\otimes
T^{\flat})$ vanishes on the frame $\\{T,X_{1},\dots,X_{n-1}\\}$ except on the
components

$g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}(T^{\flat}\otimes
T^{\flat})(X_{i},T,T,X_{i})=1,$

so that the sectional curvature of any 2-plane containing $T$ is

$\lambda-2\lambda=-\lambda,$

as opposed to $\lambda$, which would have been the case with constant
curvature $\text{Rm}=\frac{1}{2}\lambda
g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}g$.
On the other hand, the term $\nabla
T^{\flat}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}\nabla
T^{\flat}$ vanishes except on the components

$(\nabla
T^{\flat}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}\nabla
T^{\flat})(X_{i},X_{j},X_{j},X_{i})=2\lambda_{i}\lambda_{j}\hskip
14.45377pt,\hskip 14.45377pti\neq j,$

so that 2-planes spanned by $\\{X_{i},X_{j}\\}$ now have sectional curvature

$\lambda-2\lambda_{i}\lambda_{j}.$

It is in this sense that $T$ “breaks” the constant curvature of the term
$\text{Rm}=\frac{1}{2}\lambda
g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}g$
in (146).

  4. iv.

Finally, observe that Theorem 2 is uninteresting in dimension 2. Indeed,
although any Riemannian 2-manifold satisfies
$\text{Rm}=\frac{1}{2}Kg\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}g$
with $K$ the Gaussian curvature (see, e.g., [Lee18, p. 250]), and although $K$
is neither constant nor signed in general, nevertheless $K=-\lambda$ if (146)
were to be satisfied, because in dimension 2

$\nabla
T^{\flat}\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}\nabla
T^{\flat}=0\hskip 14.45377pt\text{and}\hskip
14.45377ptg\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}(T^{\flat}\otimes
T^{\flat})=\frac{1}{2}g\raisebox{0.39993pt}{\,${\scriptstyle\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}}$\,}g.$

But by the Gauss-Bonnet Theorem, if the Euler characteristic is zero (i.e.,
the manifold is a 2-torus or a Klein bottle) — recall that this must be the
case if the compact 2-manifold supports a nowhere vanishing vector field like
$T$ — then the only compact Riemannian 2-manifold with constant curvature is
the flat one.

## Appendix: the geometry of plane waves and Penrose’s limit

For the convenience of the reader, in this Appendix we provide a self-
contained introduction to the plane wave metrics $h_{\scalebox{0.4}{PW}}$ we
encountered in (36); this material already exists in the literature (e.g.,
[CBS13, LS16]), though our presentation differs slightly from these. To begin
with, recall that a vector field $N$ on a Lorentzian manifold $(M,g)$ is
_null_ (or _lightlike_) if $N\neq 0$ and $g(N,N)=0$. Here now is the
coordinate-independent definition of a plane wave metric in Lorentzian
geometry:

######  Definition 2 ([LS16]).

Let $(M,g)$ be a Lorentzian manifold endowed with a parallel, null vector
field $N$. Let $R$ denote its curvature endomorphism and $\nabla$ its Levi-
Civita connection. Then $(M,g)$ is a _pp -wave_ if

$\displaystyle R(X,Y)V=0\hskip 7.22743pt\text{for all}\hskip
7.22743ptX,Y\in\Gamma(N^{\perp})\hskip 7.22743pt\text{and}\hskip
7.22743ptV\in\mathfrak{X}(M).$ (211)

If, in addition to this, $R$ also satisfies

$\displaystyle\nabla_{\\!X}R=0\hskip 7.22743pt\text{for all}\hskip
7.22743ptX\in\Gamma(N^{\perp}),$ (212)

then $(M,g)$ is a _plane wave_.

What is not clear at the moment is why such metrics will necessarily take the
form (36). Showing this involves a number of steps, which we now detail.
First, suppose $(M,g)$ is $(n+1)$-dimensional and choose local coordinates
$(x^{0},\dots,x^{n})$ with respect to which $N=\partial_{0}.$ Because $N$ is
also parallel, we can, by a further change to a “null coordinate chart” (which
we’ll continue to denote by $(x^{0},\dots,x^{n})$), arrange it so that
$N=\partial_{0}=\text{grad}\,x^{1},$ so that $g$ takes the form that we saw in
(29):

