Results Concerning Almost Complex Structures on the Six-Sphere
For the standard metric on the six-dimensional sphere, with Levi-Civita connection ∇, we show there is no almost complex structure J such that ∇XJ and ∇JXJ commute for every X, nor is there any integrable J such that ∇JXJ = J∇XJ for every X. The latter statement generalizes a previously known result...
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| Cite this: | Results Concerning Almost Complex Structures on the Six-Sphere / S.O. Wilson // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 12 назв. — англ. |
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| description | For the standard metric on the six-dimensional sphere, with Levi-Civita connection ∇, we show there is no almost complex structure J such that ∇XJ and ∇JXJ commute for every X, nor is there any integrable J such that ∇JXJ = J∇XJ for every X. The latter statement generalizes a previously known result on the non-existence of integrable orthogonal almost complex structures on the six-sphere. Both statements have refined versions, expressed as intrinsic first-order differential inequalities depending only on J and the metric. The new techniques employed include an almost-complex analogue of the Gauss map, defined for any almost-complex manifold in Euclidean space.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 034, 21 pages
Results Concerning Almost Complex Structures
on the Six-Sphere
Scott O. WILSON
Department of Mathematics, Queens College, City University of New York,
65-30 Kissena Blvd., Queens, NY 11367, USA
E-mail: scott.wilson@qc.cuny.edu
URL: http://qcpages.qc.cuny.edu/~swilson/
Received November 20, 2017, in final form April 09, 2018; Published online April 17, 2018
https://doi.org/10.3842/SIGMA.2018.034
Abstract. For the standard metric on the six-dimensional sphere, with Levi-Civita connec-
tion ∇, we show there is no almost complex structure J such that ∇XJ and ∇JXJ commute
for every X, nor is there any integrable J such that ∇JXJ = J∇XJ for every X. The latter
statement generalizes a previously known result on the non-existence of integrable ortho-
gonal almost complex structures on the six-sphere. Both statements have refined versions,
expressed as intrinsic first order differential inequalities depending only on J and the met-
ric. The new techniques employed include an almost-complex analogue of the Gauss map,
defined for any almost complex manifold in Euclidean space.
Key words: six-sphere; almost complex; integrable
2010 Mathematics Subject Classification: 53C15; 32Q60; 53A07
1 Introduction
One knows from topology that the only spheres which admit almost complex structures are those
in dimensions two and six, [3]. On the two-sphere, every almost complex structure is integrable,
i.e., induced by a complex structure, and furthermore this is unique up to biholomorphism. The
situation for the six-sphere S6 is not well understood: while there are many almost complex
structures, Hopf’s original question from 1926, asking whether the six-sphere admits a complex
structure, remains unsolved [5]. Thus it is of fundamental importance to better understand
almost complex structures and concepts related to integrability, and in particular to determine
whether S6 has a complex structure.
There are results which forbid certain almost complex structures on S6 from being integrable.
Blanchard [1], and then independently LeBrun [7], have shown that no almost complex structure
on S6 which is orthogonal, with respect to the standard Euclidean metric on the sphere, is also
integrable. Tang has shown in [12] a more general analogous result for almost structures on S6
that preserve any metric which satisfies a certain curvature-related positivity condition. Chern,
in a result communicated by Bryant [4], has shown that no almost complex structure on S6 that
preserves a certain 2-form is integrable.
In this paper, we proceed in the following way. Given an almost complex structure J on
a manifold, and a connection ∇ on the tangent bundle, one can ask that the covariant deriva-
tive ∇XJ is J-linear in the variable X, i.e., ∇JXJ = J∇XJ . In fact, for any torsion free
connection ∇, this condition implies the integrability of J , so in this case we refer to J as being
strongly integrable with respect to ∇.
Inspired by LeBrun’s work in [7], we show using a variation of that argument that there are
no almost complex structures on S6 that are strongly integrable with respect to the Levi-Civita
connection of the standard round metric. We recover the results of [1] and [7] as a special
case since every integrable orthogonal almost complex structure on any Riemannian manifold is
mailto:scott.wilson@qc.cuny.edu
http://qcpages.qc.cuny.edu/~swilson/
https://doi.org/10.3842/SIGMA.2018.034
2 S.O. Wilson
strongly integrable with respect to ∇. On the other hand, the first order condition of J being
strongly integrable with respect to ∇ is a priori weaker than the zeroth order condition of J
being orthogonal with respect to the metric.
We also provide a sharper result that prohibits further almost complex structures on S6 from
being integrable. This is deduced from an intrinsic first order differential inequality involving J
and the metric, see Corollary 5.10.
Another result of this paper is to show that for any almost complex structure J on S6,
integrable or not, there are non-trivial global conditions upon ∇J that are dictated by topology.
We define for any such J a canonical map into a non-compact version of the Grassmannian,
which supports a symplectic structure ω determined by the curvature of a tautological bundle
with connection over this space. In fact, this map is defined for any almost complex manifold in
Euclidean space, yielding a closed 2-form associated to J and the normal bundle. For the sphere
we are able to calculate explicitly the pullback of this form, see Theorem 4.4. Several corollaries
are deduced to obtain new non-trivial conditions on ∇J . For example, there is no J on S6 such
that ∇XJ and ∇JXJ commute for all X, see Corollary 4.5 and the sharper bounds thereafter.
As a final corollary we also obtain from this viewpoint the prior result on the non-existence of
orthogonal almost complex structures on S6, see Corollary 4.7.
The contents of this paper are as follows. In Section 2 we review notions of integrability,
including a geometric characterization of integrability, that the Lie derivative LXJ of J is J-
linear in X. We study the aforementioned J-linearity of ∇XJ in the variable X, showing
it is a strong notion of integrability, and giving both a real and an equivalent complexified
formulation. Finally, we prove that on any Riemannian manifold, integrable orthogonal almost
complex structures always satisfy this condition, see Theorem 2.8.
In Section 3 we introduce models for the Grassmannian and its non-compact version, to be
used in later sections. While one can make homogeneous space or vector sub-space definitions,
we prefer to present them as spaces of idempotent matrices since this makes the canonical maps
defined in later sections most transparent. We verify that this non-compact version of the
Grassmannian also carries a symplectic structure, with explicit symplectic form tamed by many
almost complex structures. On the compact Grassmannian subspace of self-adjoint idempotent
matrices, we obtain a simple formulation of the Kähler structure in terms of the standard matrix
inner product Tr(AB∗).
In Section 4 we introduce a generalization of the Gauss map, defined for an almost complex
manifold in Euclidean space, which yields a canonical map into the non-compact Grassmannian.
We use the term “canonical” since the map is induced by the projection operator P− = 1
2(I+iJ)
and projection onto the complexified normal bundle. For the standard S6 and any J , we calculate
the pullback of the symplectic form on the target explicitly in terms of the metric and ∇J , and
the corollaries mentioned above are deduced.
In Section 5 we define for any J on S6 an immersion of S6 into the Grassmannian of self-
adjoint idempotent matrices. This is similar to the map used in [7], though modified to include
the normal bundle of the sphere in Euclidean space. We calculate ∂ of this map and show that it
vanishes if and only if ∇XJ is J-linear in the variable X. We conclude there are no such strongly
integrable almost complex structures on S6, all of which would have necessarily been integrable.
Finally, we compute the pullback of the Kähler form explicitly and, as a consequence, we deduce
an intrinsic first order differential inequality depending only on J and the metric which, if
satisfied everywhere, guarantees J to be non-integrable.
2 Integrability
An almost complex structure on a smooth manifold M is a section J of the endomorphism
bundle End(TM) which squares to − Id. Such a J is said to be integrable if it is everywhere
Results Concerning Almost Complex Structures on the Six-Sphere 3
induced by the complex structure i on Cn along coordinate charts. By a theorem of Newlander–
Nirenburg [10], J is integrable if and only if the Nijenhuis tensor
N(X,Y ) = J [X, JY ] + J [JX, Y ] + [X,Y ]− [JX, JY ] (2.1)
vanishes everywhere. A more geometric way to understand this condition is given by the follow-
ing lemma.
Lemma 2.1. An almost complex structure J is integrable if and only if the Lie derivative of J
is J-linear in the direction of any vector field X, i.e.,
LJXJ = JLXJ
for all vector fields X.
Proof. The condition given holds if and only if (LJXJ)Y = (JLXJ)Y for all vectors Y , which
is equivalent to equation (2.1) using the identity
(LXJ)Y = LX(JY )− J(LXY ) = [X, JY ]− J [X,Y ]. �
A connection ∇ on TM induces, by the Leibniz property, a connection on End(TM), which
we also denote by ∇. In particular, J2 = − Id implies
(∇XJ)J = −J(∇XJ)
for all vectors X, so that ∇J is J-anti-linear in its second argument.
