Balkan Workshop BW2013 Beyond the Standard Models 25 – 29 April, 2013, Vrnjačka Banja, Serbia M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein Spaces

1. Hidden symmetries of the
five-dimensional Sasaki-Einstein metrics
Mihai Visinescu
Department of Theoretical Physics
National Institute for Physics and Nuclear Engineering ”Horia Hulubei”
Bucharest, Romania
BALKAN WORKSHOP 2013
– Beyond the Standard Models –
Vrnjaˇcka Banja, Serbia, 25 – 29 April, 2013

2. Outline
1. Symmetries and conserved quantities
2. Killing forms
3. K¨ahler, Sasaki manifolds
4. Killing forms on Kerr-NUT-(A)dS spaces
5. Y(p, q) spaces
6. Killing forms on mixed 3-Sasakian manifolds
7. Outlook

3. Symmetries and conserved quantities (1)
Let (M, g) be a n-dimensional manifold equipped with a
(pseudo-)Riemmanian metric g and denote by
H =
1
2
gij
pipj ,
the Hamilton function describing the motion in a curved space.
In terms of the phase-space variables(xi, pi) the Poisson
bracket of two observables P, Q is
{P, Q} =
∂P
∂xi
∂Q
∂pi
−
∂P
∂pi
∂Q
∂xi
.

4. Symmetries and conserved quantities (2)
A conserved quantity of motions expanded as a power series in
momenta:
K = K0 +
p
k=1
1
k!
Ki1···ik (x)pi1
· · · pik
.
Vanishing Poisson bracket with the Hamiltonian, {K, H} = 0,
implies
K(i1···ik ;i)
= 0 ,
Such symmetric tensor Ki1···ik
k is called a St¨ackel-Killing (SK)
tensor of rank k

5. Killing forms (1)
A vector field X on a (pseudo-)Riemannian manifold (M, g) is
said to be a Killing vector field if the Levi-Civita connection
of g satisfies
g( Y X, Z) + g(Y, Z X) = 0,
for all vector fields Y, Z on M.

6. Killing forms (2)
A natural generalization of Killing vector fields is given by the
conformal Killing vector fields , i.e. vector fields with a flow
preserving a given conformal class of metrics.
More general, a conformal Killing-Yano tensor
( also called conformal Yano tensor or conformal Killing
form or twistor form ) of rank p on a (pseudo-) Riemannian
manifold (M, g) is a p -form ω which satisfies:
X ω =
1
p + 1
X−| dω −
1
n − p + 1
X∗
∧ d∗
ω,
for any vector field X on M, where n is the dimension of M , X∗
is the 1-form dual to the vector field X with respect to the metric
g , −| is the operator dual to the wedge product and d∗ is the
adjoint of the exterior derivative d.

7. Killing forms (3)
If ω is co-closed then we obtain the definition of a Killing-Yano
tensor, also called Yano tensor or Killing form :
ωi1...ik−1(ik ;j) = 0 .
Moreover, a Killing form ω is said to be a special Killing form if
it satisfies for some constant c the additional equation
X (dω) = cX∗
∧ ω ,
for any vector field X on M .
These two generalizations of the Killing vectors could be
related. Given two Killing-Yano tensors ωi1,...,ir and σi1,...,ir it is
possible to associate with them a St¨ackel-Killing tensor of rank
2 :
K
(ω,σ)
ij = ωii2...ir
σ i2...ir
j + σii2...ir
ω i2...ir
j .

8. K¨ahler, Sasaki manifolds (1)
Even dimensions (1)
An almost Hermitian structure on a smooth manifold M is a
pair (g, J), where g is a Riemannian metric on M and J is an
almost complex structure on M, which is compatible with g, i.e.
g(JX, JY) = g(X, Y),
for all vector fields X, Y on M. In this case, the triple (M, J, g) is
called an almost Hermitian manifold.
If J is parallel with respect to the Levi-Civita connection of g,
then (M, J, g) is said to be a K¨ahler manifold.
On a K¨ahler manifold, the associated K¨ahler form, i.e the
alternating 2-form Ω defined by
Ω(X, Y) = g(JX, Y)
is closed.

