On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric over Kählerian Manifold
We propose a fiber-wise deformation of the Sasaki metric on slashed and unit tangent bundles over the Kalerian manifold based on the Berger deformation of metric on a unit sphere. The geodesics of this metric have different projections on a base manifold for the slashed and unit tangent bundles in...
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irk-123456789-1067172016-10-04T03:02:21Z On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric over Kählerian Manifold Yampolsky, A. We propose a fiber-wise deformation of the Sasaki metric on slashed and unit tangent bundles over the Kalerian manifold based on the Berger deformation of metric on a unit sphere. The geodesics of this metric have different projections on a base manifold for the slashed and unit tangent bundles in contrast to usual Sasaki metric. Nevertheless, the projections of geodesics of the unit tangent bundle over the locally symmetric K ahlerian manifold still preserve the property to have all geodesic curvatures constant. Предложена послойная деформация метрики Сасаки касательного без базы расслоения и единичного касательного расслоения Кэлерова многообразия, основанная на деформации Берже метрики на единичной сфере. В отличие от классической метрики Сасаки, геодезические этой деформированной метрики имеют разные проекции на базу касательного и единичного касательного расслоений. Однако проекции геодезических единичного расслоения над кэлеровим локально симметрическим многообразием все еще сохраняют свойство проектироваться в кривые с постоянными кривизнами. 2012 Article On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric over Kählerian Manifold / A. Yampolsky // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 2. — С. 117-189. — Бібліогр.: 10 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106717 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We propose a fiber-wise deformation of the Sasaki metric on slashed and unit tangent bundles over the Kalerian manifold based on the Berger deformation of metric on a unit sphere. The geodesics of this metric have different projections on a base manifold for the slashed and unit tangent bundles in contrast to usual Sasaki metric. Nevertheless, the projections of geodesics of the unit tangent bundle over the locally symmetric K ahlerian manifold still preserve the property to have all geodesic curvatures constant. |
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Yampolsky, A. |
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Yampolsky, A. On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric over Kählerian Manifold Журнал математической физики, анализа, геометрии |
| author_facet |
Yampolsky, A. |
| author_sort |
Yampolsky, A. |
| title |
On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric over Kählerian Manifold |
| title_short |
On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric over Kählerian Manifold |
| title_full |
On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric over Kählerian Manifold |
| title_fullStr |
On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric over Kählerian Manifold |
| title_full_unstemmed |
On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric over Kählerian Manifold |
| title_sort |
on geodesics of tangent bundle with fiberwise deformed sasaki metric over kählerian manifold |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/106717 |
| citation_txt |
On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric over Kählerian Manifold / A. Yampolsky // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 2. — С. 117-189. — Бібліогр.: 10 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT yampolskya ongeodesicsoftangentbundlewithfiberwisedeformedsasakimetricoverkahlerianmanifold |
| first_indexed |
2025-07-07T18:53:45Z |
| last_indexed |
2025-07-07T18:53:45Z |
| _version_ |
1837015423136038912 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2012, vol. 8, No. 2, pp. 177�189
On Geodesics of Tangent Bundle with Fiberwise
Deformed Sasaki Metric over K�ahlerian Manifold
A. Yampolsky
Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv 61077, Ukraine
E-mail: alexymp@gmail.com
Received February 24, 2011
We propose a �ber-wise deformation of the Sasaki metric on slashed
and unit tangent bundles over the K�alerian manifold based on the Berger
deformation of metric on a unit sphere. The geodesics of this metric have
di�erent projections on a base manifold for the slashed and unit tangent
bundles in contrast to usual Sasaki metric. Nevertheless, the projections of
geodesics of the unit tangent bundle over the locally symmetric K�ahlerian
manifold still preserve the property to have all geodesic curvatures constant.
Key words: Sasaki metric, K�ahlerian manifold, tangent bundle, geode-
sics.
Mathematics Subject Classi�cation 2000: 53B20, 53B25 (primary); 53B21
(secondary).
Introduction
Let (M, g) be a Riemannian manifold. Denote by TM and T1M the tangent
bundle and the unit tangent bundle of (M, g) with the Sasaki metric. It is easy
to prove that if π is a bundle projection π : TM → M and Γ(σ) is a non-vertical
geodesic on TM or T1M, then the projected curve γ(σ) = (π ◦Γ)(σ) on M is the
same. In other words, the non-vertical geodesic lines on TM or T1M are generated
by di�erent vector �elds along the same set of the curves on the base manifold.