This paper concerns the J-linearity of ∇J in its first argument, which is stronger than
integrability when ∇ is torsion free, cf. Lemma 2.3. Recall that a connection ∇ on the tangent
bundle of a manifold M is said to be torsion free if
∇XY −∇YX = [X,Y ]
for all vector fields X and Y .
Definition 2.2. Let ∇ be a torsion free connection on the tangent bundle of an almost complex
manifold (M,J). We say J is strongly integrable with respect to ∇ if ∇XJ is J-linear in the
variable X, i.e.,
∇JXJ = J∇XJ
for all vectors X.
Lemma 2.3. If J is strongly integrable with respect to some torsion free connection ∇, then J
is integrable.
Proof. If ∇ is torsion free, then using the identity (∇XJ)Y = ∇X(JY )− J∇XY , we have
N(X,Y ) = (∇JY J − J∇Y J)X − (∇JXJ − J∇XJ)Y, (2.2)
and that proves the lemma. �
In two real dimensions every almost complex structure is integrable, so the next example
shows that an integrable J need not be strongly integrable with respect to every connection. At
the end of the section we will conclude that any integrable J is strongly integrable with respect
to some torsion free connection.
4 S.O. Wilson
Example 2.4. Let M = {(x, y) ∈ R× R |x 6= 0} and let J on M be given by
J(x, y) =
[
0 x
−1
x 0
]
.
With the standard metric and Levi-Civita connection ∇ we have ∇ ∂
∂y
J(x, y) = 0 and
∇ ∂
∂x
J(x, y) =
[
0 1
1
x2 0
]
.
So, J is not strongly integrable with respect to ∇ since
∇J ∂
∂y
J(x, y) = ∇x ∂
∂x
J(x, y) =
[
0 x
1
x 0
]
,
J∇ ∂
∂y
J(x, y) = 0.
The proof of Lemma 2.3 suggests considering, for any torsion free connection ∇, the deviation
from ∇XJ being J-linear in X, i.e.,
m(X,Y ) := (∇JXJ − J∇XJ)Y. (2.3)
It is straightforward to check that m(X,Y ) is a tensor (i.e., linear over functions in each variable),
and that
m(JX, Y ) = m(X, JY ) = −Jm(X,Y ), (2.4)
so that m is J anti-linear. One can regard m as a smoothly varying family of bi-linear operations
on the tangent spaces, so we will refer to this as the tangent algebra associated to J and ∇. We
can recast the previous lemmas as
Corollary 2.5. An almost complex structure is integrable if and only if, for any torsion free
connection, the tangent algebra associated to J and the connection is everywhere commutative.
An almost complex structure is strongly integrable with respect to a torsion free connection if
and only if the associated tangent algebra is everywhere vanishing.
Proof. Using the definition in equation (2.3), the first statement follows from equation (2.2)
and the second statement follows from Definition (2.2). �
Remark 2.6. A tangent algebra on any surface defines (at any point) an operation which is
either zero or a division algebra. In fact, one can see from equation (2.4) that the tangent
algebra is uniquely determined at any point by its value m(X,X) for some single vector X. In
higher dimensions this is far from the case, as we have direct sums of (trivial and non-trivial)
rank two examples, such as on the vector space C with product (a, b) 7→ εab, for any ε ∈ R. It
is perhaps worth noting that there are no division algebras in dimension six.
Recall that given (M,J) we define the holomorphic and anti-holomorphic tangent spaces to
be the +i and −i eigenspaces of J on TM ⊗ C, i.e.,
T 1,0
x = {v ∈ TxM ⊗ C | Jv = iv},
T 0,1
x = {v ∈ TxM ⊗ C | Jv = −iv}.
The following lemma shows that strong integrability is equivalent to the connection respecting
the T 0,1 summand.
Results Concerning Almost Complex Structures on the Six-Sphere 5
Lemma 2.7. Let (M,J) be an almost complex manifold and ∇ any torsion free connection.
The following are equivalent:
• J is strongly integrable with respect to ∇, i.e., the tangent algebra vanishes.
• The C-linear extension of ∇ satisfies ∇ : T 0,1 ⊗ T 0,1 → T 0,1.
The second condition should be compared with the integrability condition that T 0,1 is closed
under the Lie bracket, thus giving another verification of Lemma 2.3.
Proof. Any V,W ∈ T 0,1 can be written uniquely as V = X + iJX and W = Y + iJY for some
real X and Y . Then
∇X+iJX(Y + iJY ) = (∇XY −∇JX(JY )) + i(∇X(JY ) +∇JXY ).
The right hand side is in T 0,1 if and only if
J(∇XY −∇JX(JY )) = (∇X(JY ) +∇JXY ).
Using the identity ∇JXY + J∇JX(JY ) = J(∇JXJ)Y , this is equivalent to
Jm(X,Y ) = (∇XJ + J(∇JXJ))Y = 0
for all X and Y , which holds for all X and Y if and only if J is strongly integrable with respect
to ∇. �
In fact, for orthogonal almost complex structures, the two conditions of the previous lemma
are equivalent to integrability (see [11, p. 267]). For completeness here, we give our own proof
in one direction, inspired by an argument in [8, Appendix C.7]. There a stronger hypothesis
(namely, that the form ω below is also closed) is used to conclude the stronger consequence (that
∇J ≡ 0).
Theorem 2.8. Let M be a smooth manifold with Riemannian metric 〈−,−〉, Levi-Civita con-
nection ∇, and integrable almost complex structure J . If J is compatible with the metric, i.e.,
〈JX, JY 〉 = 〈X,Y 〉 for all X,Y , then J is strongly integrable with respect to ∇.
Proof. Since J is orthogonal, there is an induced 2-form ω defined by
ω(X,Y ) = 〈JX, Y 〉.
For any given vectors X, Y and Z at a fixed point p ∈ M , choose extensions of these vectors
to vector fields so that all six covariant derivatives of the form ∇VW vanish at the point p.
Then, at the point p, all Lie brackets also vanish, and using the formula for d in terms of the
Lie bracket, and the fact that ∇ respects the metric, we have
dω(X,Y, Z) = 〈(∇XJ)Y,Z〉+ 〈(∇Y J)Z,X〉+ 〈(∇ZJ)X,Y 〉.
Using equation (2.2) and the previous equation we calculate
〈X,N(Y, Z)〉 = 〈X, (∇JZJ)Y 〉 − 〈X, J(∇ZJ)Y 〉 − 〈X, (∇JY J)Z〉+ 〈X, J(∇Y J)Z〉
= −〈X, J(∇ZJ)Y 〉+ 〈X, J(∇Y J)Z〉 − 〈JZ, (∇Y J)X〉+ 〈Z, (∇XJ)(JY )〉
− 〈Y, (∇XJ)(JZ)〉+ 〈JY, (∇ZJ)X〉+ dω(X, JZ, Y )− dω(X, JY, Z)
= −2〈J(∇XJ)Y, Z〉+ dω(X, JZ, Y )− dω(X, JY, Z),
6 S.O. Wilson
where in the last equality we have used the fact that J and ∇J are both skew-adjoint. So,
2〈J(∇XJ)Y, Z〉 = −〈X,N(Y,Z)〉+ dω(X, JZ, Y )− dω(X, JY, Z),
2〈(∇JXJ)Y, Z〉 = 2〈J(∇JXJ)Y, JZ〉
= −〈JX,N(Y, JZ)〉+ dω(JX, J2Z, Y )− dω(JX, JY, JZ)
= 〈X,N(Y,Z)〉 − dω(JX,Z, Y )− dω(JX, JY, JZ).
Subtracting these equations we have
2〈(J(∇XJ)− (∇JXJ))Y,Z〉 = −2〈X,N(Y,Z)〉
+ dω(X,JZ, Y )− dω(X,JY, Z) + dω(JX,Z, Y ) + dω(JX, JY, JZ).
Using N ≡ 0 we have
2〈((∇JXJ)− J(∇XJ))Y,Z〉
= dω(X,Y, JZ) + dω(X, JY, Z) + dω(JX, Y, Z)− dω(JX, JY, JZ).
The right hand side of this equation is alternating in X and Y , but by Corollary 2.5, the left hand
side is symmetric in X and Y , so both sides vanish for all Z, and therefore ((∇JXJ)−J(∇XJ))
≡ 0. �
The argument also shows that, under the hypotheses of the theorem,
dω(JX, JY, JZ) = dω(JX, Y, Z) + dω(X, JY, Z) + dω(X,Y, JZ).