$\displaystyle(g_{ij})=\begin{pmatrix}0&1&0&0&\cdots&0\\\
1&g_{11}&g_{12}&g_{13}&\cdots&g_{1n}\\\
0&g_{21}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{22}}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{23}}&\cdots&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{2n}}\\\
0&g_{31}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{32}}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{33}}&\cdots&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{3n}}\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&g_{n1}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{n2}}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{n3}}&\cdots&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g_{nn}}\end{pmatrix}\cdot$
(213)

Consider now the red submatrix $(g_{ij})_{i,j=2,\dots,n}$; it is positive-
definite because the coordinate vectors $\partial_{2},\dots,\partial_{n}$,
being orthogonal to the null vector field $\partial_{0}$ and linearly
independent to it, must span a positive-definite subspace ([O’N83, Lemma 28,
p. 142]). In other words, each codimension 2 embedded submanifold

$\displaystyle\Lambda_{\scalebox{0.5}{\emph{b,c}}}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\left\\{\big{(}b,c,x^{2},\dots,x^{n}\big{)}\right\\},$
(214)

with induced metric

$\displaystyle g_{\scalebox{0.4}{\emph{b,c}}}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\sum_{i=2}^{n}g_{ij}(b,c,x^{2},\dots,x^{n})dx^{i}\otimes
dx^{j},$ (215)

is a Riemannian manifold. Furthermore, notice the following fact, a basic one
from linear algebra:

$\text{$\text{det}\,g=-\text{det}\,(\text{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}red
block}})$};$

i.e., the fact that $g$ in (213) is a Lorentzian metric in no way depends on
the components $g_{11},g_{12},\dots,g_{1n}$, in the following sense: Changing
any of these will change the metric, but it cannot change the nondegeneracy or
the Lorentzian signature of the metric. It is precisely this fact that Penrose
exploited when he took his plane wave limit (recall (32) and (36)), for these
are precisely the terms that vanished in the limit $\varepsilon\to 0$ — the
novelty of Penrose’s limit, therefore, is that, via the homothety (32), he
managed to do this while _preserving_ the geometry. But in fact there is much
more geometry here: Because $N=\partial_{0}$ is also parallel, the Riemann
curvature tensor $\text{Rm}_{\scalebox{0.5}{\emph{b,c}}}$ of
$g_{\scalebox{0.4}{\emph{b,c}}}$ is intimately related to that of the ambient
Lorentzian metric $g$:

######  Proposition 5.

The null vector field $\partial_{0}$ in (213) is parallel if and only if all
components of the metric (213) are independent of $x^{0}$. In such a case, the
Levi-Civita connection $\nabla^{\scalebox{0.4}{b,c}}$ of
$(\Lambda_{\scalebox{0.5}{b,c}},g_{\scalebox{0.4}{b,c}})$ in (214) is related
to the Levi-Civita connection $\nabla$ of $g$ as follows _:_ for any
$X,Y\in\mathfrak{X}(\Lambda_{\scalebox{0.5}{b,c}})$,

$\displaystyle\nabla_{\\!X}{Y}=\nabla^{\scalebox{0.4}{b,c}}_{\\!X}\,Y+\alpha\partial_{0},$
(216)

for some smooth function $\alpha$. As a result, their curvature tensors
satisfy

$\displaystyle\emph{\text{Rm}}_{\scalebox{0.5}{b,c}}=\emph{\text{Rm}}\hskip
14.45377pton\hskip 14.45377pt\mathfrak{X}(\Lambda_{\scalebox{0.5}{b,c}}).$
(217)

###### Proof.

If the null vector field $\partial_{0}=\text{grad}\,x^{1}$ is parallel, then

$\nabla_{\\!\partial_{0}}{\partial_{i}}=\nabla_{\\!\partial_{i}}{\partial_{0}}=0\hskip
14.45377pt,\hskip 14.45377pti=0,1,2,\dots,n,$

which immediately implies that each $\partial_{0}(g_{ij})=0$. Conversely, if
each $\partial_{0}(g_{ij})=0$, then each of the Christoffel symbols
$\Gamma_{i0}^{k}$ vanishes, hence each
$\nabla_{\\!\partial_{i}}{\partial_{0}}=0$. Next, consider each embedded
2-submanifold $\Lambda_{\scalebox{0.5}{\emph{b,c}}}$ given by (214), with its
corresponding induced Riemannian metric

$g_{\scalebox{0.4}{\emph{b,c}}}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\sum_{i,j=2}^{n}g_{ij}(c,x^{2},\dots,x^{n})dx^{i}\otimes
dx^{j}.$