For the case of the base manifold of constant curvature, S. Sasaki [1] and K. Sato
[2] gave a complete description of the curves and vector �elds along them which
generated non-vertical geodesics on T1M
n and TMn, respectively. They proved
that the projected curves have constant (possibly zero) �rst and second geodesic
curvatures while the others vanish. P. Nagy [3] generalized these results to the
c© A. Yampolsky, 2012
A. Yampolsky
case of locally symmetric base manifold and proved that the projected curves have
all geodesic curvatures constant.
The Sasaki metric weakly inherits the base manifold properties. Under most
considerations it behaves just as a general Riemannian metric. That is why a
number of authors proposed to deform the Sasaki metric in order to get some
kind of "�exibility" of its properties (see [4�8] and others).
Using the concept of natural transformation of the Riemannian metric on the
manifold to its tangent bundle, M.T.K. Abbassi and M. Sarih [6] proposed a much
more general metric on the tangent and the unit tangent bundles which includes
the Sasaki metric, the Cheeger�Gromoll metric and some others as partial cases.
This metric uses some kind of "deformation" of the Sasaki metric in the direction
of the "tangent bundle point".
In present paper we propose another natural way of deforming the Sasaki
metric in the presence of almost complex structure J . If the base manifold (M, g)
of dimension 2n is endowed with almost complex structure J , then the unit sphere
S2n−1
x in the tangent space TxM carries the Hopf vector �eld Jξ, where ξ is a
unit normal vector �eld on the sphere. Applying the Berger metric deformation
to each tangent sphere, we get the unit tangent bundle over M with the Berger
metric spheres as �bers. In a wider scope, one can deform the Sasaki metric on the
slashed manifold TM0 := TM\M in the direction of Jξ such that the restriction of
the deformed metric on the unit tangent bundle gives the construction described.
The main result of the paper is the following.
Theorem 2.1.Let Γ be a geodesic on the unit tangent bundle with the Berger-
type deformed Sasaki metric over K�ahlerian locally symmetric manifold M and
γ = π ◦ Γ be its projection to the base. Then all geodesic curvatures of γ are
constant.
If Γ is a geodesic on the slashed tangent bundle TM0, then the projected curve
γ = π ◦ Γ does not possess this property.
For the speci�c case of the K�ahlerian manifold of constant holomorphic cur-
vature, Theorem 2.1 can be improved.
Theorem 2.2.Let Γ be a geodesic of the unit tangent bundle with the Berger-
type deformed Sasaki metric over K�ahlerian manifold M2n (n ≥ 3) of constant
holomorphic curvature. Then the geodesic curvatures of γ = π ◦Γ are all constant
and k6 = · · · = kn−1 = 0.
1. Basic Properties of the Berger-Type Deformed Sasaki Metric
Let (M, g) be an n-dimensional Riemannian manifold with metric g. Denote
by
〈·, ·〉 a scalar product with respect to g.
178 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2
On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric
It is well known that at each point Q = (q, ξ) ∈ TM the tangent space TQTM
splits into vertical and horizontal parts:
TQTM = HQTM ⊕ VQTM.
The vertical part VQTM is tangent to the �ber, while the horizontal part is
transversal to it. Denote by (x1, . . . , xn; ξ1, . . . , ξn) the natural induced local
coordinate system on TM . Denote ∂i = ∂/∂xi, ∂n+i = ∂/∂ξi. Then for X̃ ∈
TQTMn we have
X̃ = X̃i∂i + X̃n+i∂n+i.
Denote by π : TM → M the tangent bundle projection. The mapping π∗
de�nes a point-wise linear isomorphism between HQTM and TqM . Notice that
kerπ∗|Q = VQ.
The so-called connection mapping K : TQTM → TqM acts on X̃ by
KX̃ = (X̃n+i + Γi
jkξ
jX̃k)∂i
and de�nes a point-wise linear isomorphism between VQTM and TqM . Here Γi
jk
are the Christo�el symbols of g. Notice that kerK|Q = HQ.
The images π∗X̃ and KX̃ are called horizontal and vertical projections of X̃,
respectively. The operations inverse to the projections are called lifts. Namely, if
X ∈ TqM
n, then
Xh = Xi∂i − Γi
jkξ
jXk∂n+i
is in HQTM and is called the horizontal lift of X, and
Xv = Xi∂n+i
is in VQTM and is called the vertical lift of X.