9. K¨ahler, Sasaki manifolds (2)
Even dimensions (2)
In local holomorphic coordinates (z1, ..., zm), the associated
K¨ahler form Ω can be written as
Ω = igj¯k dzj
∧ d¯zk
= X∗
j ∧ Y∗
j =
i
2
Z∗
j ∧ ¯Z∗
j ,
where (X1, Y1, ..., Xm, Ym) is an adapted local orthonormal field
(i.e. such that Yj = JXj ), and (Zj, ¯Zj) is the associated complex
frame given by
Zj =
1
2
(Xj − iYj), ¯Zj =
1
2
(Xj + iYj) .

10. K¨ahler, Sasaki manifolds (3)
Even dimensions (3)
Volume form is
dV =
1
m!
Ωm
,
where dV denotes the volume form of M, Ωm is the wedge
product of Ω with itself m times, m being the complex dimension
of M . Hence the volume form is a real (m, m)-form on M.
If the volume of a K¨ahler manifold is written as
dV = dV ∧ d ¯V
then dV is the complex volume holomorphic (m, 0) form of M.
The holomorphic form of a K¨ahler manifold can be written in a
simple way with respect to any (pseudo-)orthonormal basis,
using complex vierbeins ei + Jei. Up to a power factor of the
imaginary unit i, the complex volume is the exterior product of
these complex vierbeins.

11. K¨ahler, Sasaki manifolds (4)
Odd dimensions (1)
Let M be a smooth manifold equipped with a triple (ϕ, ξ, η),
where ϕ is a field of endomorphisms of the tangent spaces, ξ is
a vector field and η is a 1-form on M.
If we have:
ϕ2
= −I + η ⊗ ξ , η(ξ) = 1
then we say that (ϕ, ξ, η) is an almost contact structure on M.

12. K¨ahler, Sasaki manifolds (5)
Odd dimensions (2)
A Riemannian metric g on M is said to be compatible with the
almost contact structure (ϕ, ξ, η) iff the relation
g(ϕX, ϕY) = g(X, Y) − η(X)η(Y)
holds for all pair of vector fields X, Y on M.
An almost contact metric structure (ϕ, ξ, η, g) is a Sasakian
structure iff the Levi-Civita connection of the metric g
satisfies
( X ϕ)Y = g(X, Y)ξ − η(Y)X
for all vector fields X, Y on M

13. K¨ahler, Sasaki manifolds (6)
Metric cone (1)
A Sasakian structure may also be reinterpreted and
characterized in terms of the metric cone as follows. The metric
cone of a Riemannian manifold (M, g) is the manifold
C(M) = (0, ∞) × M
with the metric given by
¯g = dr2
+ r2
g,
where r is a coordinate on (0, ∞).
M is a Sasaki manifold iff its metric cone C(M) is K¨ahler.
The cone C(M) is equipped with an integrable complex
structure J and a K¨ahler 2-form Ω, both of which are parallel
with respect to the Levi-Civita connection ¯ of ¯g.

14. K¨ahler, Sasaki manifolds (7)
Metric cone (2)
M has odd dimension 2n + 1, where n + 1 is the complex
dimension of the K¨ahler cone.
The Sasakian manifold (M, g) is naturally isometrically
embedded into the cone via the inclusion
{r = 1} × M ⊂ C(M)
and the K¨ahler structure of the cone (C(M), ¯g) induces an
almost contact metric structure (φ, ξ, η, g) on M.

15. K¨ahler, Sasaki manifolds (8)
Einstein manifolds
An Einstein-Sasaki manifold is a Riemannian manifold (M, g)
that is both Sasaki and Einstein, i.e.
Ricg = λg
for some real constant λ, where Ricg denotes the Ricci tensor
of g. Einstein manifolds with λ = 0 are called Ricci-flat
manifolds.
An Einstein-K¨ahler manifold is a Riemannian manifold (M, g)
that is both K¨ahler and Einstein. The most important subclass
of K¨ahler-Einstein manifolds are the Calabi-Yau manifolds
(i. e. K¨ahler and Ricci-flat).
A Sasaki manifold M is Einstein iff the cone metric C(M) is
K¨ahler Ricci-flat. In particular the K¨ahler cone of a
Sasaki-Einstein manifold has trivial canonical bundle and the
restricted holonomy group of the cone is contained in SU(m),
where m denotes the complex dimension of the K¨ahler cone.