(Since each $g_{ij}$ is independent of $x^{0}$, the components $g_{ij}$ are
$x^{0}$-independent.) By the Gauss Equation ([Lee18, Theorem 8.5, p. 230]), we
know that the components
$\text{Rm}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{i},\partial_{j},\partial_{k},\partial_{l})$
of the curvature tensor of $g_{\scalebox{0.4}{\emph{b,c}}}$ are related to
that of $g$ by

$\displaystyle\text{Rm}(\partial_{i},\partial_{j},\partial_{k},\partial_{l}\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\text{Rm}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{i},\partial_{j},\partial_{k},\partial_{l})$
$\displaystyle-\,g(\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{i},\partial_{l}),\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{j},\partial_{k}))+g(\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{i},\partial_{k}),\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{j},\partial_{l})),$

where $\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}$ is the second fundamental
form of $g_{\scalebox{0.4}{\emph{b,c}}}$. But when $\partial_{0}$ is parallel,
this equation simplifies considerably, because in such a case any vector field
$\nabla_{\\!\partial_{i}}{\partial_{j}}$, for $i,j=2,\dots,n$, has no
$\partial_{1}$-component; i.e.,

$\displaystyle\nabla_{\\!\partial_{i}}{\partial_{j}}=\alpha_{ij}\partial_{0}+\sum_{k=2}^{n}\beta_{ij}^{k}\partial_{k},$
(219)

for some smooth functions $\alpha_{ij},\beta_{ij}^{2},\dots,\beta_{ij}^{n}$.
Thus, since $\partial_{0}$ is orthogonal to
$\Lambda_{\scalebox{0.5}{\emph{b,c}}}$, the normal component of (219) is

$\displaystyle\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{i},\partial_{j})\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=(\nabla_{\\!\partial_{i}}{\partial_{j}})^{\perp}=\alpha_{ij}\partial_{0}.$

This verifies (216). (Since the subspace $S\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\text{span}\\{\partial_{2},\dots,\partial_{n}\\}$
is positive-definite, its orthogonal complement $S^{\perp}$ is necessarily
timelike (i.e., of Lorentzian index), and we have the direct sum
$T_{p}M=S_{p}\oplus S_{p}^{\perp}$ (see, e.g., [O’N83, p. 141]). Therefore,
$\alpha_{ij}\partial_{0}$ in (219) is indeed the (unique) normal component of
$\nabla_{\\!\partial_{i}}{\partial_{j}}$.) But as $\partial_{0}$ is _null_ —
and here is where Lorentzian geometry plays the crucial role — the last two
terms in (Proof.) each vanish identically:

$\underbrace{\,g(\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{i},\partial_{l}),\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{j},\partial_{k}))\,}_{\alpha_{il}\alpha_{jk}g_{00}}=\underbrace{\,g(\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{i},\partial_{k}),\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{j},\partial_{l}))\,}_{\alpha_{ik}\alpha_{jl}g_{00}}=0.$

This confirms (217) and completes the proof. ∎

When endowed with parallel null vector field, a Lorentzian manifold $(M,g)$ is
called a _Brinkmann spacetime_ ; such spacetimes were first introduced in
[Bri25] and later generalized in [Wal50]. Proposition 5 is what will allow us
to express plane waves in terms of the “Rosen coordinates” of (36):

######  Proposition 6.

An $(n+1)$-dimensional Lorentzian manifold $(M,g)$ is a pp-wave if and only if
there exist local coordinates $(v,u,x^{2},\dots,x^{n})$ about any point of $M$
in which $g$ takes the form

$\displaystyle(g_{ij})\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\begin{pmatrix}0&1&0&0&\cdots&0\\\
1&H&0&0&\cdots&0\\\ 0&0&1&0&\cdots&0\\\ 0&0&0&1&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&0&\cdots&1\end{pmatrix},$
(220)

where $H(u,x^{2},\dots,x^{n})$ is a smooth function independent of $v$_;_
$(M,g)$ will be a plane wave if and only if $H$ is a quadratic polynomial in
$x^{2},\dots,x^{n}$. In the latter case, there also exist coordinates
$(\tilde{x}^{0},\tilde{x}^{1},\dots,\tilde{x}^{n})$ in which $g$ takes the
form (36).