Let X̃, Ỹ ∈ TQTM. The standard Sasaki metric on TM is de�ned at each
point Q = (q, ξ) ∈ TM by the scalar product
〈〈
X̃, Ỹ
〉〉∣∣
Q
=
〈
π∗X̃, π∗Ỹ
〉∣∣
q
+
〈
KX̃,KỸ
〉∣∣
q
.
The horizontal and vertical subspaces are mutually orthogonal with respect to the
Sasaki metric. The Sasaki metric can be completely de�ned by a scalar product
of various combinations of lifts by
〈〈
Xh, Y h
〉〉
=
〈
X, Y
〉
,
〈〈
Xh, Y v
〉〉
= 0,
〈〈
Xv, Y v
〉〉
=
〈
X,Y
〉
.
Let (M, g, J) be a Hermitian manifold of dimension 2n with an almost complex
structure J , i.e. the (1, 1)-tensor �eld satisfying J2 = −id. Denote by TM0
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2 179
A. Yampolsky
a slashed tangent bundle, i.e. the tangent bundle with zero section deleted. De�ne
a �ber-wise Berger-type deformation of the Sasaki metric on TM0 by
〈〈
Xh, Y h
〉〉
=
〈
X,Y
〉
,
〈〈
Xh, Y v
〉〉
= 0,
〈〈
Xv, Y v
〉〉
=
〈
X,Y
〉
+ δ2
〈
X, Jξ
〉〈
Y, Jξ
〉
,
(1)
where δ is some constant.
In what follows we restrict the considerations to the case of the K�ahlerian base
manifold. In this case J has no torsion and ∇J = 0.
The following formulas are independent from the choice of the tangent bundle
metric and are known as Dombrowski formulas.
Lemma 1.1. At each point (q, ξ) ∈ TM the brackets of lifts of vector �elds
from M to TM are
[
Xh, Y h
]
=
[
X, Y
]h − (
R(X, Y )ξ
)v
,
[
Xh, Y v
]
=
(∇XY
)v
,
[
Xv, Y v
]
= 0,
where ∇ is the connection on M and R is its curvature tensor.
Denote by ∇̃ the Levi�Civita connection of metric (1). The following lemma
contains the Kowalski-type formulas [9] and it is the main tool for further consid-
erations.
Lemma 1.2. Let (M, g, J) be a K�ahlerian manifold. The Levi�Civita connec-
tion of the Berger-type deformed Sasaki metric (1) on the slashed tangent bundle
TM0 is completely de�ned by
∇̃XhY h =
(∇XY
)h − 1
2
(
R(X, Y )ξ
)v
,
∇̃XhY v = 1
2
(
R(ξ, Y )X + δ2 〈Y, Jξ〉R(ξ, Jξ)X
)h
+
(
∇XY
)v
,
∇̃XvY h = 1
2
(
R(ξ, X)Y + δ2 〈X,Jξ〉R(ξ, Jξ)Y
)h
∇̃XvY v = δ2
(
〈X,Jξ〉 JY + 〈Y, Jξ〉 JX−
δ2
1+δ2|ξ|2
( 〈Y, ξ〉 〈X, Jξ〉+ 〈X, ξ〉 〈Y, Jξ〉 )Jξ
)v
,
where ∇ is the Levi-Civita connection on M and R is its curvature tensor.
P r o o f. The proof is based on the following rules of di�erentiations:
Xh
〈〈
Y h, Zh
〉〉
=
〈∇XY, Z
〉
+
〈
Y,∇XZ
〉
,
Xh
〈〈
Y v, Zv
〉〉
=
〈〈
(∇XY )v, Zv
〉〉
+
〈〈
Y v, (∇XZ)v
〉〉
,
Xv
〈〈
Y h, Zh
〉〉
= 0,
Xv
〈〈
Y v, Zv
〉〉
= δ2
(〈
Y, JX
〉〈
Z, Jξ
〉
+
〈
Y, Jξ
〉〈
Z, JX
〉)
.
(2)
180 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2
On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric
(2)1: Indeed, keeping in mind (1), we have
Xh
〈〈
Y h, Zh
〉〉
= Xh
〈
Y, Z
〉
=
〈∇XY, Z
〉
+
〈
Y,∇XZ
〉
.