Remark 2.9. According to LeBrun [7], if J on S2n is integrable and orthogonal with respect
to the standard Euclidean metric, then for V,W ∈ T 0,1 and the Levi-Civita connection ∇, we
have
∇VW =
1
2
[V,W ] ∈ T 0,1,
where the containment follows from integrability. The previous theorem implies in this case
that J is strongly integrable with respect to∇, (as also does Lemma 2.7). Note that∇VW ∈ T 0,1
is a priori weaker than the condition ∇VW = 1
2 [V,W ] ∈ T 0,1.
Corollary 2.10. If J is integrable then there exists a torsion free metric preserving connection ∇
such that J is strongly integrable with respect to ∇.
To contrast with the Kähler case, of course there need not exist a torsion free metric preserving
connection ∇ such that ∇J = 0.
Proof. Choose any metric compatible with J (for example, one may average any given metric
with respect to J) and then let ∇ be the Levi-Civita connection associated to this metric. �
3 Grassmannian of idempotent matrices
We begin with the definitions of the Grassmannian and its non-compact version, both presented
as a space of idempotent matrices (see, e.g., [9, Problem 5-C]).
Definition 3.1. For n ∈ N and 0 ≤ k ≤ n let
Ik,n =
{
P ∈Mn×n(C) |P 2 = P, rank(P ) = k
}
Results Concerning Almost Complex Structures on the Six-Sphere 7
be the space of idempotent n by n complex matrices of rank k, and let Gk,n ⊂ Ik,n be the
subspace of self-adjoint matrices, given by
Gk,n =
{
P ∈Mn×n(C) |P 2 = P, rank(P ) = k, P ∗ = P
}
,
where the adjoint is taken with respect to the standard complex inner product on Cn.
Note that P ∈ Gk,n if and only if P is an orthogonal projection of rank k. So, the latter
space Gk,n can be identified with the Grassmannian of k-planes in Cn, since there is a unique
orthogonal projection whose image is a given k-dimensional subspace. The space Ik,n can be
described as the space of two complementary subspaces of Cn, of dimensions k and n− k.
The relation P 2 = P implies, for any path Pt ∈ Ik,n, that
P ′tPt + PtP
′
t = P ′t ,
so that (I−Pt)P
′
t = P ′tPt and PtP
′
t = P ′t(I−Pt). For short we may write this as dPP = (I−P )dP
and PdP = dP (I − P ). The tangent spaces are
TPIk,n =
{
X ∈Mn×n(C) |XP = (I − P )X andPX = X(I − P )
}
and
TPGk,n =
{
X ∈Mn×n(C) |XP = P⊥X, PX = XP⊥, andX∗ = X
}
,
of dimensions 2k(n− k), and k(n− k), respectively.
Notice that P = [ 1 0
0 0 ] in any basis respecting the splitting Cn = Im(P ) ⊕ Ker(P ), so that
X ∈ TPIk,n if and only if X =
[
0 B
C 0
]
for some linear operators B : Ker(P ) → Im(P ) and
C : Im(P ) → Ker(P ). If X ∈ TPGk,n then Im(P ) and Ker(P ) are orthogonal and we have
C = B∗.
The subspaces Im(P ) and Ker(P ) form the fibers of two vector bundles Im → Ik,n and
Ker → Ik,n, of ranks k and n − k, respectively. By the above, the tangent bundle of Ik,n is
simply the bundle Hom(Ker, Im)⊕Hom(Im,Ker), and the tangent bundle of Gk,n is isomorphic
to Hom(Im,Ker), as expected. Note all of these bundles are contained inside a trivial complex
bundle.
In particular, the bundle Im → Ik,n is a sub-bundle of the trivial Cn bundle, so it inherits
a connection given by the trivial connection d on the trivial Cn bundle, followed (at each point P )
by the projection P onto Im(P ) ⊂ Cn. In short, ∇Im = P ◦ d. The curvature of this connection
(cf. [6, p. 344]) is given by the matrix-valued 2-form
R = P ∧ dP ∧ dP,
which we may write for short as PdP 2.
We will be interested in the closed C-valued 2-form1
ω =
1
2i
Tr
(
PdP 2
)
∈ Ω2(Ik,n;C).
Concretely, for X,Y ∈ TPIk,n given as above by X =
[
0 B
C 0
]
and Y =
[
0 D
E 0
]
, we have
ω(X,Y ) =
1
2i
Tr(PXY − PY X) =
1
2i
Tr(BE −DC).
1It is customary to include a factor of π in the denominator so that the resulting class is integral. In order to
simplify the presentation we drop this factor since it has no impact on the results here.
8 S.O. Wilson
Note that for X,Y ∈ TPGk,n we have
ω(X,Y ) =
1
2i
Tr(BD∗ −DB∗) = Im Tr(BD∗).
Now it is clear that ω is a Kähler form on Gk,n since for any X ∈ TPGk,n given by X =
[
0 B
B∗ 0
]
,
and the J operator on TPGk,n defining the complex structure on Gk,n given explicitly by
J
([
0 A
A∗ 0
])
=
[
0 −iA
iA∗ 0
]
,
we have
ω(X, JY ) = Im Tr
(
B(−iD)∗
)
= Re Tr(BD∗),
which is a real inner product on TPGk,n.
For completeness we record that Ik,n is a symplectic manifold, whose symplectic form Re(ω)
is tamed by many (possibly non-integrable) almost complex structures J , which by definition
means that Re(ω)(X, JX) > 0 for all X 6= 0. The form will be used in subsequent sections,
though the operators J considered in the proof will not.
Theorem 3.2. For each 0 < k < n, the closed real valued 2-form Re(ω) is a symplectic form
on the manifold Ik,n whose restriction to Gk,n is the Kähler form ω.
Proof. For any choice of inner product on Cn, the 2-form Re(ω) is tamed by the almost complex
structure on Ik,n given by
JX = J
([
0 B
C 0
])
=
[
0 −iC∗
iB∗ 0
]
,
since for X 6= 0
Re(ω)(X, JX) = Re
(
1
2i
Tr(B(iB∗)− (−iC∗)C)
)
=
1
2
Tr(BB∗ + C∗C) > 0. �
The imaginary part of the 2-form ω, defined on Ik,n, is d-exact. To see this, consider the
retraction Ik,n → Gk,n which assigns to a pair of complementary planes the unique orthogonal
pair which keeps the first plane fixed. This is a fibration with contractible fibers, therefore
a cohomology isomorphism, and the restriction of the complex form ω to Gk,n is real.
4 The canonical map
Definition 4.1. Let M be a 2k-dimensional submanifold of Rn, and let J be an almost complex
structure on M . At any point x ∈M we have
Cn = Rn ⊗ C = (TxM ⊕Nx)⊗ C =
(
T 0,1
x ⊕ (Nx ⊗ C)
)
⊕ T 1,0
x ,
where Nx is the fiber at x of the normal bundle N of M with respect to the standard Euclidean
metric.
The canonical map P : M → In−k,n is defined for x ∈M so that P (x) is the unique projection
on Cn with image equal to
(
T 0,1
x ⊕ (Nx ⊗ C)
)
and kernel equal to T 1,0
x . We’ll write
P = P− ⊕ πNx⊗C,
where πNx⊗C : Cn → Cn is orthogonal projection onto Nx ⊗ C.
Results Concerning Almost Complex Structures on the Six-Sphere 9
First we remark that we are using the non-compact Grassmannian in this section to obtain
Theorem 4.4 and its first two corollaries, which concern all almost complex structures. In the
next section, for results concerning the non-existence of strongly integrable almost complex
structures, it suffices to work with the compact Grassmannian.
Next we note there is obviously some choice here in defining the canonical map, where we
have chosen the map to have image Im(P−)⊕ (N ⊗ C), versus Im(P+)⊕ (N ⊗ C), or Im(P±).
The choice of Im(P−) = T 0,1 is motivated by Lemma 2.7, and we include the normal bundle
since the covariant derivative on the sphere with standard metric has a normal component, as
given in equation (4.1) below.
Remark 4.2. The decomposition TM ⊗ C = Im(P+) ⊕ Im(P−) is an orthogonal direct sum
decomposition if and only if J is orthogonal. It follows that the canonical map P : M → In−k,n
factors through Gn−k,n if and only if J is orthogonal.
Lemma 4.3. Let n = 1 or n = 3. For any J on S2n ⊂ R2n+1 the induced canonical map
P : S2n → In+1,2n+1 is an immersion.