16. K¨ahler, Sasaki manifolds (9)
Progression from Einstein-K¨ahler to Einstein-Sasaki to Calabi-Yau manifolds (1)
Suppose we have an Einstein-Sasaki metric gES on a manifold
M2n+1 of odd dimension 2n + 1. An Einstein-Sasaki manifold
can always be written as a fibration over an Einstein-K¨ahler
manifold M2n with the metric gEK twisted by the overall U(1)
part of the connection
ds2
ES = (dψn + 2A)2
+ ds2
EK ,
where dA is given as the K¨ahler form of the Einstein-K¨ahler
base. The metric of the cone manifold M2n+2 = C(M2n+1) is
ds2
cone = dr2
+ r2
ds2
ES = dr2
+ r2
(dψn + 2A)2
+ ds2
EK .

17. K¨ahler, Sasaki manifolds (10)
Progression from Einstein-K¨ahler to Einstein-Sasaki to Calabi-Yau manifolds (2)
The cone manifold is Calabi-Yau and its K¨ahler form is
Ωcone = rdr ∧ (dψn + 2A) + r2
ΩEK ,
and the K¨ahler condition dΩcone = 0 implies
dA = ΩEK ,
where ΩEK is K¨ahler form of the Einstein-K¨ahler base manifold
M2n.
The Sasakian 1-form of the Einstein-Sasaki metric is
η = 2A + dψn ,
which is a special unit-norm Killing 1-form obeying for all vector
fields X
X η =
1
2
X−| dη , X (dη) = −2X∗
∧ η .

18. K¨ahler, Sasaki manifolds (11)
Hidden symmetries (1)
The hidden symmetries of the Sasaki manifold M2n+1 are
described by the special Killing (2k + 1)−forms
Ψk = η ∧ (dη)k
, k = 0, 1, · · · , n.
Besides these Killing forms, there are n closed conformal
Killing forms ( also called ∗-Killing forms)
Φk = (dη)k
, k = 1, · · · , n.

19. K¨ahler, Sasaki manifolds (12)
Hidden symmetries (2)
Moreover, in the case of holonomy SU(n + 1) , i.e. the cone
M2n+2 = C(M2n+1) is K¨ahler and Ricci-flat, or equivalently
M2n+1 is Einstein-Sasaki, it follows that we have two additional
Killing forms of degree n + 1 on the manifold M2n+1. These
additional Killing forms are connected with the additional
parallel forms of the Calabi-Yau cone manifold M2n+2 given by
the complex volume form and its conjugate.

20. K¨ahler, Sasaki manifolds (13)
Hidden symmetries (3)
In order to extract the corresponding additional Killing forms on
the Sasaki-Einstein space we make use of the fact that for any
p -form ω on the space M2n+1 we can define an associated
(p + 1) -form ωC on the cone C(M2n+1)
ωC
:= rp
dr ∧ ω +
rp+1
p + 1
dω .
The 1-1-correspondence between special Killing p -forms on
M2n+1 and parallel (p + 1) -forms on the metric cone C(M2n+1)
allows us to describe the additional Killing forms on
Sasaki-Einstein spaces.

21. Killing forms on Kerr-NUT-(A)dS spaces (1)
In the case of Kerr-NUT-(A)dS spacetime the Einstein-K¨ahler
metric gEK and the K¨ahler potential A are
gEK =
∆µdx2
µ
Xµ
+
Xµ
∆µ


n−1
j=0
σ(j)
µ dψj


2
,
Xµ = −4
n+1
i=1
(αi − xµ) − 2bµ ,
A =
n−1
k=0
σ(k+1)
dψk ,

22. Killing forms on Kerr-NUT-(A)dS spaces (2)
with
∆µ =
ν=µ
(xν − xµ) ,
σ(k)
µ =
ν1<···<νk
νi =µ
xν1
. . . xνk
, σ(k)
=
ν1<···<νk
xν1
. . . xνk
.
Here, coordinates xµ (µ = 1, . . . , n) stands for the Wick rotated
radial coordinate and longitudinal angles and the Killing
coordinates ψk (k + 0, . . . , n − 1) denote time and azimuthal
angles with Killing vectors ξ(k) = ∂ψk
.
αi (i = 1, . . . , n + 1) and bµ are constants related to the
cosmological constant, angular momenta, mass and NUT
parameters.

23. Killing forms on Kerr-NUT-(A)dS spaces (3)
Write the metric gEK on the Einstein-K¨ahler manifold M2n in the
form
gEK = oˆµ
oˆµ
+ ˜oˆµ˜oˆµ
,
and the K¨ahler 2-form
Ω = dA = oˆµ
∧ ˜oˆµ
.
where
oˆµ
=
∆µ
Xµ(xµ)
dxµ ,
˜oˆµ
=
Xµ(xµ)
∆µ
n−1
j=0
σ(j)
µ dψj .