(2)2: In a similar way,
Xh
〈〈
Y v, Zv
〉〉
= Xh
(〈
Y, Z
〉
+ δ2
〈
Y, Jξ
〉〈
Z, Jξ
〉)
=
〈∇XY,Z
〉
+
〈
Y,∇XZ
〉
+ δ2Xh
(〈
Y, Jξ
〉〈
Z, Jξ
〉)
.
As M is K�ahlerian and hence ∇XJ = 0, we have
Xh
〈
Y, Jξ
〉
= −Xh
〈
JY, ξ
〉
= −Xi∂i
〈
JY, ξ
〉
+ Γs
jkξ
jXk∂n+s
〈
JY, ξ
〉
= −Xi
〈
J∇iY, ξ
〉−Xiξk
〈
JY, Γs
ki∂s
〉
+ Γs
kiξ
kXi
〈
JY, ∂s
〉
=
〈∇XY, Jξ
〉
.
Therefore,
Xh
〈〈
Y v, Zv
〉〉
=
〈∇XY, Z
〉
+
〈
Y,∇XZ
〉
+ δ2
〈∇XY, Jξ
〉〈
Z, Jξ
〉
δ2
〈
Y, Jξ
〉〈∇XZ, Jξ
〉
= +
〈〈
(∇XY )v, Zv
〉〉
+
〈〈
Y v, (∇XZ)v
〉〉
.
(2)3: We have Xv
〈〈
Y h, Zh
〉〉
= Xv
〈
Y, Z
〉
= Xi∂n+i
〈
Y,Z
〉
= 0.
(2)4: Finally,
Xv
〈
Y, Jξ
〉
= Xi∂n+i
〈
Y, Jξ
〉
=
〈
Y, JX
〉
,
and therefore
Xv
〈〈
Y v, Zv
〉〉
= Xv
(〈
Y, Z
〉
+ δ2
〈
Y, Jξ
〉〈
Z, Jξ
〉)
= δ2
(〈
Y, JX
〉〈
Z, Jξ
〉
+
〈
Y, Jξ
〉〈
Z, JX
〉)
.
Now we can prove the lemma relatively easy by applying Lemma 1.1 and the
Kozsul formula for the Levi�Civita connection
2
〈∇AB,C
〉
= A
〈
B, C
〉
+ B
〈
A, C
〉− C
〈
A,B
〉
+
〈
[A,B], C
〉
+
〈
[C,A], B
〉− 〈
[B, C], A
〉
.
• Take A = Xh, B = Y h, C = Zh. Then
2
〈〈∇̃XhY h, Zh
〉〉
= 2
〈∇XY, Z
〉
= 2
〈〈
(∇XY )h, Zh
〉〉
.
Take A = Xh, B = Y h, C = Zv. Then
2
〈〈∇̃XhY h, Zv
〉〉
= −Zv
〈〈
Xh, Y h
〉〉
+
〈〈
[Xh, Y h], Zv
〉〉
= −〈〈
(R(X, Y )ξ)v, Zv
〉〉
.
Hence
∇̃XhY h =
(∇XY
)h − 1
2
(
R(X, Y )ξ
)v
.
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2 181
A. Yampolsky
• Take A = Xh, B = Y v, C = Zh. Then
2
〈〈∇̃XhY v, Zh
〉〉
=
〈〈
[Zh, Xh], Y v
〉〉
=
〈〈
(R(X,Z)ξ)v, Y v
〉〉
=
〈
R(X, Z)ξ, Y
〉
+ δ2
〈
R(X, Z)ξ, Jξ
〉〈
Y, Jξ
〉
=
〈
R(ξ, Y )X, Z
〉
+ δ2
〈
Y, Jξ
〉〈
R(ξ, Jξ)X, Z
〉
=
〈〈(
R(ξ, Y )X + δ2
〈
Y, Jξ
〉
R(ξ, Jξ)X
)h
, Zh
〉〉
.
Take A = Xh, B = Y v, C = Zv. Then by (2), we have
2
〈〈∇̃XhY v, Zv
〉〉
= Xh
〈〈
Y v, Zv
〉〉
+
〈〈
[Xh, Y v], Zv
〉〉
+
〈〈
[Zv, Xh], Y v
〉〉
=
〈〈
(∇XY )v, Zv
〉〉
+
〈〈
Y v, (∇XZ)v
〉〉
+
〈〈
(∇XY )v, Zv
〉〉− 〈〈
(∇XZ)v, Y v
〉〉
= 2
〈〈
(∇XY )v, Zv
〉〉
.