Proof. The idea is to use the geometric fact that normal direction of the sphere moves tangen-
tially. Explicitly, if X ∈ TxS2n then dXP ∈ TP (x)In+1,2n+1 is a non-zero matrix since, for the
real normal unit vector nx to S2n at x, we have
(dXP )(nx) = dX(P (nx))− PdX(nx) = (I − P )dX(nx) = P+(X) =
1
2
(X − iJX).
So, dXP is non-zero whenever X is non-zero. �
Before stating the main result of this section, we make the following remarks. An almost
complex structure J satisfies Tr(J) = 0 at every point, Tr(∇XJ) = 0 for every tangent vector X,
and of course, Tr([∇XJ,∇Y J ]) = 0 for any two tangent vectors X and Y , where [−,−] denotes
the commutator. These also hold for the C-linear extension of these operators to TM ⊗ C.
Now, for any X and Y , the commutator [∇XJ,∇Y J ] preserves the splitting TM⊗C = T 1,0⊕
T 0,1 since it commutes with J . The statement below pertains to the restriction of [∇XJ,∇Y J ]
to T 0,1 which is denoted by
[∇XJ,∇Y J ]
∣∣
T 0,1 : T 0,1 → T 0,1.
Note that [∇XJ,∇Y J ]
∣∣
T 0,1 is the difference of two compositions T 0,1 → T 1,0 → T 0,1, in opposite
order, so the trace need not be zero, and is a priori an arbitrary complex number.
Recall the form Re(ω) = Re
(
1
2i Tr
(
PdP 2
))
∈ Ω2(In+1,2n+1;C) from Section 3, and the
canonical map P : S2n → In+1,2n+1 induced by an arbitrary J .
Theorem 4.4. Let 〈−,−〉 be the Euclidean metric of R2n+1 where n = 1 or n = 3. Let J be any
almost complex structure on the unit sphere S2n ⊂ R2n+1, with induced metric and Levi-Civita
connection ∇. Let P : S2n → In+1,2n+1 be the induced canonical map. The pullback of Re(ω)
along P is given by
P ∗Re(ω)(X,Y ) = −1
8
Im Tr
(
[∇XJ,∇Y J ]
∣∣
T 0,1
)
− 1
4
〈X, JY 〉+
1
4
〈Y, JX〉.
Proof. Let X and Y be fixed tangent vectors at a point x. As in Section 3, in any complex
basis of (TxM ⊗C)⊕ (Nx⊗C) for which the projection P (x) is of the form P (x) = [ 1 0
0 0 ] we have
dXP =
[
0 B
C 0
]
for complex matrices B : T 1,0
x → T 0,1
x ⊕ (Nx⊗C) and C : T 0,1
x ⊕ (Nx⊗C)→ T 1,0
x ,
which are the restriction of dXP to their appropriate domains.
10 S.O. Wilson
On the sphere with the standard Euclidean metric we have
dXW = ∇XW − 〈X,W 〉nx, (4.1)
where nx is the outward unit normal vector at x. This equation also holds for complex vector
fields W by extending the structures linearly over i ∈ C. For the remainder of this proof (only),
we will reserve the notation 〈−,−〉 for the extension of the standard metric in R2n+1 to a bilinear
map which is C-linear in both entries.
We first show that
B =
i
2
(∇XJ) + 〈X, −〉nx,
which is not surprising since P = 1
2(I+ iJ)⊕πNx⊗C. Explicitly, for any Z− iJZ ∈ T 1,0 we have
B(Z − iJZ) = (dXP )(Z − iJZ) = dX(P (Z − iJZ))− P (dX(Z − iJZ))
= −P (dX(Z − iJZ)) = −P (∇X(Z − iJZ)− 〈X,Z − iJZ〉nx)
= −P−∇X(Z − iJZ) + 〈X,Z − iJZ〉nx,
where the last equality follows from the definition P
∣∣
TM⊗C = P− and P (nx) = nx. Now using
P− = 1
2(I + iJ) we have
−P−∇X(Z − iJZ) = −P−∇X(P+Z) = ∇X(−P−P+Z) +∇X(P−)(P+Z)
=
1
2
∇X(I + iJ)(Z − iJZ) =
i
2
(∇XJ)(Z − iJZ),
which completes the proof that B = i
2(∇XJ) + 〈X, −〉nx.
Also, C
∣∣
T 0,1 = i
2(∇XJ) since
C(Z + iJZ) = (dXP )(Z + iJZ) = dX(P (Z + iJZ))− P (dX(Z + iJZ))
= dX(Z + iJZ)− P (dX(Z + iJZ)) = (I − P )dX(Z + iJZ)
= (I − P )(∇X(Z + iJZ)− 〈X,Z + iJZ〉nx)
= (I − P )(∇X(Z + iJZ)) = P+(∇X(Z + iJZ))
and
P+(∇X(Z + iJZ)) = P+(∇X(P−Z))
= ∇X
(
P+P−Z
)
−∇X
(
P+
)
(P−Z) =
i
2
(∇XJ)(Z + iJZ).
Finally,
C(nx) =
1
2
(X − iJX)
as in the proof of Lemma 4.3.
It follows that if dY P =
[
0 D
E 0
]
then
D(Z − iJZ) =
i
2
(∇Y J)(Z − iJZ) + 〈Y, Z − iJZ〉nx,
E(Z + iJZ) =
i
2
(∇Y J)(Z + iJZ),
E(nx) =
1
2
(Y − iJY ),
Results Concerning Almost Complex Structures on the Six-Sphere 11
so that
BE(Z + iJZ) = −1
4
(∇XJ)(∇Y J)(Z + iJZ) +
〈
X,
i
2
(∇Y J)(Z + iJZ)
〉
nx,
BE(nx) =
i
4
(∇XJ)(Y − iJY ) +
1
2
〈X, (Y − iJY )〉nx,
DC(Z + iJZ) = −1
4
(∇Y J)(∇XJ)(Z + iJZ) +
〈
Y,
i
2
(∇XJ(Z + iJZ))
〉
nx
DC(nx) =
i
4
(∇Y J)(X − iJX) +
1
2
〈Y,X − iJX〉nx.
The matrix BE−DC is a C-linear endomorphism of Im(P ) = T 0,1
x ⊕(Nx⊗C). Let Zk + iJZk
for k = 1, . . . , n be an orthonormal basis over C for T 0,1
x , so that this set along with the vector nx
is an orthonormal basis over C for Im(P ). For the remainder of this proof, let 〈〈−,−〉〉 denote
the standard inner product on C2n+1, which is C-linear in the first entry, and conjugate linear
in the second entry.
Then
Tr(BE −DC) =
∑
k
〈〈(BE −DC)(Zk + iJZk), Zk + iJZk〉〉+ 〈〈(BE −DC)nx, nx〉〉
=
∑
k
〈〈(BE −DC)(Zk + iJZk), Zk + iJZk〉〉+
1
2
〈X, (Y − iJY )〉 − 1
2
〈Y,X − iJX〉
=
∑
k
〈〈
−1
4
(∇XJ)(∇Y J)(Zk + iJZk) +
1
4
(∇Y J)(∇XJ)(Zk + iJZk), Zk + iJZk
〉〉
+
1
2
〈X,Y 〉 − 1
2
〈Y,X〉+
i
2
(〈Y, JX〉 − 〈X, JY 〉)
=
1
4
∑
k
〈〈[∇Y J,∇XJ ](Zk + iJZk), Zk + iJZk〉〉+
i
2
(〈Y, JX〉 − 〈X, JY 〉).
Since Re(ω) = Re
(
1
2i Tr(BE −DC)
)
we have
P ∗(Reω)(X,Y ) = −1
8
Im Tr
(
[∇XJ,∇Y J ]
∣∣
T 0,1
)
− 1
4
〈X, JY 〉+
1
4
〈Y, JX〉. �
For the following corollaries we continue to consider the round sphere with standard metric
〈−,−〉 and Levi-Civita connection ∇.
Corollary 4.5. No almost complex structure J on S6 satisfies2 that ∇XJ and ∇JXJ are eve-
rywhere commuting for all vectors X.
Proof. By Theorem 4.4, if ∇XJ and ∇JXJ always commute then
P ∗(Reω)(X, JX) =
1
4
〈X,X〉+
1
4
〈JX, JX〉 > 0,
implying S6 has a closed non-degenerate 2-form, which is a contradiction since H2
(
S6
)
= 0. �
More generally, we conclude that the topology of S6 restricts the possible imaginary trace of
[∇XJ,∇JXJ ]
∣∣
T 0,1 .
2This result, and all of those below that rely on H2
(
S6
)
= 0, can be sharpened to state that the “disqualifying
condition” cannot happen along the image of any closed J-holomorphic curve Σ → S6 which is an immersion
on a full measure subset, by Stokes’ theorem. One can show that for an integrable J on S6, there is at most
a discrete set of (image) curves which are not multiply covered. The sharpened statements will not be mentioned
in the subsequent corollaries.