###### Proof.

First, suppose that $(M,g)$ is a pp-wave; by Definition 2, we know that
$R(X,Y)Z=0$ for all sections $X,Y,Z\in\Gamma(\partial_{0}^{\perp})$ (where we
are using the coordinates (213), in which $N=\partial_{0}$). By (217) in
Proposition 5, it follows that each codimention 2 submanifold
$(\Lambda_{\scalebox{0.5}{\emph{b,c}}},g_{\scalebox{0.4}{\emph{b,c}}})$ is
flat. Thus there exist local coordinates
$\\{r_{\scalebox{0.5}{\emph{c}}}^{i}(x^{2},\dots,x^{n})\\}_{i=2,\dots,n}$ on
each $\Lambda_{\scalebox{0.5}{\emph{b,c}}}$ with respect to which the induced
metric $g_{\scalebox{0.4}{\emph{b,c}}}$ (215) takes the form

$\displaystyle
g_{\scalebox{0.4}{\emph{b,c}}}=dr_{\scalebox{0.5}{\emph{c}}}^{2}\otimes
dr_{\scalebox{0.5}{\emph{c}}}^{2}+\cdots+dr_{\scalebox{0.5}{\emph{c}}}^{n}\otimes
dr_{\scalebox{0.5}{\emph{c}}}^{n}.$ (221)

At each $x^{1}=c$, we thus have the tuple
$(c,r_{\scalebox{0.5}{\emph{c}}}^{2},\dots,r_{\scalebox{0.5}{\emph{c}}}^{n})$;
considering $-r_{\scalebox{0.5}{\emph{c}}}^{n}$ if necessary, we may assume
that each
$\\{\partial_{r_{\scalebox{0.4}{\emph{c}}}^{2}},\dots,\partial_{r_{\scalebox{0.4}{\emph{c}}}^{n}}\\}$
is positively oriented. We now form a smooth coordinate chart
$(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})$, as follows.
First, at each point $(b,c,0,\dots,0)\in\Lambda_{\scalebox{0.5}{\emph{b,c}}}$,
rotate
$\\{\partial_{r_{\scalebox{0.4}{\emph{c}}}^{2}},\dots,\partial_{r_{\scalebox{0.4}{\emph{c}}}^{n}}\\}$
so that $\partial_{r_{\scalebox{0.4}{\emph{c}}}^{2}}$ points in the direction
of $\partial_{x^{2}}$ (i.e., so that
$\partial_{r_{\scalebox{0.4}{\emph{c}}}^{2}}=\frac{1}{\sqrt{\bar{g}_{22}}}\partial_{x^{2}}\big{|}_{(b,c,0,\dots,0)}$);
taken together, these $\partial_{r_{\scalebox{0.4}{\emph{c}}}^{2}}$’s thus
comprise a smooth vector field $X_{2}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\frac{1}{\sqrt{\bar{g}_{22}}}\partial_{x^{2}}$
on the submanifold $\Sigma\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\\{(\tilde{x}^{0},\tilde{x}^{1},0,\dots,0)\\}$.
As each $\partial_{r_{\scalebox{0.4}{\emph{c}}}^{3}}$ is orthogonal to its
corresponding $\partial_{r_{\scalebox{0.4}{\emph{c}}}^{2}}$, and since an
orientation has been fixed, we may arrange that the
$\partial_{r_{\scalebox{0.4}{\emph{c}}}^{3}}$’s also comprise a smooth vector
field $X_{3}$ on $\Sigma$, namely, the unique unit length vector field
orthogonal to $X_{2}$ and such that $\\{X_{2},X_{3}\\}$ is positively oriented
and spans the same 2-plane as $\\{\partial_{x^{2}},\partial_{x^{3}}\\}$.
Proceeding inductively in this way, we arrive at orthonormal vector fields
$\\{X_{2},\dots,X_{n}\\}$. Now, at each
$(b,c,0,\dots,0)\in\Sigma\cap\Lambda_{\scalebox{0.5}{\emph{b,c}}}$, parallel
transport $X_{2},\dots,X_{n}$ along the integral curve
$\gamma^{\scalebox{0.5}{\emph{b,c}}}$ of
$\partial_{r_{\scalebox{0.4}{\emph{c}}}^{2}}$ through $(b,c,0,\dots,0)$, via
the connection $\nabla^{\scalebox{0.4}{\emph{b,c}}}$ compatible with
$g_{\scalebox{0.4}{\emph{b,c}}}$. Then, at each point along
$\gamma^{\scalebox{0.5}{\emph{b,c}}}$, parallel transport the resulting vector
fields along the integral curve of
$\partial_{r_{\scalebox{0.4}{\emph{c}}}^{3}}$ on
$\Lambda_{\scalebox{0.5}{\emph{b,c}}}$. Repeat this process inductively along
the integral curves of
$\partial_{r_{\scalebox{0.4}{\emph{c}}}^{4}},\dots,\partial_{r_{\scalebox{0.4}{\emph{c}}}^{n}}$,
and let $X_{2},\dots,X_{n}$ continue to denote the resulting vector fields,
now smoothly defined on the domain of our original coordinates
$(x^{1},\dots,x^{n})$ as a whole (since we began with a smooth vector field on
$\Sigma$).222Note that by the flatness of each
$(\Lambda_{\scalebox{0.5}{\emph{b,c}}},g_{\scalebox{0.4}{\emph{b,c}}})$, the
equality
$\nabla^{\scalebox{0.4}{\emph{b,c}}}_{\\!\partial_{r_{\scalebox{0.4}{\emph{c}}}^{2}}}\\!X_{2}=0$
holds along _all_ integral curves of
$\partial_{r_{\scalebox{0.4}{\emph{c}}}^{2}}$, not just
$\gamma^{\scalebox{0.5}{\emph{b,c}}}$; e.g.,
$\nabla^{\scalebox{0.4}{\emph{b,c}}}_{\\!\partial_{r_{\scalebox{0.4}{\emph{c}}}^{3}}}\nabla^{\scalebox{0.4}{\emph{b,c}}}_{\\!\partial_{r_{\scalebox{0.4}{\emph{c}}}^{2}}}\\!X_{2}\overset{\eqref{eqn:Pi}}{=}\nabla^{\scalebox{0.4}{\emph{b,c}}}_{\\!\partial_{r_{\scalebox{0.4}{\emph{c}}}^{2}}}\cancelto{0}{\nabla^{\scalebox{0.4}{\emph{b,c}}}_{\\!\partial_{r_{\scalebox{0.4}{\emph{c}}}^{3}}}\\!X_{2}}=0.$
Thus, since
$\nabla^{\scalebox{0.4}{\emph{b,c}}}_{\\!\partial_{r_{\scalebox{0.4}{\emph{c}}}^{2}}}\\!X_{2}$
is parallel along integral curves of
$\partial_{r_{\scalebox{0.4}{\emph{c}}}^{3}}$, and since it vanished at its
starting point on $\gamma^{\scalebox{0.5}{\emph{b,c}}}$, by uniqueness of
parallel transport it must vanish identically; likewise for
$\partial_{r_{\scalebox{0.4}{\emph{c}}}^{4}},\dots,\partial_{r_{\scalebox{0.4}{\emph{c}}}^{n}}$.
By the Gauss Formula (see, e.g., [Lee18, Theorem 8.2, p. 228]) and the
flatness condition (221), we have, on each
$(\Lambda_{\scalebox{0.5}{\emph{b,c}}},g_{\scalebox{0.4}{\emph{b,c}}})$, that