So, we see that
∇̃XhY v =
1
2
(
R(ξ, Y )X + δ2 〈Y, Jξ〉R(ξ, Jξ)X
)h
+
(
∇XY
)v
.
• Take A = Xv, B = Y h, C = Zh. Then
2
〈〈∇̃XvY h, Zh
〉〉
= Xv
〈〈
Y h, Zh
〉〉
+
〈〈
[Xv, Y h], Zh
〉〉
+
〈〈
[Zh, Xv], Y h
〉〉
− 〈〈
[Y h, Zh], Xv
〉〉
=
〈〈
(R(Y,Z)ξ)v, Xv
〉〉
=
〈
R(Y, Z)ξ,X
〉
+ δ2
〈
R(Y, Z)ξ, Jξ
〉〈
X, Jξ
〉
=
〈
R(ξ, X)Y, Z
〉
+ δ2
〈
X,Jξ
〉〈
R(ξ, Jξ)Y, Z
〉
=
〈〈
(R(ξ,X)Y + δ2
〈
X, Jξ
〉
R(ξ, Jξ)Y )h, Zh
〉〉
.
Take A = Xv, B = Y h, C = Zv. Then
2
〈〈∇̃XvY h, Zv
〉〉
= Y h
〈〈
Zv, Xv
〉〉
+
〈〈
[Xv, Y h], Zv
〉〉− 〈〈
[Y h, Zv], Xv
〉〉
=
〈〈
(∇Y Z)v, Xv
〉〉
+
〈〈
Zv, (∇Y X)v
〉〉
− 〈〈
(∇Y X)v, Zv
〉〉− 〈〈
(∇Y Z)v, Xv
〉〉
= 0.
So, we have
∇̃XvY h =
1
2
(
R(ξ, X)Y + δ2 〈X,Jξ〉R(ξ, Jξ)Y
)h
.
• Take A = Xv, B = Y v, C = Zh. Then we have
2
〈〈∇̃XvY v, Zh
〉〉
= −Zh
〈〈
Xv, Y v
〉〉
+
〈〈
[Zh, Xv], Y v
〉〉
− 〈〈
[Y v, Zh], Xv
〉〉
= −〈〈
(∇ZX)v, Y v
〉〉− 〈〈
Xv, (∇ZY )v
〉〉
+
〈〈
(∇ZX)v, Y v
〉〉
+
〈〈
(∇ZY )v, Xv
〉〉
= 0.
182 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2
On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric
Finally, take A = Xv, B = Y v, C = Zv. Then
2
〈〈∇̃XvY v, Zv
〉〉
= Xv
〈〈
Y v, Zv
〉〉
+ Y v
〈〈
Xv, Zv
〉〉− Zv
〈〈
Xv, Y v
〉〉
= δ2
(〈
Y, JX
〉〈
Z, Jξ
〉
+
〈
Y, Jξ
〉〈
Z, JX
〉
+
〈
X, JY
〉〈
Z, Jξ
〉
+
〈
X, Jξ
〉〈
Z, JY
〉− 〈
X, JZ
〉〈
Y, Jξ
〉− 〈
X, Jξ
〉〈
Y, JZ
〉)
= 2δ2
(〈
Y, Jξ
〉〈
JX, Z
〉
+
〈
X, Jξ
〉〈
JY, Z
〉)
.
Thus, we see that
〈〈∇̃XvY v, Zv
〉〉
= δ2
(〈
Y, Jξ
〉〈
JX,Z
〉
+
〈
X,Jξ
〉〈
JY, Z
〉)
.
On the other hand,
〈〈
(JY )v, Zv
〉〉
=
〈
JY, Z
〉
+ δ2
〈
Y, ξ
〉〈
Z, Jξ
〉
and
〈〈
(Jξ)v, Zv
〉〉
=
〈
Jξ, Z
〉
+ δ2
〈
Z, Jξ
〉|ξ|2 = (1 + δ2|ξ|2)〈Z, Jξ
〉
.
Therefore, 〈
Z, Jξ
〉
=
1
1 + δ2|ξ|2
〈〈
(Jξ)v, Zv
〉〉
and, as a consequence,
〈
JY, Z
〉
=
〈〈
(JY )v, Zv
〉〉− δ2
〈
Y, ξ
〉 1
1 + δ2|ξ|2
〈〈
(Jξ)v, Zv
〉〉
=
〈〈
(JY )v − δ2
1 + δ2|ξ|2
〈
Y, ξ
〉
(Jξ)v, Zv
〉〉
.
