ORTHOGONAL COMPLEX STRUCTURES

S. M. Salamon

Key words: Hermitian metric, Weyl tensor, twistor space,
          anti-self-dual 4-manifold.

1991 Math. Subject Classification: 53C55 (53C15, 32L25).

Introduction

Let $M$ be a smooth manifold of dimension $2n$, and let $g$ be a
Riemannian metric on $M$.

An almost-complex structure (abbreviated \acs) $J$ on $M$ is an
endomorphism of the tangent bundle $TM$, or equivalently the cotangent
bundle $T^*M$, of $M$ such that $J^2=-1$. Such a tensor induces an
orientation on $M$ by taking the $2n$-form $e_1\we Je_1\we\cdots\we
e_n\we Je_n$ to always be a positive multiple of the volume form. The
triple $(M,g,J)$ is called {\sl almost-Hermitian} if $J$ is an
orthogonal transformation relative to $g$, i.e.\ if
\[g(JX,JY)=g(X,Y)\] for all tangent vectors $X,Y$. This equation
implies that the tensor $\w$ defined by $\w(X,Y)=g(JX,Y)$ is
anti-symmetric; it is called the fundamental 2-form of the
almost-Hermitian structure. Any two of the tensors $g,J,\w$ determine
the third.

The \acs\ $J$ is said to be {\sl integrable} if the Nijenhuis tensor
\begin{equation}\label{N}
N_J(X,Y)= [JX,JY]-J[JX,Y]-J[X,JY]-[X,Y]
\end{equation}
vanishes. Indeed,
the Newlander-Nirenberg theorem \cite{NN} implies that $N_J=0$ if and
only if $(M,J)$ is a complex manifold in the sense that there exist
local complex coordinates $z^1,\ldots, z^n$ such that $Jdz^k=i\,dz^k$,
$1\le k\le n$. In these circumstances one also says that $(M,g,J)$ is
a {\sl Hermitian manifold}.\vs

We shall be concerned with the problem of finding different Hermitian
structures on a given Riemannian manifold $(M,g)$, and the following
terminology will be convenient.

\pr{Definition} An orthogonal complex structure (OCS) on $(M,g)$ is an
integrable \acs\ $J$ on $M$ such that $g(JX,JY)=g(X,Y)$.\vs

\n If $M$ is already oriented then an OCS $J$ may or may not induce the
chosen orientation, and according to case we say that $J$ is
positively or negatively oriented.

The purpose of this note is to investigate the following

\pr{Problem} Given a Riemannian manifold $(M,g)$, does there exist an
OCS? If so, describe the set of all OCS's.\vs

The question can be asked either (i) globally, or (ii) locally, and
the corresponding questions can be rather different in nature. Given
an OCS $J$ defined over a compact manifold, one may ask to what extent
it is unique (at least up to sign) and it may be appropriate to
consider separately the case in which $J$ is positively or
negatively oriented. In the case in which $M$ is a symmetric space,
the question has been successfully tackled by Burstall and Rawnsley
\cite{BR} by introducing the twistor space of $M$, and in
general this is a valuable tool for characterizing the existence of
OCS's.

Note that an OCS $J$ remains orthogonal if the metric $g$ is replaced
by a conformally equivalent one $e^f g$. The above problem therefore
relates more accurately to the conformal class $[g]$ determined by
$g$. Any 2-dimensional oriented conformal structure uniquely
determines an integrable complex structure, so we shall always suppose
that $n\ge2$. Posing the above question in another way leads to the

\pr{Problem} Find conformal structures $(M,[g])$ which admit an abundance,
at least locally, of OCS's.\vs

We shall see that continuous families of such OCS's arise from the
partial integrability of the twistor space, which is determined by
properties of the Weyl conformal curvature tensor $W$. The study of
twistor spaces of Riemannian 4-manifolds has advanced considerably
since their inception in \cite{AHS}, and the most important aspect of
the preceding problem for $n=2$ is the classification of self-dual
structures. A summary of the state of this art is included in
Section~4, and provides motivation for work on higher-dimensional
situations. We shall see that a pre-requisite for progress here
is a more complete algebraic understanding of $W$, and the
extent to which it may be constrained.\vs

The next series of examples illustrate and clarify the above problems.

\pr{Example 1} (i) If $n>1$ then the sphere $S^{2n}$ has no OCS $J$
relative to its standard metric. Indeed, it is well known that for
$n\ne1,3$, the sphere $S^{2n}$ does not even admit an {\sl almost}
complex structure. But it is also true that $S^6$ has no OCS (see
e.g.\ \cite{LS6}), even though it does have a $G_2$-invariant
non-integrable \acs.

\n(ii) Let $x\in S^{2n}$; the induced metric on $S^{2n}\setminus\{x\}$ is
conformally equivalent to the flat metric on $\R^{2n}$. The latter admits
infinitely many OCS's including the constant ones parametrized by the
homogeneous space $O(2n)/U(n)$.\vs

\pr{Example 2} (i) The complex projective space $\CP^n$ with the Fubini-Study
metric admits a standard OCS that we denote by $J_0$, but there are no
others apart from $-J_0$ (a result of D.~Burns and the authors of
\cite{BMGR}). Note that $\pm J_0$ induce the same orientation on
$\CP^n$ if and only if $n$ is even.

\n(ii) In fact on {\sl any} open set $U\subset\CP^2$, the only OCS's
inducing the standard orientation are $\pm J_0$. By contrast, if $x$
is a point and $L$ a projective line in $\CP^2$ then
$\CP^2\setminus\{x\}$ has exactly one pair $\pm J_x$ of
negatively-oriented OCS's, and $\CP^2\setminus L$ has infinitely many
OCS's.\vs

The last examples show that there is an interesting interface between
the global and local existence questions. Below, we shall explain the
above statements and raise a number of related questions in the course
of a general survey of relevant material.