24. Killing forms on Kerr-NUT-(A)dS spaces (4)
We introduce the following complex vierbeins on Einstein-
K¨ahler manifold M2n
ζµ = oµ
+ i˜oˆµ
, µ = 1, · · · , n .
On the Calabi-Yau cone manifold M2n+2 the complex vierbeins
are Λµ = rζµ for µ = 1, · · · , n and
Λn+1 =
dr
r
+ iη .
The standard complex volume form of the Calabi-Yau cone
manifold M2n+2 is
dV = Λ1 ∧ Λ2 ∧ · · · ∧ Λn+1 .

25. Killing forms on Kerr-NUT-(A)dS spaces (5)
As real forms we obtain the real part respectively the imaginary
part of the complex volume form. For example, writing
Λj = λ2j−1 + iλ2j, j = 1, ..., n + 1,
the real part of the complex volume is given by
Re dV =
[n+1
2
]
p=0 1≤i1<i2<...<in+1≤2n+2 (C)
(−1)p
λi1
∧λi2
∧...∧λin+1
(1)
where the condition (C) means that in the second sum are
taken only the indices i1, ..., in+1 such that
i1 + ... + in+1 = (n + 1)2 + 2p and (ik , ik+1) = (2j − 1, 2j), for all
k ∈ {1, ..., n} and j ∈ {1, ..., n + 1}.

26. Killing forms on Kerr-NUT-(A)dS spaces (6)
The imaginary part of the complex volume is given by
Im dV =
[n
2
]
p=0 1≤i1<i2<...<in+1≤2n+2 (C )
(−1)p
λi1
∧λi2
∧...∧λin+1
(2)
where the condition (C ) in means that in the second sum are
considered only the indices i1, i2, ..., in+1 such that
i1 + ... + in+1 = (n + 1)2 + 2p + 1 and (ik , ik+1) = (2j − 1, 2j), for
all k ∈ {1, ..., n} and j ∈ {1, ..., n + 1}.

27. Killing forms on Kerr-NUT-(A)dS spaces (7)
Finally, the Einstein-Sasaki manifold M2n+1 is identified with the
submanifold {r = 1} of the Calabi-Yau cone manifold
M2n+2 = C(M2n+1) and the additional (n + 1)-Killing forms are
accordingly acquired.
The 1-1-correspondence between special Killing
p -forms on M2n+1 and parallel (p + 1) -forms on the metric
cone C(M2n+1) allows us to describe the additional Killing
forms on Einstein-Sasaki spaces.
In order to find the additional Killing forms on the Einstein -
K´ahler manifold M2n+1 we must identify the ωM form in the
complex volume form of the Calabi-Yau cone.

28. Y(p,q) spaces (1)
Infinite family Y(p, q) of Einstein-Sasaki metrics on S2 × S3
provides supersymmetric backgrounds relevant to the AdS/CFT
correspondence. The total space Y(p, q) of an S1-fibration over
S2 × S2 with relative prime winding numbers p and q is
topologically S2 × S3.
Explicit local metric of the 5-dimensional Y(p, q) manifold given
by the line element
ds2
ES =
1 − c y
6
(dθ2
+ sin2
θ dφ2
) +
1
w(y)q(y)
dy2
+
q(y)
9
(dψ − cos θ dφ)2
+ w(y) dα +
ac − 2y + c y2
6(a − y2)
[dψ − cos θ dφ]
2
,
where

29. Y(p,q) spaces (2)
w(y) =
2(a − y2)
1 − cy
, q(y) =
a − 3y2 + 2cy3
a − y2
and a, c are constants. The constant c can be rescaled by a
diffeomorphism and in what follows we assume c = 1.
The coordinate change α = −1
6 β − 1
6 c ψ , ψ = ψ takes the line
element to the following form ( with p(y) = w(y) q(y) )
ds2
ES =
1 − y
6
(dθ2
+ sin2
θ dφ2
) +
1
p(y)
dy2
+
p(y)
36
(dβ + cos θ dφ)2
+
1
9
[dψ − cos θ dφ + y(dβ + cos θ dφ)]2
,

30. Y(p,q) spaces (3)
One can write
ds2
ES = dS2
EK + (
1
3
dψ + σ)2
The Sasakian 1-form of the Y(p, q) space is
η =
1
3
dψ + σ ,
with
σ =
1
3
[− cos θ dφ + y(dβ + cos θ dφ)] .
connected with local K¨ahler form ΩEK .
This form of the metric with the 1-form η is the standard one for
a locally Einstein-Sasaki metric with ∂
∂ψ the Reeb vector field.