So, we have
〈〈∇̃XvY v, Zv
〉〉
= δ2
〈〈 [〈
X, Jξ
〉(
JY − δ2
1 + δ2|ξ|2
〈
Y, ξ
〉
Jξ
)
+
〈
Y, Jξ
〉(
JX − δ2
1 + δ2|ξ|2
〈
X, ξ
〉
Jξ
)]v
, Zv
〉〉
.
Finally,
∇̃XvY v = δ2
(〈
X, Jξ
〉
JY +
〈
Y, Jξ
〉
JX
− δ2
1 + δ2|ξ|2
(〈
Y, ξ
〉〈
X,Jξ
〉
+
〈
X, ξ
〉〈
Y, Jξ
〉)
Jξ
)v
.
The lemma is proved.
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2 183
A. Yampolsky
2. Geodesics of the Berger-Type Deformed Sasaki Metric
Consider a non-vertical curve Γ on the tangent bundle with metric (1). Geometri-
cally, Γ = {x(σ), ξ(σ)}, where x(σ) is a curve on M and ξ(σ) is a vector �eld along
this curve. Let σ be an arc length parameter on Γ. Then Γ′ =
(
dx
dσ
)h
+ (∇ dx
dσ
ξ)v.
Introduce the notations x′ = dx
dσ and ξ′ = ∇ dx
dσ
ξ. Then
Γ′ = (x′)h + (ξ′)v.
Now we can easily derive the di�erential equations of geodesic lines of metric (1).
Lemma 2.1. Let (M, g, J) be a K�ahlerian manifold and TM0 be its slashed
tangent bundle with the Berger-type deformed Sasaki metric. The curve Γ =
{x(σ), ξ(σ)} is a geodesic on TM0 if and only if x(σ) and ξ(σ) satisfy
x′′ +R(ξ, ξ′)x′ = 0
ξ′′ + 2δ2
〈
ξ′, Jξ
〉(
Jξ′ − δ2
1+δ2|ξ|2
〈
ξ′, ξ
〉
Jξ
)
= 0,
(3)
where R(ξ, ξ′) = R(ξ, ξ′) + δ2
〈
ξ′, Jξ
〉
R(ξ, Jξ), and R is the curvature operator of
the base manifold M .
P r o o f. Using Lemma 1.2, �nd the derivative Γ′′ and equalize it to zero
Γ′′ = ∇̃(x′)h+(ξ′)v
(
(x′)h + (ξ′)v
)
=
(
x′′ + R(ξ, ξ′)x′ + δ2
〈
ξ′, Jξ
〉
R(ξ, Jξ)x′
)h
+
(
ξ′′ + 2δ2
(〈
ξ′, Jξ
〉
Jξ′ − δ2
1 + δ2|ξ|2
〈
ξ′, ξ
〉〈
ξ′, Jξ
〉)
Jξ
)v
=
(
x′′ +
(
R(ξ, ξ′) + δ2
〈
ξ′, Jξ
〉
R(ξ, Jξ)
)
x′
)h
+
(
ξ′′ + 2δ2
〈
ξ′, Jξ
〉(
Jξ′ − δ2
1 + δ2|ξ|2
〈
ξ′, ξ
〉)
Jξ
)v
= 0.
The lemma is proved.
Consider now the unit tangent bundle T1M .
Lemma 2.2. Let (M2n, g, J) be a K�ahlerian manifold and T1M be its unit
tangent bundle with the Berger-type deformed Sasaki metric. Set c = |ξ′|, µ =〈
ξ′, Jξ
〉
. The curve Γ = {x(σ), ξ(σ)} is a geodesic on T1M if and only if
184 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2
On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric
(a) c = const, µ = const;
(b) x(σ) and ξ(σ) satisfy the equations
x′′ +R(ξ, ξ′)x′ = 0
ξ′′ + c2ξ + 2δ2µ(Jξ′ + µξ) = 0,
(4)
where R(ξ, ξ′) = R(ξ, ξ′) + δ2µR(ξ, Jξ) and R is the curvature operator of
the base manifold M .
P r o o f. At each point (q, ξ) ∈ T1M , the unit normal to T1M is ξv. Indeed,
with respect to metric (1), we have
〈〈
Xh, ξv
〉〉
= 0 for all X tangent to M,〈〈
Xv, ξv
〉〉
= 0 for all X ∈ ξ⊥ .