$\nabla_{\\!X_{2}}{X_{2}}=\cancelto{0}{\nabla^{\scalebox{0.4}{\emph{b,c}}}_{\\!X_{2}}\,X_{2}}\,+\,\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(X_{2},X_{2});$

similarly for $\nabla_{\\!X_{i}}{X_{j}}$ for all $i,j=2,\dots,n$.333That each
$\nabla^{\scalebox{0.4}{\emph{b,c}}}_{\\!X_{i}}\,X_{j}=0$ follows from the
fact that the $X_{i}$’s are parallel on
$(\Lambda_{\scalebox{0.5}{\emph{b,c}}},g_{\scalebox{0.4}{\emph{b,c}}})$.
Indeed, writing
$X_{i}=\sum_{k=2}^{n}\alpha_{i}^{k}\partial_{r_{\scalebox{0.4}{\emph{c}}}^{k}}$
for some smooth functions $\alpha_{i}^{k}$, we have that
$\nabla^{\scalebox{0.4}{\emph{b,c}}}_{X_{i}}X_{j}=\sum_{k=2}^{n}\alpha_{i}^{k}\cancelto{0}{\nabla^{\scalebox{0.4}{\emph{b,c}}}_{\partial_{r_{\scalebox{0.4}{\emph{c}}}}}X_{j}}=0.$
But just as in (Proof.), each
$\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\cdot,\cdot)$ is directly
proportional to $\partial_{0}$. Because of this, all Lie brackets of the frame
$\\{\partial_{0},\partial_{1},X_{2},\dots,X_{n}\\}$ will have at most a
$\partial_{0}$-component, and this in turn implies that each
$X_{i}^{\flat}=g(X_{i},\cdot)$ is closed; indeed, for any pair of vector
fields $A,B\in\\{\partial_{0},\partial_{1},X_{2},\dots,X_{n}\\}$,