12 S.O. Wilson
Corollary 4.6. For any almost complex structure J on S6, there is some non-zero tangent
vector X at some point in S6 such that
Im Tr
(
[∇XJ,∇JXJ ]
∣∣
T 0,1
)
≥ 2
(
‖X‖2 + ‖JX‖2
)
.
In particular, there is a unit tangent vector X for which
Im Tr
(
[∇XJ,∇JXJ ]
∣∣
T 0,1
)
≥ 2.
Proof. If the inequality were false for all X, then by Theorem 4.4 P ∗(Reω) would be non-
degenerate, thereby contradicting H2
(
S6
)
= 0. The second case follows from the first with
‖X‖ = 1. �
Corollary 4.7 (Blanchard [1], Lebrun [7]). No orthogonal almost complex structure on the
round sphere S6 is integrable.
Both [1] and [7] use in their arguments a certain J-holomorphic map, which is closely related
to the canonical map we’re using here. A similar approach will be taken in the next section to
conclude a stronger result. Here we’ll re-prove the claim by calculation, using Theorem 4.4.
Proof. If J were integrable and orthogonal, then by Theorem 2.8, J is strongly integrable with
respect to ∇, i.e., ∇JXJ = J∇XJ for any X. Then
[∇XJ,∇JXJ ] = (∇XJJ∇XJ − J∇XJ∇XJ) = −2J(∇XJ)2,
so that(
P ∗Reω
)
(X,JX) =
1
4
Im Tr
(
J(∇XJ)2
∣∣
T 0,1
)
+
1
4
〈X,X〉+
1
4
〈JX, JX〉.
Now, letting Zk + iJZk for k = 1, . . . , n be an orthonormal basis for T 0,1 we have
Im Tr
(
J(∇XJ)2
∣∣
T 0,1
)
= Im
∑
k
〈〈
J(∇XJ)2(Zk + iJZk), Zk + iJZk
〉〉
=
∑
k
(〈〈
J(∇XJ)2JZk, Zk
〉〉
−
〈〈
J(∇XJ)2Zk, JZk
〉〉)
= −2
∑
k
〈〈
(∇XJ)2Zk, Zk
〉〉
= 2
∑
k
‖(∇XJ)Zk‖2,
where in the last step we used that J is skew, so that (∇XJ)∗ = (∇XJ
∗) = −∇XJ . This shows
(P ∗Reω)(X, JX) > 0, which again contradicts H2
(
S6
)
= 0. �
It does not appear that using Theorem 4.4 alone we can relax the previous theorem’s hypothe-
ses (of integrable and orthogonal) to strong integrability; it is possible that Im Tr
(
J(∇XJ)2
∣∣
T 0,1
)
is negative and large in absolute value. In the next section we provide a more conceptual way
to show that there are no almost complex structures on S6 which are strongly integrable with
respect to the Levi-Civita connection on the round sphere.
5 Grassmann map
In this section we let n = 1 or n = 3, S2n ⊂ R2n+1 be the unit sphere with respect to the
standard Euclidean metic, and let ∇ be the Levi-Civita connection. Recall that at any point
x ∈ S2n we have
C2n+1 =
(
T 0,1
x ⊕ (Nx ⊗ C)
)
⊕ T 1,0
x .
Results Concerning Almost Complex Structures on the Six-Sphere 13
Definition 5.1. For any J on S2n, the induced map P⊥ : S2n → Gn+1,2n+1 is defined so
that P⊥(x) is the orthogonal projection of C2n+1 onto T 0,1
x ⊕ (Nx ⊗ C).
Lebrun [7] considers the closely related map which is given by orthogonal projection onto T 0,1
x ,
and shows it is an embedding. We’ll see below that by including the normal bundle in the new
map above, the prohibited condition on S6 becomes “strongly integrable” rather than “integrable
and orthogonal”. This again is motivated by the normal component in equation (4.1).
Theorem 5.3 and Corollary 5.4 below are very much inspired by the author’s reading of [7].
First, we show this map is an immersion, which is sufficient for our purposes.
Lemma 5.2. For any J on S2n, the map P⊥ : S2n → Gn+1,2n+1 is an immersion.
Proof. If X ∈ TxS2n then dXP ∈ TP (x)Gn+1,2n+1 is a non-zero matrix since, for the real normal
unit vector nx to S2n at x, we have(
dXP
⊥)(nx) = dX
(
P⊥(nx)
)
− P⊥dX(nx) =
(
I − P⊥
)
dX(nx) =
(
I − P⊥
)
(X).
This shows dXP is non-zero whenever X is non-zero since X ∈ TxM is real, but P⊥(X) ∈ T 0,1
is not real. �
Recall that the tangent algebra m associated to J and ∇ is given by
m(X,Y ) = ∇JX(J)Y − J∇X(J)Y,
and that J is strongly integrable with respect to ∇ if and only if ∇XJ is J-linear in the vari-
able X, i.e., m vanishes.
For a map f : (M,J)→ (N,K) between almost complex manifolds, we define
∂Xf = dJXf −K ◦ dXf.
Theorem 5.3. For any J on S2n, the map P⊥ : S2n → Gn+1,2n+1 induced by J satisfies
∂XP
⊥(nx) = 0,
∂XP
⊥(Y + iJY ) =
(
I − P⊥
)
(Jm(X,Y )),
where m(X,Y ) = ∇JX(J)Y − J∇X(J)Y is the tangent algebra.
Furthermore, ∂P⊥ ≡ 0 if and only if J is strongly integrable with respect to ∇, i.e., the
tangent algebra m vanishes.
Proof. Recall from Section 3 that the complex structure on TPGn+1,2n+1 is given by multipli-
cation by +i on Hom
(
Im
(
P⊥
)
,Ker
(
P⊥
))
, and by self-adjointness, is given by multiplication
by −i on Hom
(
Ker
(
P⊥
)
, Im
(
P⊥
))
.
As in the previous lemma we have
∂XP
⊥(nx) = −i
(
dX+iJXP
⊥)(nx) = −i
(
I − P⊥
)
(X + iJX) = 0,
and, for arbitrary Z ∈ T 0,1
x , we have(
dX+iJXP
⊥)(Z) =
(
I − P⊥
)
dX+iJX(Z) =
(
I − P⊥
)
(∇X+iJX(Z)− 〈X + iJX,Z〉nx)
=
(
I − P⊥
)
(∇X+iJX(Z)),
since P⊥(nx) = nx. Now, if Z = Y + iJY , then the last expression is equal to(
I − P⊥
)(
∇XY −∇JX(JY ) + i(∇X(JY ) +∇JXY )
)
=
(
I − P⊥
)
i
(
∇X(JY ) +∇JXY − J(∇XY −∇JX(JY ))
)
= i
(
I − P⊥
)(
J(∇JX(J)Y − J∇X(J)Y )
)
= i
(
I − P⊥
)
(Jm(X,Y )).
14 S.O. Wilson
So,
∂XP
⊥(Y + iJY ) = −i
(
dX+iJXP
⊥)(Y + iJY ) = (I − P⊥)(Jm(X,Y )),
as claimed. Since Jm(X,Y ) is real, the right hand side vanishes if and only if m vanishes, i.e.,
J is strongly integrable with respect to ∇.
The remainder of the proof, showing that ∂P⊥ ≡ 0 when m vanishes, is entirely formal since
the restriction dP⊥ :
(
T 0,1
)⊥ → T 0,1 is the adjoint of the restriction dP⊥ : T 0,1 →
(
T 0,1
)⊥
.
Explicitly, for any W + iJZ =
(
T 0,1
)⊥
we have with respect to the complex inner product
〈−,−〉,〈(
dX−iJXP
⊥)(W + iJZ), Y + iJY
〉
=
〈
W + iJZ,
(
dX+iJXP
⊥)(Y + iJY )
〉
= 0
since P⊥ is always self adjoint. Also, for U + iV ∈
(
T 0,1
)⊥
, we have〈(
dX−iJXP
⊥)(W + iJZ), U + iJV
〉
= 0,
since dX−iJXP
⊥ :
(
T 0,1
)⊥ → T 0,1, because P⊥ is a projection operator. �
Note that by Theorem 2.8, any integrable orthogonal J on S6 is strongly integrable with
respect to ∇, whereas locally there are non-orthogonal almost complex structures on S6 which
are strongly integrable with respect to a torsion free connection. For example, from Euclidean
space we have those non-orthogonal almost complex structures which are covariantly constant.