As T1M is a hypersurface in TM , the curve on T1M is geodesic i� its second
covariant derivative in TM is collinear to the unit normal, i.e. to ξv. That is why,
to �nd the equations of geodesics on T1M , it is su�cient to set |ξ| = 1 in (3) and
to suppose the left-hand side of (3)2 to be collinear ξ. Thus, we get
x′′ +R(ξ, ξ′)x′ = 0
ξ′′ + 2δ2
〈
ξ′, Jξ
〉
Jξ′ = ρξ,
(5)
where ρ is some function.
Put c = |ξ′|. Then c = const, since
〈
ξ′′, ξ′
〉
= 0 directly from (5)2. Put
µ =
〈
ξ′, Jξ
〉
. Then µ = const, since µ′ =
〈
ξ′′, Jξ
〉
= 0 . Multiplying (5)2 by ξ,
we �nd
−ρ = c2 + 2δ2µ2 = const.
After substituting it into (5), we get what was claimed.
The di�erence in the description of solutions of (3) and (4) follows from dif-
ferent behavior of the operator R(ξ, ξ′) along π ◦ Γ.
Proposition 2.1. Let Γ be a geodesic of the slashed or unit tangent bundle
over K�ahlerian locally symmetric manifold M and γ = π ◦ Γ. Then R(ξ, ξ′) is
parallel along γ for the case of T1M and non-parallel for the case of TM0.
P r o o f. First, consider the case of T1M . Using (4), we get
R′(ξ, ξ′) = R(ξ, ξ′′) + δ2µR(ξ′, Jξ) + δ2µR(ξ, Jξ′)
= −2δ2µR(ξ, Jξ′)− δ2µR(Jξ′, ξ) + δ2µR(ξ, Jξ′) = 0.
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2 185
A. Yampolsky
Here we also used the fact that R(JX, JY ) = R(X, Y ).
A similar but longer calculation shows that for the of case of TM0
R′(ξ, ξ′) =
2δ6
〈
ξ′, Jξ
〉〈
ξ′, ξ
〉(
1− |ξ|2)
1 + δ2|ξ|2 R(ξ, Jξ),
which completes the proof.
Theorem 2.1. Let Γ be a geodesic of the unit tangent bundle with the Berger-
type deformed Sasaki metric over the K�ahlerian locally symmetric manifold M,
and γ = π ◦ Γ. Then all geodesic curvatures of γ are constant.
P r o o f. For the case of T1M, Proposition 2.1 implies that if Γ is a geodesic
on T1M , then along each curve γ = π ◦ Γ
x(p+1)(σ) = −R(ξ, ξ′) x(p)(σ) p ≥ 1. (6)
On the other hand, it is rather evident that
〈R(ξ, ξ′)X, Y
〉
= −〈R(ξ, ξ′)Y, X
〉
.
This fact and (6) imply
|x(p)(σ)| = const for all p ≥ 1. (7)
Indeed,
d
dσ |x(p)(σ)| 2 = 2
〈
x(p+1)(σ), x(p)(σ)
〉
= −2
〈R(ξ, ξ′) x(p)(σ), x(p)(σ)
〉
= 0.
Denote by s an arc length parameter on γ. Then x′σ = x′s
ds
dσ , and therefore
1 = ‖Γ′‖2 =
∣∣∣ ds
dσ
∣∣∣
2
+ |ξ′|2 + δ2
〈
ξ′, Jξ
〉2 =
∣∣∣ ds
dσ
∣∣∣
2
+ c2 + δ2µ2.
Hence
ds
dσ
=
√
1− c2 − δ2µ2 =
√
1− λ2, (8)
where λ2 = c2 + δ2µ2 = const.
Denote by ν1, . . . , ν2n−1 the Frenet frame along γ and by k1, . . . , k2n−1 the
geodesic curvatures of γ. Then, keeping in mind (8), we have
x′ =
√
1− λ2 ν1,
x′′ = (1− λ2)k1ν2.
Now (7) implies k1 = const. Next,
x(3) = (1− λ2)3/2 k1(−k1ν1 + k2ν3),
and again (7) implies k2 = const. By continuing the process, we �nish the proof.
186 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2
On Geodesics of Tangent Bundle with Fiberwise Deformed Sasaki Metric
As proven in [10], for the case of T1CPn and TCPn with the Sasaki metric, the
curvatures of γ = π◦Γ are zeroes starting from k6. It is rather remarkable that this
property is still valid for the case of the Berger-deformed Sasaki metric on the unit
tangent bundle over the K�ahlerian manifold of constant holomorphic curvature.