$dX_{i}^{\flat}(A,B)=\cancelto{0}{A(g(X_{i},B))}-\cancelto{0}{B(g(X_{i},A))}-\cancelto{0}{g(X_{i},[A,B])}=0.$

By the Poincaré Lemma, these 1-forms are locally exact,
$X_{i}^{\flat}=d\tilde{x}^{i}$, for some smooth functions
$\tilde{x}^{2},\dots,\tilde{x}^{n}$. Thus, with respect to the coordinate
basis
$\\{\partial_{0},\partial_{1},\partial_{\tilde{x}^{2}},\dots,\partial_{\tilde{x}^{n}}\\}$,
the Lorentzian metric $g$ takes the form

$\displaystyle(g_{ij})\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\begin{pmatrix}0&1&0&0&\cdots&0\\\
1&\bar{g}_{11}&0&0&\cdots&0\\\ 0&0&1&0&\cdots&0\\\ 0&0&0&1&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&0&\cdots&1\end{pmatrix},$

with $\bar{g}_{11}=\bar{g}_{11}(\tilde{x}^{1},\dots,\tilde{x}^{n})$.
Conversely, suppose that at every point on an $(n+1)$-dimensional Lorentzian
manifold $(M,g)$ there exist local coordinates $(v,u,x^{2},\dots,x^{n})$ in
which $g$ takes the form (220); observe that $\partial_{v}$ is a parallel,
null vector field. For such a metric, the nonvanishing Christoffel symbols are

$\displaystyle\nabla_{\\!\partial_{i}}{\partial_{u}}=\nabla_{\\!\partial_{u}}{\partial_{i}}=\frac{H_{i}}{2}\partial_{v}\hskip
14.45377pt,\hskip
14.45377pt\nabla_{\\!\partial_{u}}{\partial_{u}}=\frac{H_{u}}{2}\partial_{v}-\frac{1}{2}\sum_{i=2}^{n}H_{i}\partial_{i},$
(222)

for $i=x^{2},\dots,x^{n}$. It follows that
$R(\partial_{i},\partial_{j})\partial_{k}=0$ for all
$i,j,k=x^{2},\dots,x^{n}$, and also that

$\displaystyle
R(\partial_{i},\partial_{j})\partial_{u}=\nabla_{\\!\partial_{i}}\nabla_{\\!\partial_{j}}\partial_{u}-\nabla_{\\!\partial_{j}}\nabla_{\\!\partial_{i}}\partial_{u}=\frac{H_{ij}}{2}\partial_{v}-\frac{H_{ji}}{2}\partial_{v}=0.$
(223)

Thus $R(X,Y)V=0$ for all $X,Y\in\Gamma(\partial_{v}^{\perp})$ and
$V\in\mathfrak{X}(M)$. Finally, if it is also the case that $\nabla_{\\!X}R=0$
for all $X\in\Gamma(\partial_{v}^{\perp})$, then

$\displaystyle\underbrace{\,(\nabla_{\\!\partial_{i}}R)(\partial_{u},\partial_{j},\partial_{u})\,}_{0}\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\partial_{i}(R(\partial_{u},\partial_{j})\partial_{u})-R(\nabla_{\\!\partial_{i}}{\partial_{u}},\partial_{j})\partial_{u}$
$\displaystyle\hskip
36.135pt-\cancelto{0}{R(\partial_{u},\nabla_{\\!\partial_{i}}{\partial_{j}})\partial_{u}}-R(\partial_{u},\partial_{j})\nabla_{\\!\partial_{i}}{\partial_{u}}$
$\displaystyle=$
$\displaystyle\\!\\!\partial_{i}(\nabla_{\\!\partial_{u}}\nabla_{\\!\partial_{j}}\partial_{u}-\nabla_{\\!\partial_{j}}\nabla_{\\!\partial_{u}}\partial_{u})-\frac{H_{i}}{2}\cancelto{0}{R(\partial_{v},\partial_{i})\partial_{u}}$
$\displaystyle\hskip
120.69077pt-\frac{H_{i}}{2}\cancelto{0}{R(\partial_{u},\partial_{i})\partial_{v}}$
$\displaystyle=$
$\displaystyle\\!\\!\cancelto{0}{\frac{H_{jui}}{2}\partial_{v}-\frac{H_{uji}}{2}\partial_{v}}+\frac{1}{2}\sum_{k=2}^{n}H_{kji}\partial_{k},$