31. Y(p,q) spaces (4)
The local K¨ahler and holomorphic (2, 0) form for ds2
EK are
ΩEK =
1 − y
6
sin θdθ ∧ dφ +
1
6
dy ∧ (dβ + cos θdφ)
dVEK =
1 − y
6p(y)
(dθ + i sin θdφ) ∧ dy + i
p(y)
6
(dβ + cos θdφ)

32. Y(p,q) spaces (5)
From the isometries SU(2) × U(1) × U(1) the momenta
Pφ, Pψ, Pα and the Hamiltonian describing the geodesic
motions are conserved. Pφ is the third component of the SU(2)
angular momentum, while Pψ and Pα are associated with the
U(1) factors. Additionally, the total SU(2) angular momentum
given by
J2
= P2
θ +
1
sin2
θ
(Pφ + cos θPψ)2
+ P2
Ψ ,
is also conserved.

33. Y(p,q) spaces (6)
Specific conserved quantities for Einstein-Sasaki spaces (1)
First of all from the 1-form η
Ψ = η ∧ dη
=
1
9
[(1 − y) sin θ dθ ∧ dφ ∧ dψ + dy ∧ dβ ∧ dψ
+ cos θ dy ∧ dφ ∧ dψ − cos θ dy ∧ dβ ∧ dφ
+ (1 − y)y sin θ dβ ∧ dθ ∧ dφ] .
is a special Killing form. Let us note also that
Ψk = (dη)k
, k = 1, 2 ,
are closed conformal Killing forms ( -Killing forms).

34. Y(p,q) spaces (7)
Specific conserved quantities for Einstein-Sasaki spaces (2)
On the Calabi-Yau manifold the K¨ahler form is
Ωcone = rdr ∧ η + r2
ΩEK .
and the holomorphic (3, 0) form is
dVcone = eψ
r2
dVEK ∧ [dr + ir ∧ η]
= eψ
r2 1 − y
6p(y)
dθ + i sin θdφ
∧ dy + i
p(y)
6
(dβ + cos θdφ)
∧ dr + i
r
3
[ydβ + dψ − (1 − y) cos θdφ

35. Y(p,q) spaces (8)
Specific conserved quantities for Einstein-Sasaki spaces (3)
The additional Killing 3-forms of the Y(p, q) spaces are
extracted from the volume form dVcone.
Using the the 1-1-correspondence between special Killing p
-forms on M2n+1 and parallel (p + 1) -forms on the metric cone
C(M2n+1) for p = 2 we get the following additional Killing
2-forms of the Y(p, q) spaces written as real forms:
Ξ = Re ωM
=
1 − y
6 p(y)
× cos ψ −dy ∧ dθ +
p(y)
6
sin θ dβ ∧ dφ
− sin ψ − sin θ dy ∧ dφ −
p(y)
6
dβ ∧ dθ
+
p(y)
6
cos θ dθ ∧ dφ

36. Y(p,q) spaces (9)
Specific conserved quantities for Einstein-Sasaki spaces (4)
Υ = Im ωM
=
1 − y
6 p(y)
× sin ψ −dy ∧ dθ +
p(y)
6
sin θ dβ ∧ dφ
+ cos ψ − sin θ dy ∧ dφ −
p(y)
6
dβ ∧ dθ
+
p(y)
6
cos θ dθ ∧ dφ

37. Y(p,q) spaces (10)
Specific conserved quantities for Einstein-Sasaki spaces (5)
The St¨ackel-Killing tensors associated with the Killing forms
Ψ , Ξ , Υ are constructed as usual. Together with the Killing
vectors Pφ, Pψ, Pα and the total angular momentum J2 these
St¨ackel-Killing tensors provide the superintegrability of the
Y(p, q) geometries.

38. Killing forms on mixed 3-Sasakian manifolds (1)
3-Sasakian manifolds (1)
A Riemannian manifold (M, g) of real dimension m is
3-Sasakian if the holonomy group of the metric cone
(C(M) , ¯g) = (R+ × M , dr2
+ r2
g)
reduces to a subgroup of Sp m+1
4 . In particular,
m = 4n + 3 , n ≥ 1 and (C(M) , ¯g) is hyperK¨ahler.
A 3-Sasakian manifold admits three characteristic vector fields
(ξ1 , ξ2 , ξ3), satisfying any of the corresponding conditions of
the Sasakian structure, such that
g(ξα , ξβ) = δαβ ,
and
[ξα , ξβ] = 2 αβγξγ .