It is well known that the complete simply connected K�ahlerian manifold of the
constant holomorphic sectional curvature k is isometric to: the complex projective
space CPn for k > 0; the open ball Dn ⊂ Cn for k < 0; Cn for k = 0.
Theorem 2.2. Let Γ be a geodesic of the unit tangent bundle with the Berger-
type deformed Sasaki metric over K�ahlerian manifold M2n (n ≥ 3) of the constant
holomorphic curvature. Then the geodesic curvatures of γ = π◦Γ are all constant,
and k6 = · · · = k2n−1 = 0.
P r o o f. For the case of the K�ahlerian manifold of constant holomorphic
curvature k we have
R(X, Y )Z =
k
4
(〈
Y,Z
〉
X − 〈
X, Z
〉
Y
+
〈
JY, Z
〉
JX − 〈
JX,Z
〉
JY + 2
〈
X, JY
〉
JZ
)
.
Rewrite equations (4) as follows:
x′′ = −k
4
(〈
ξ′, x′
〉
ξ − 〈
ξ, x′
〉
ξ′ +
〈
Jξ′, x′
〉
Jξ − 〈
Jξ, x′
〉
Jξ′
+ 2
〈
ξ, Jξ′
〉
Jx′
)
− 1
2
δ2µ
(〈
Jξ, x′
〉
ξ − 〈
ξ, x′
〉
Jξ − Jx′
)
(9)
ξ′′ = −(c2 + 2δ2µ2)ξ − 2δ2µJξ′. (10)
Equation (9) shows that x′′ is a linear combination of at most ξ, ξ′, Jξ, Jξ′ and
Jx′. Therefore, Jx′′ is a linear combination of at most Jξ, Jξ′, ξ, ξ′, x′. Equation
(10) shows that ξ′′ is a linear combination of at most ξ and Jξ′. Therefore, Jξ′′
is a linear combination of at most Jξ and ξ′.
For the sake of brevity, denote a point-wise linear combination of the corre-
sponding vectors by l.c.(·, ·, . . . ). Then we can write
x′′ = l.c.(ξ, ξ′, Jξ, Jξ′, Jx′), Jx′′ = l.c.(Jξ, Jξ′, ξ, ξ′, x′),
ξ′′ = l.c.(ξ, Jξ′), Jξ′′ = l.c.(Jξ, ξ′).
Hence
x′′′ = l.c.(ξ′, ξ′′, Jξ′, Jξ′′, Jx′′)
= l.c.
(
ξ′, l.c.(ξ, Jξ′), Jξ′, l.c.(Jξ, ξ′), l.c.(Jξ, Jξ′, ξ, ξ′, x′)
)
= l.c.(ξ, ξ′, Jξ, Jξ′, x′).
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2 187
A. Yampolsky
In a similar way,
x(4) = l.c.(ξ, ξ′, Jξ, Jξ′, Jx′).
Continuing the process, we conclude that
x(p) = l.c.(ξ, ξ′, x′, Jξ, Jξ′, Jx′)
for all p. This means that no more than the �rst six derivatives could be linearly
independent, and hence at least
x(7) = l.c.(x′, x′′, . . . , x(6)). (11)
On the other hand, the Frenet formulas yield
x′ =
√
1− λ2 ν1 x′′ = (1− λ2)k1ν2, x′′′ = (1− λ2)3/2(−k2
1ν1 + k1k2ν3)
and, in general,
x(2m) = l.c.(ν2, . . . , ν2m−2) + (1− λ2)m k1 . . . k2m−1 ν2m,
x(2m+1) = l.c.(ν1, . . . , ν2m−1) + (1− λ2)m+1/2 k1 . . . k2m ν2m+1,
where λ is given by (8). For m = 3, we have
x(7) = l.c.(ν1, . . . , ν5) + (1− λ2)7/2 k1 . . . k6 ν7.
Notice that for all p ≤ 6,
x(p) = l.c.(ν1, . . . , ν6).
From (11) it follows that
(1− λ2)7/2 k1 . . . k6 ν7 = l.c.(ν1, . . . , ν6),
and therefore, at least k6 = 0.
Acknowledgement. The author thanks Professor P.T. Nagy for the idea of
considering the deformation proposed and a hospitality during the visit to the
University of Debrecen (Hungary).
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