which implies that $H_{kji}=0$ for all $i,j,k=2,\dots,n$. Thus
$H(u,x^{2},\dots,x^{n})$ is quadratic in $x^{2},\dots.x^{n}$ if and only if
$\nabla_{\\!X}R=0$ for all $X\in\Gamma(\partial_{v}^{\perp})$. Finally, we
define an isometry converting a plane wave in “Rosen coordinates” (36) to one
in “Brinkmann coordinates” (220) (see also, e.g., [Bla11]); in what follows,
we will employ the Einstein summation connection for convenience. Define an
$(n-2)\times(n-2)$ matrix $C$, with entries $C^{i}_{j}(\tilde{x}^{0})$ being
smooth functions of $\tilde{x}^{0}$, via the equation

$C^{i}_{k}(\tilde{x}^{0})g_{ij}(\tilde{x}_{0},0,\dots,0)C^{j}_{l}(\tilde{x}^{0})=\delta_{kl}\hskip
14.45377pt,\hskip 14.45377pti,j,k,l=2,\dots,n,$

and such that

$(\partial_{0}C^{i}_{k})g_{ij}(\tilde{x}_{0},0,\dots,0)C^{j}_{l}=C^{i}_{k}g_{ij}(\tilde{x}_{0},0,\dots,0)(\partial_{0}C^{j}_{l}),$

where $\partial_{0}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\frac{\partial}{\partial\tilde{x}^{0}}$.
Now define the coordinates

$\displaystyle\left\\{\begin{array}[]{ccc}u&\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=&\tilde{x}^{0},\\\
x^{i}&\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=&C^{i}_{j}\tilde{x}^{j},\\\
v&\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=&\tilde{x}^{1}+\frac{1}{2}g_{ij}(\tilde{x}^{0},0,\dots,0)(\partial_{0}C^{i}_{k})C^{j}_{l}\tilde{x}^{k}\tilde{x}^{l}.\end{array}\right.$
(227)

These will be Brinkmann coordinates (220), with

$\displaystyle H(u,x^{2},\dots,x^{n})\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=-\partial_{u}\big{(}g_{ij}(u,0,\dots,0)\partial_{u}(C^{i}_{k}(u))\big{)}C^{j}_{l}(u)x^{k}x^{l}.$
(228)

This completes the proof. ∎

Finally, we mentioned that there is an even more geometric way in which to
define pp-waves, via the following vector bundle that naturally sits above any
Brinkmann spacetime (see also [CBS13] and [LS16]):

######  Proposition 7.

Let $(M,g)$ be a Lorentzian manifold and $N$ a parallel null vector field,
with orthogonal complement $N^{\perp}\subseteq TM$. Then the vector bundle
$N^{\perp}/N$ admits a positive-definite inner product $\bar{g}$,

$\bar{g}([X],[Y])\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=g(X,Y)\hskip
14.45377pt\text{for all}\hskip 14.45377pt[X],[Y]\in\Gamma(N^{\perp}/N),$

and a corresponding linear connection
$\overline{\nabla}\colon\mathfrak{X}(M)\times\Gamma(N^{\perp}/N)\longrightarrow\Gamma(N^{\perp}/N)$,

$\overline{\nabla}_{\\!V}{[Y]}\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=[\nabla_{\\!W}{Y}]\hskip
14.45377pt\text{for all}\hskip 14.45377ptV\in\mathfrak{X}(M)\hskip
14.45377pt\text{and}\hskip 14.45377pt[Y]\in\Gamma(N^{\perp}/N).$

This connection is flat if and only if $(M,g)$ is a pp-wave.