39. Killing forms on mixed 3-Sasakian manifolds (2)
3-Sasakian manifolds (2)
Let (M, g) be a 3-Sasakian manifold and (ϕα, ξα, ηα),
α ∈ {1, 2, 3} , be its 3-Sasakian structure. then
ηα(ξβ) = δαβ ,
ϕα(ξβ) = − αβγξγ ,
ϕαϕβ − ξα ⊗ ηβ = − αβγϕγ − δαβI .
Theorem
Every 3-Sasakian manifold (M , g) of dimension 4n + 3 is
Einstein with Einstein constant λ = 4n + 2.

40. Killing forms on mixed 3-Sasakian manifolds (3)
A mixed 3-structure on a smooth manifold M is a triple of
structures (ϕα, ξα, ηα), α ∈ {1, 2, 3} , which are almost
paracontact structures for α = 1, 2 and almost contact structure
for α = 3 , satisfying the following compatibility conditions
ηα(ξβ) = 0,
ϕα(ξβ) = τβξγ, ϕβ(ξα) = −ταξγ,
ηα ◦ ϕβ = −ηβ ◦ ϕα = τγηγ ,
ϕαϕβ − ταηβ ⊗ ξα = −ϕβϕα + τβηα ⊗ ξβ = τγϕγ ,
where (α, β, γ) is an even permutation of (1, 2, 3) and
τ1 = τ2 = −τ3 = −1.

41. Killing forms on mixed 3-Sasakian manifolds (4)
If a manifold M with a mixed 3-structure (ϕα, ξα, ηα)α=1,3 admits
a semi-Riemannian metric g such that:
g(ϕαX, ϕαY) = τα[g(X, Y) − εαηα(X)ηα(Y)],
for all X, Y ∈ Γ(TM) and α = 1, 2, 3 , where
εα = g(ξα, ξα) = ±1
then we say that M has a metric mixed 3-structure and g is
called a compatible metric.

42. Killing forms on mixed 3-Sasakian manifolds (5)
A manifold M endowed with a (positive/negative) metric mixed
3-structure (ϕα, ξα, ηα, g) is said to be a (positive/negative)
mixed 3-Sasakian structure if (ϕ3, ξ3, η3, g) is a Sasakian
structure, while both structures (ϕ1, ξ1, η1, g) and (ϕ2, ξ2, η2, g)
are para-Sasakian, i.e.
( X ϕα)Y = τα[g(X, Y)ξα − αηα(Y)X]
for all vector fields X, Y on M and α = 1, 2, 3.
Theorem
Any (4n + 3)−dimensional manifold endowed with a mixed
3-Sasakian structure is an Einstein space with Einstein
constant λ = (4n + 2)θ, with θ = 1, according as the metric
mixed 3-structure is positive or negative, respectively.

43. Killing forms on mixed 3-Sasakian manifolds (6)
In (mixed) 3-Sasakian case any linear combination of the forms
Ψk1,k2,k3
defined by
Ψk1,k2,k3
=
k1
k1 + k2 + k3
[η1 ∧ (dη1)k1−1
] ∧ (dη2)k2
∧ (dη3)k3
+
k2
k1 + k2 + k3
(dη1)k1
∧ [η2 ∧ (dη2)k2−1
] ∧ (dη3)k3
+
k3
k1 + k2 + k3
(dη1)k1
∧ (dη2)k2
∧ [η3 ∧ (dη3)k3−1
]
for arbitrary positive integers k1, k2, k3 , is a special Killing form
on M.

44. Outlook
Complete integrability of geodesic equations
Separability of Hamilton-Jacobi, Klein-Gordon, Dirac
equations
Hidden symmetries of other spacetime structures

45. References
M. Visinescu, Mod. Phys. Lett. A 27, 1250217 (2012)
M. Visinescu, G. E. Vˆılcu, SIGMA 8, 058 (2012)
M. Visinescu, Mod. Phys. Lett. A 26, 2719 (2011)
M. Visinescu, SIGMA 7, 037 (2011)
M. Visinescu, J. Phys.: Conf. Series 411, 012030 (2013)