Nevertheless, we can conclude:
Corollary 5.4. There are no almost complex structures on S6 which are strongly integrable with
respect to the Levi-Civita connection ∇ of the standard metric.
Proof. By the Theorem 5.3, if J is strongly integrable with respect to ∇, then P⊥ : S6 → G4,7
is a J-holomorphic immersion. Since G4,7 is a Kähler manifold, this shows the pullback
(
P⊥
)∗
ω
is closed and non-degenerate, which contradicts H2
(
S6
)
= 0. �
Note that in Theorem 5.3 and Corollary 5.4 we did not use the explicit formula for the
Kähler form ω, but rather only the existence of such a structure compatible with the complex
structure on Gk,n. While this was sufficient to conclude the pullback
(
P⊥
)∗
ω is non-degenerate,
it is far from necessary. We next give a necessary and sufficient condition for
(
P⊥
)∗
ω to be
non-degenerate, which is intrinsic in that it depends only on J , the metric and the covariant
derivatives of J .
To do so, we calculate the pullback form
(
P⊥
)∗
ω explicitly, but first we need a lemma. This
lemma should be thought of as the non-orthogonal analogue of the standard way in which any
orthogonal J determines a compatible 2-form tamed by J .
Lemma 5.5. Let V be a real vector space with complex structure J and real inner product
〈−,−〉. Extend 〈−,−〉 to a complex inner product on W = V ⊗C ∼= V ⊕ iV and let Q : W →W
be orthogonal projection onto T 0,1(V, J), i.e., the −i eigenspace of J . Then Q = R + iM for
unique real linear operators R and M on V , and we have
(1) R−MJ = Id;
(2) R∗ = R and M∗ = −M ;
(3) Tr(Q) = dim(V )/2;
(4) R2 −M2 = R and RM +MR = M .
Results Concerning Almost Complex Structures on the Six-Sphere 15
Conversely, any real operators R and M on V satisfying the above properties determine the
operator Q = R+ iM to be the orthogonal projection onto T 0,1, and satisfy
(a) MJ = −J∗M ;
(b) M2J − J∗M2 = −M ;
(c) ν(X,Y ) := 〈MX,Y 〉 is a 2-form, tamed by and compatible with J , i.e., there is an asso-
ciated positive definite inner product
(X,Y ) := ν(X, JY ),
and J is orthogonal with respect to ν and ( , ).
Proof. Let R = Re(Q) and M = Im(Q). The first condition follows since Q is the identity
on T 0,1
(R+ iM)(Z + iJZ) = (R−MJ)Z + i(RJ +MZ) = Z + iJZ.
The second condition follows since Q is self adjoint, the third condition since Q is a projection
of rank dim(V )/2, and the fourth since Q2 = Q. The converse holds since orthogonal projection
onto T 0,1 is the unique self adjoint projection of rank dim(V )/2 and image T 0,1. Condition (a)
follows from (1) and (2) since R− Id is self adjoint, and M is skew-adjoint. Next, condition (b)
is given by(
M2J − J∗M2) = MMJ +MJM = M(R− I) + (R− I)M = MR+RM − 2M = −M.
The inequality
0 ≤ ‖X‖2 − ‖QX‖2 = 〈(I −Q)X,X〉 = 〈(I −R)X,X〉+ i〈MX,X〉
shows 〈MX,X〉 = 0 and
(X,X) = 〈−MJX,X〉 = 〈(I −R)X,X〉 ≥ 0.
Finally, η is alternating by (2), and the rest follows
(X,Y ) = 〈MX,JY 〉 = 〈X, J∗MY 〉 = 〈JX,MY 〉 = (Y,X),
and
ν(JX, JY ) = 〈MJX, JY 〉 = 〈−J∗MX,JY 〉 = ν(X,Y ),
and
(JX, JY ) = −ν(JX, Y ) = ν(Y, JX) = (Y,X) = (X,Y ). �
Example 5.6. If J is orthogonal on V with respect to a given inner product, then R = 1
2 Id
and M = 1
2J . The last condition in the lemma is then a multiple of the usual assignment
ω(X,Y ) := (JX, Y ).
We can apply the preceding algebraic lemma at every tangent space to give an explicit
description of the pullback form
(
P⊥
)∗
ω along the map P⊥ : S2n → Gn+1,2n+1.
Proposition 5.7. The pullback of the Kähler form ω on Gn+1,2n+1 along P⊥ is given by(
P⊥
)∗
ω(X,Y ) =
∑
k
ν
(
(∇XJ)Zk, (∇Y J)Zk
)
+ ν(X,Y ), (5.1)
where the sum is over any Zk such that {Zk + iJZk} is an orthonormal basis of T 0,1, and
ν(α, β) = 〈Mα,β〉 is the associated 2-form from Lemma 5.5.
16 S.O. Wilson
Proof. We first show that
dXP
⊥∣∣
T 0,1 = M∇XJ.
Letting Z + iJZ ∈ T 0,1 be arbitrary, we calculate
dXP
⊥(Z + iJZ) =
(
I − P⊥
)
dX(Z + iJZ) =
(
I − P⊥
)(
∇X(Z + iJZ)− 〈X,Z + iJZ〉nx
)
=
(
I − P⊥
)
(∇X(Z + iJZ)).
By definition P⊥ = Q
⊥
⊕ πN where Q is the orthogonal projection of TS2n⊗C onto T 0,1 and πN
is orthogonal projection onto the normal bundle, so
dXP
⊥(Z + iJZ) = (I −Q⊕ πN )(∇X(Z + iJZ))
= (I −Q)(∇X(Z + iJZ)) = (∇XQ)(Z + iJZ).
Now, letting Q = R+ iM , as in the previous lemma,
(∇XR+ i∇XM)(Z + iJZ) = (∇XR)Z − (∇XM)(JZ) + i
(
(∇XM)(Z) + (∇XR)(JZ)
)
.
By the lemma, R−MJ = Id so that
∇XR− (∇XM)J −M(∇XJ) = 0,
so we conclude
dXP
⊥(Z + iJZ) = M∇X(J)Z + iM∇X(J)(JZ) = M∇X(J)(Z + iJZ).
Recall that the Kähler form ω(X,Y ) on Gk,n is given by Im Tr(BD∗), so that the pull-
back
(
P⊥
)∗
ω is given by(
P⊥
)∗
ω(X,Y ) = Im Tr
(
dXP
⊥∣∣
(T 0,1⊕N)⊥
◦ dY P⊥
∣∣
T 0,1⊕N
)
,
where we have made the substitutions B = dXP
⊥∣∣
(T 0,1⊕N)⊥
: Ker
(
P⊥
)
→ Im
(
P⊥
)
and D∗ =
dY P
⊥∣∣
T 0,1⊕N : Im
(
P⊥
)
→ Ker
(
P⊥
)
.
For any orthonormal basis {Zk + iJZk} ∪ {n} of T 0,1 ⊕N with respect to the complex inner
product 〈−,−〉, which is conjugate linear in the second variable, we use the fact that the adjoint
of B is B∗ = dXP
⊥∣∣
T 0,1⊕N : Im
(
P⊥
)
→ Ker
(
P⊥
)
, and have(
P⊥
)∗
ω(X,Y ) = Im
〈(
dY P
⊥∣∣
T 0,1⊕N
)
n,
(
dXP
⊥∣∣
T 0,1⊕N
)
n
〉
+
∑
k
Im
〈(
dY P
⊥∣∣
T 0,1⊕N
)
(Zk + iJZk),
(
dXP
⊥∣∣
T 0,1⊕N
)
(Zk + iJZk)
〉
.
For the first summand on the right hand side we use(
dXP
⊥∣∣
T 0,1⊕N
)
n =
(
I − P⊥
)
∇Xn =
(
I − P⊥
)
(X) = X −RX − iMX,
and Lemma 5.5 to obtain
Im
〈(
dY P
⊥∣∣
T 0,1⊕N
)
n,
(
dXP
⊥∣∣
T 0,1⊕N
)
n
〉
= 〈Y −RY − iMY,X −RX − iMX〉
= 〈(I −R)Y,MX〉 − 〈MY, (I −R)X〉 = 〈MX,Y 〉.