###### Proof.

The metric $\bar{g}$ will be well defined, and positive definite, whenever $V$
is null; indeed, every $X\in\Gamma(N^{\perp})$ not proportional to $V$
satisfies $g(X,X)>0$, so that $\bar{g}$ is nondegenerate (and positive-
definite), and if $[X]=[X^{\prime}]$ and $[Y]=[Y^{\prime}]$, so that
$X^{\prime}=X+fN$ and $Y^{\prime}=Y+kN$ for some smooth functions $f,k$, then

$\bar{g}([X^{\prime}],[Y^{\prime}])=g(X^{\prime},Y^{\prime})=g(X,Y)=\bar{g}([X],[Y]).$

On the other hand, the connection $\overline{\nabla}$ requires $N$ to be
parallel or else it is not well defined:
$\nabla_{\\!V}{Y}\in\Gamma(N^{\perp})$ if and only if $N$ is parallel, in
which case

$\overline{\nabla}_{\\!V}{[Y^{\prime}]}=[\nabla_{\\!V}{Y^{\prime}}]=[\nabla_{\\!V}{Y}]+\cancelto{0}{[V(k)N]}+\cancelto{0}{[k\nabla_{\\!V}{N}]}\,=\overline{\nabla}_{\\!V}{[Y]}.$

That $\overline{\nabla}$ is indeed a linear connection follows easily. Now, if
this connection is flat, then by definition its curvature endomorphism

$\displaystyle\bar{R}\colon\mathfrak{X}(M)\times\mathfrak{X}(M)\\!\\!\\!\\!$
$\displaystyle\times$
$\displaystyle\\!\\!\\!\\!\Gamma(N^{\perp}/N)\longrightarrow\Gamma(N^{\perp}/N),~{}\text{given
by}$ $\displaystyle\bar{R}(V,W)[X]\\!\\!\\!\\!$ $\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=$
$\displaystyle\\!\\!\\!\\!\overline{\nabla}_{\\!V}[\overline{\nabla}_{\\!W}[X]]-\overline{\nabla}_{\\!W}[\overline{\nabla}_{\\!V}[X]]-\overline{\nabla}_{\\![V,W]}{[X]}$

vanishes, for any section $[X]\in\Gamma(N^{\perp}/N)$ and vector fields
$V,W\in\mathfrak{X}(M)$. Using $\bar{g}$, this flatness condition is
equivalent to

$\bar{g}(\bar{R}(V,W)[X],[Y])=0\hskip 7.22743pt\text{for all}\hskip
7.22743ptV,W\in\mathfrak{X}(M)\hskip 7.22743pt,\hskip
7.22743pt[X],[Y]\in\Gamma(N^{\perp}/N).$

But if we unpack the definitions of $\overline{\nabla}$ and $\bar{g}$, we see
that

$\bar{g}(\bar{R}(V,W)[X],[Y])=\text{Rm}(V,W,X,Y)=\text{Rm}(X,Y,V,W).$

It follows that $\bar{R}=0$ if and only if $R(X,Y)V=0$ for all
$X,Y\in\Gamma(N^{\perp})$ and $V\in\mathfrak{X}(M)$, which, by Definition 2,
is precisely the condition to be a pp-wave. ∎

An additional consequence of Proposition 7 is its implication that every
Riemannian $n$-manifold $(M,g)$ locally admits such a vector bundle. Indeed,
express $g$ locally in any choice of smooth coordinate chart
$(x^{1},\dots,x^{n})$, and then define the following $(n+2)$-dimensional
Lorentzian metric $g_{\scalebox{0.4}{\emph{L}}}$ in the coordinates
$(v,u,x^{1},\dots,x^{n})$:

$\displaystyle((g_{\scalebox{0.4}{\emph{L}}})_{ij})\mathrel{\hbox
to0.0pt{\raisebox{1.03334pt}{$\cdot$}\hss}\raisebox{-1.03334pt}{$\cdot$}}=\begin{pmatrix}0&1&0&0&0&\cdots&0\\\
1&0&0&0&0&\cdots&0\\\ 0&0&g_{11}&g_{12}&g_{13}&\cdots&g_{1n}\\\
0&0&g_{21}&g_{22}&g_{23}&\cdots&g_{2n}\\\
0&0&g_{31}&g_{32}&g_{33}&\cdots&g_{3n}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&g_{n1}&g_{n2}&g_{n3}&\cdots&g_{nn}\end{pmatrix}\hskip 14.45377pt,\hskip
14.45377ptg_{ij}=g_{ij}(x^{1},\dots,x^{n}).$

This is a special case of a Lorentzian metric — in fact, a Brinkmann
spacetime, since the null vector field $\partial_{v}$ is parallel — of the
form (213) ($g$ is precisely the red block of the latter). By (217) of
Proposition 5, note that
$\text{Rm}_{\scalebox{0.5}{\emph{g}}}=\text{Rm}_{\scalebox{0.4}{\emph{L}}}$ on
$M$ (viewed as a codimension 2 submanifold of
$\\{(v,u,x^{1},\dots,x^{n})\\}$).

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