For the latter summation term we have
Im
〈(
dY P
⊥∣∣
T 0,1
)
(Zk + iJZk),
(
dXP
⊥∣∣
T 0,1
)
(Zk + iJZk)
〉
= 〈(M∇Y J)(JZk), (M∇XJ)(Zk)〉 − 〈(M∇Y J)(Zk), (M∇XJ)(JZk)〉
=
〈(
M2J∇Y J
)
(Zk), (∇XJ)(Zk)
〉
− 〈(M∇Y J)(Zk),−MJ(∇XJ)(Zk)〉
=
〈(
M2J − J∗M2
)
(∇Y J)(Zk), (∇XJ)(Zk)
〉
= 〈M(∇XJ)(Zk), (∇Y J)(Zk)〉,
where in the last step we have we have used Lemma 5.5. �
Results Concerning Almost Complex Structures on the Six-Sphere 17
Remark 5.8. Proposition 5.7 also verifies (in another way) Corollary 5.4, for if ∇JXJ = J∇XJ ,
then by Lemma 5.5 part (c) we have(
P⊥
)∗
ω(X, JX) =
∑
k
‖(∇XJ)Zk‖2 + ‖X‖2 > 0,
where ‖ · ‖is the norm induced by the inner product (α, β) := ν(α, Jβ). In the case that J is
orthogonal, this agrees with the proof in Corollary 4.7, by Example 5.6.
For the remainder of this section, consider the 2-form given by
η(X,Y ) =
∑
k
ν
(
(∇XJ)Zk, (∇Y J)Zk
)
,
where the sum is over any Zk such that {Zk + iJZk} is an orthonormal basis of T 0,1, and
ν(α, β) = 〈Mα,β〉 is the associated 2-form from Lemma 5.5. It is an artifact of the previous
proof that this form η on S6 does not depend on the choice of {Zk}, and we have(
P⊥
)∗
ω(X,Y ) = η(X,Y ) + ν(X,Y ).
The necessary degeneracy of this form on S6 implies
Corollary 5.9. For any J on S6, integrable or not, there are some tangent vectors X, Y at
some point for which
η(X,Y ) + ν(X,Y ) = 0.
Let (−,−) be the induced positive definite inner product from Lemma 5.5 defined by (X,Y ) =
ν(X, JY ) = 〈MX,JY 〉, with associated norm ‖ · ‖.
Corollary 5.10. If J on S6 satisfies the first order differential inequality∑
k
(
(∇XJ)Zk, Jm(Zk, X)
)
6=
∑
k
‖(∇XJ)Zk‖2 + ‖X‖2 (5.2)
for all non-zero X, then J is not integrable.
In particular, if the tangent algebra m satisfies(
(∇XJ)Z, Jm(Z,X)
)
≤ ‖(∇XJ)Z‖2, (5.3)
for all X and Z, or the weaker condition
‖m(Z,X)‖ ≤ ‖(∇XJ)Z‖, (5.4)
for all X and Z, then J is not integrable.
There are a plethora of local almost complex structures in dimension 6 for which equation (5.2)
does hold, since it is an open condition that holds for every local integrable orthogonal almost
complex structure. Note that geometrically, condition equation (5.3) is an open half-space
restriction on m in terms of ∇J , whereas equation (5.4) is an open sub-disk restriction.
Proof. Recall from equation (2.2), J is integrable if and only if
(∇Y J)Z = −J(∇JY J)Z + J(∇JZJ − J∇ZJ)Y = −J(∇JY J)Z + Jm(Z, Y )
18 S.O. Wilson
for all vectors Y and Z. This condition gives
η(X,Y ) =
∑
k
ν
(
(∇XJ)Zk,−J(∇JY J)Zk
)
+
∑
k
ν
(
(∇XJ)Zk, Jm(Zk, Y )
)
.
Now letting Y = JX we have
η(X,JX) =
∑
k
‖(∇XJ)Zk‖2 −
∑
k
(
(∇XJ)Zk, Jm(Zk, X)
)
,
where we have used m(Zk, JX) = −Jm(Zk, X). Since(
P⊥
)∗
ω(X, JX) = η(X, JX) + ν(X,JX) = η(X,JX) + ‖X‖2,
we conclude that if∑
k
‖(∇XJ)Zk‖2 −
∑
k
(
(∇XJ)Zk, Jm(Zk, X)
)
+ ‖X‖2 6= 0
for all X, then J is not integrable. The final claim follows from the Cauchy–Schwarz inequa-
lity. �
The remainder of this section is meant to explain the meaning and applicability of the previous
corollary. First, we have already seen an example that distinguishes between the conditions in
Corollary 5.10.
Example 5.11. The above Example 2.4 in dimension 2 provides a J which is not strongly
integrable with respect to the standard metric, the bound in equation (5.4) does not hold, but
the bound in equation (5.3) holds for 0 < x < 1, and therefore equation (5.2) holds for such x as
well. Therefore, the bound in equation (5.4) is a priori weaker than the bound in equation (5.2),
which is a priori weaker than being strongly integrable.
To gain some further perspective on the meaning of equation (5.2) in Corollary 5.10, it is
instructive to consider the situation for a general J on S2, with the standard metric, for which
we have the following surprisingly pleasant geometric result.
Proposition 5.12. Let J be an arbitrary almost complex structure on S2. The pullback form(
P⊥
)∗
ω ∈ Ω2(S2) is non-degenerate at p ∈ S2 if and only if for some vector Xp at p,
det(dFp) 6= ‖Xp + iJXp‖2,
where dFp is the Jacobian of the smooth mapping Fp : TpS
2 → TpS
2 defined by Fp(w) = J̃w(Xp),
and the almost complex structure J̃ on the manifold TpS
2 is given by the pullback of J on S2 by
stereographic projection.
Since every almost complex structure in dimension two is integrable, and for integrable almost
complex structures (5.2) holds precisely when
(
P⊥
)∗
ω is non-degenerate, this determines all
almost complex structures on S2 for which (5.2) holds.
Proof. We perform a local calculation of the form
(
P⊥
)∗
ω at a given point p in the sphere
using stereographic projection. Any J on S2 can be pulled back via these coordinates to TpS
2,
which can be written uniquely in the standard basis {∂x, ∂y} as
J̃ =
[
f −1−f2
g
g −f
]
Results Concerning Almost Complex Structures on the Six-Sphere 19
for some functions f, g : R2 → R, with g nowhere zero. Moreover, on any fixed disk about the
origin in TpS
2, every such J̃ occurs in this way.
Without loss of generality, we assume p ∈ S2 is the south pole of the sphere, with stereographic
projection defined by projection on the tangent plane at the south pole. The standard metric
on the sphere in these stereographic coordinates is given by
ds2 = h(x, y)
(
dx2 + dy2
)
, where h(x, y) =
1(
1 + x2 + y2
)2 .
Using the expression for the Christoffel symbols in terms of the metric,
Γ`
jk =
1
2
(
ds2
)lr(
∂k
(
ds2
)
rj
+ ∂j
(
ds2
)
rk
− ∂r
(
ds2
)
jk
)
,
we have
Γx
xx = Γy
xy = Γy
yx = −Γx
yy =
hx
2h
, Γy
yy = Γx
xy = Γx
yx = −Γy
xx =
hy
2h
.
One then calculates
(∇∂xJ)∂x =
(
fx +
hy
2h
(
g − 1 + f2
g
))
∂x +
(
gx −
fhy
h
)
∂y, (5.5)
(∇∂yJ)∂x =
(
fy −
hx
2h
(
g −
(
1 + f2
)
g
))
∂x +
(
gy +
fhx
h
)
∂y. (5.6)
Note that since this is dimension two,
(
P⊥
)∗
ω is non-degenerate at p if and only if
(
P⊥
)∗
ω(X, JX)
6= 0 for some X at p. We next calculate
(
P⊥
)∗
ω(X,JX), and without loss of generality we may
assume X = ∂x.
From equation (5.1) of Proposition 5.7 we have(
P⊥
)∗
ω(X, JX) = −〈M(∇∂xJ)Z, (∇J∂xJ)Z〉+ 〈M∂x, J∂x〉,
where Z is any vector such that Z+ iJZ is unit vector spanning T 0,1, and 〈−,−〉 is the standard
round metric, which equals the metric ds2 above at p. We choose
Z = c∂x, where c =
1√
1 + f2 + g2
so that c(∂x + i(f∂x + g∂y)) is a unit vector.
We have P⊥(Y ) = 〈Y, Z + iJZ〉(Z + iJZ), and M = ImP⊥, so that
P⊥(∂x) =
(
c2∂x + c2f(f∂x + g∂y)
)
+ ic2g∂y,
and
M(∂x) = c2g∂y and M(∂y) = −c2g∂x.
Next we have (∇J)Z = c(∇J)∂x, so using h(0, 0) = 1 and hx(0, 0) = hy(0, 0) = 0, we have from
equation (5.5) that
M(∇∂xJ)∂x = M(fx∂x + gx∂y) = c2g(fx∂y − gx∂x).
Since
(
P⊥
)∗
ω is alternating and J∂x = f∂x + g∂y, we have from equation (5.6)(
P⊥
)∗
ω(X, JX) = −c2g〈M(∇∂xJ)∂x, (∇∂yJ)∂x〉+ c2g2〈∂y, ∂y〉
= −c4g2〈fx∂y − gx∂x, fy∂x + gy∂y〉+ c2g2 = −c4g2(fxgy − gxfy) + c2g2.
20 S.O. Wilson
Since g is nowhere zero, this is non-zero if and only if
fxgy − gxfy 6= 1 + f2 + g2, i.e., det(dF ) 6= ‖∂x + iJ∂x‖2,
where F (x, y) = (f(x, y), g(x, y)) = J̃(∂x). �
Thus, on S2, the condition that the pullback form
(
P⊥
)∗
ω is degenerate is quite rare, though
one can construct almost complex structures on S2 such that equation (5.2) of Corollary 5.10
fails to hold.
By a similar local calculation for R6, we see that the pullback form
(
P⊥
)∗
ω of an arbitrary J
on S6 can be degenerate at a point, even if J is integrable near that point. In fact, one can
consider an integrable J in a neighborhood of the origin in R6 which splits into the direct sum
of an almost complex structure K on R2 of the form the proof of Proposition 5.12, together with
two standard almost complex structures on R2, making this J orthogonal at the origin of R6
(only). Then the same calculation shows the transport of this local J to S6 (extended smoothly
to the whole sphere) satisfies that
(
P⊥
)∗
ω ∈ Ω2
(
S6
)
is degenerate if K is degenerate.
We next show that the bound in (5.3) in Corollary 5.10 allows us to conclude considerably
more than one would a priori know for an arbitrary manifold. To see this, we first consider the
left hand side of the inequality in (5.3) as a quadratic form on each tangent space.
Definition 5.13. Let (M,J, 〈−,−〉) be an almost complex Riemmannian manifold, with asso-
ciated Levi-Civita connection ∇ and tangent algebra m. Let (−,−) be the induced metric from
Lemma 5.5.
For any Z ∈ TxM define
QZ(X) =
(
(∇XJ)Z, Jm(Z,X)
)
.
Note that QJZ(X) = QZ(X) for all Z and X.
Note that this quadratic form vanishes identically whenever J is strongly integrable with
respect to ∇. Next we show that the condition that QZ(X) is negative definite, even at any one
point, already guarantees J to be non-integrable on any manifold.
Proposition 5.14. Let (M,J, 〈−,−〉) be as above, and let QZ be the associated quadratic form
for any fixed Z. If there are any two non-zero tangent vectors X and Z at any point of M for
which QZ(X) < 0 and QZ(JX) < 0, then J is not integrable.
Proof. We show that if J is integrable then, the form
K(X) = QZ(X) +QZ(JX)
is positive semi-definite, so that for each X,
QZ(X) ≥ 0 or QZ(JX) ≥ 0.
Sincem is J-anti-linear and J is orthogonal with respect to the metric (−,−) from Lemma 5.5,
we have
QZ(JX) +QZ(X) =
(
(∇JXJ)Z, Jm(Z, JX)
)
+
(
(∇XJ)Z, Jm(Z,X)
)
=
(
(∇JXJ)Z,m(Z,X)
)
−
(
J(∇XJ)Z,m(Z,X)
)
=
(
m(X,Z),m(Z,X)
)
.
Recall, J is integrable if and only if m(X,Z) = m(Z,X) for all X and Z, so in this case
QZ(JX) +QZ(X) = ‖m(X,Z)‖2 ≥ 0. �
Results Concerning Almost Complex Structures on the Six-Sphere 21
Thus, what is gained from Corollary 5.10 when (5.3) holds on S6, beyond what is a priori
true for all manifolds, is the case that the left hand side of (5.3) is nowhere negative definite.
Finally, we mention that in [2] it was shown that there is an open set, containing all orthogonal
almost complex structures, none of which are integrable. It would be interesting to see how this
open set compares to the set of almost complex structures which are forbidden to be integrable
by Corollary 5.10.
Acknowledgements
I gratefully acknowledge the Queens College sabbatical/fellowship leave program, which provided
me with time to conduct some of this research. I thank Arthur Parzygnat for comments on
a preliminary version of this paper, and also thank the referees for their suggestions, which have
improved this paper.
References
[1] Blanchard A., Recherche de structures analytiques complexes sur certaines variétés, C. R. Acad. Sci. Paris
236 (1953), 657–659.
[2] Bor G., Hernández-Lamoneda L., The canonical bundle of a Hermitian manifold, Bol. Soc. Mat. Mexicana
5 (1999), 187–198.
[3] Borel A., Serre J.P., Détermination des p-puissances réduites de Steenrod dans la cohomologie des groupes
classiques. Applications, C. R. Acad. Sci. Paris 233 (1951), 680–682.
[4] Bryant R., S.-S. Chern’s study of almost-complex structures on the six-sphere, arXiv:1405.3405.
[5] Hopf H., Zur Topologie der komplexen Mannigfaltigkeiten, in Studies and Essays Presented to R. Courant
on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, 1948, 167–185.
[6] Karoubi M., Leruste C., Algebraic topology via differential geometry, London Mathematical Society Lecture
Note Series, Vol. 99, Cambridge University Press, Cambridge, 1987.
[7] LeBrun C., Orthogonal complex structures on S6, Proc. Amer. Math. Soc. 101 (1987), 136–138.
[8] McDuff D., Salamon D., J-holomorphic curves and symplectic topology, American Mathematical Society
Colloquium Publications, Vol. 52, 2nd ed., Amer. Math. Soc., Providence, RI, 2012.
[9] Milnor J.W., Stasheff J.D., Characteristic classes, Annals of Mathematics Studies, Vol. 76, Princeton Uni-
versity Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1974.
[10] Newlander A., Nirenberg L., Complex analytic coordinates in almost complex manifolds, Ann. of Math. 65
(1957), 391–404.
[11] Salamon S.M., Hermitian geometry, in Invitations to Geometry and Topology, Oxford Graduate Texts in
Mathematics, Vol. 7, Oxford University Press, Oxford, 2002, 233–291.
[12] Tang Z., Curvature and integrability of an almost Hermitian structure, Internat. J. Math. 17 (2006), 97–105,
.
https://arxiv.org/abs/1405.3405
https://doi.org/10.1017/CBO9780511629372
https://doi.org/10.1017/CBO9780511629372
https://doi.org/10.2307/2046564
https://doi.org/10.2307/1970051
https://doi.org/10.1142/S0129167X0600331X
1 Introduction
2 Integrability
3 Grassmannian of idempotent matrices
4 The canonical map
5 Grassmann map
References
|
| id | nasplib_isofts_kiev_ua-123456789-209538 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:56:54Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Wilson, S.O. 2025-11-24T10:46:47Z 2018 Results Concerning Almost Complex Structures on the Six-Sphere / S.O. Wilson // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 12 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53C15; 32Q60; 53A07 arXiv: 1610.09620 https://nasplib.isofts.kiev.ua/handle/123456789/209538 https://doi.org/10.3842/SIGMA.2018.034 For the standard metric on the six-dimensional sphere, with Levi-Civita connection ∇, we show there is no almost complex structure J such that ∇XJ and ∇JXJ commute for every X, nor is there any integrable J such that ∇JXJ = J∇XJ for every X. The latter statement generalizes a previously known result on the non-existence of integrable orthogonal almost complex structures on the six-sphere. Both statements have refined versions, expressed as intrinsic first-order differential inequalities depending only on J and the metric. The new techniques employed include an almost-complex analogue of the Gauss map, defined for any almost-complex manifold in Euclidean space. I gratefully acknowledge Queens College's sabbatical/fellowship leave program, which provided me with time to conduct some of this research. I thank Arthur Parzygnat for comments on a preliminary version of this paper, and also thank the referees for their suggestions, which have improved this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Results Concerning Almost Complex Structures on the Six-Sphere Article published earlier |
| spellingShingle | Results Concerning Almost Complex Structures on the Six-Sphere Wilson, S.O. |
| title | Results Concerning Almost Complex Structures on the Six-Sphere |
| title_full | Results Concerning Almost Complex Structures on the Six-Sphere |
| title_fullStr | Results Concerning Almost Complex Structures on the Six-Sphere |
| title_full_unstemmed | Results Concerning Almost Complex Structures on the Six-Sphere |
| title_short | Results Concerning Almost Complex Structures on the Six-Sphere |
| title_sort | results concerning almost complex structures on the six-sphere |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209538 |
| work_keys_str_mv | AT wilsonso resultsconcerningalmostcomplexstructuresonthesixsphere |