Filomat 29:10 (2015), 2367–2379
DOI 10.2298/FIL1510367B
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
F−geodesics on Manifolds
Cornelia-Livia Bejan
a
, Simona-Luiza Drut¸˘a-Romaniuc
a
Dedicated to Academician Professor Mileva Prvanovi
´
c on her birthday
a
Universitatea Tehnic˘a ”Gheorghe Asachi” din Ias¸i
Postal address: Seminarul Matematic, Universitatea ”Alexandru Ioan Cuza” din Ias¸i, Bd. Carol I, No. 11, 700506 Ias¸i, ROMANIA
Abstract. The notion of F−geodesic, which is slightly different from that of F−planar curve (see [13], [17],
and [18]), generalizes the magnetic curves, and implicitly the geodesics, by using any (1,1)-tensor field
on the manifold (in particular the electro-magnetic field or the Lorentz force). We give several examples
of F−geodesics and the characterizations of the F−geodesics w.r.t. Vranceanu connections on foliated
manifolds and adapted connections on almost contact manifolds. We generalize the classical projective
transformation, holomorphic-projective transformation and C−projective transformation, by considering
a pair of symmetric connections which have the same F−geodesics. We deal with the transformations
between such two connections, namely F−planar diffeomorphisms ([18]). We obtain a Weyl type tensor
field, invariant under any F−planar diffeomorphism, on a 1−codimensional foliation.
1. Introduction
Recently, in mathematics literature, a series of papers on magnetic curves, inspired from theoretical
physics (see [1]-[3], [5], [8], [11], [21]) have appeared. The Lorentz force, the electro-magnetic tensor field,
as well as some special forces involved in the Euler-Lagrange equations from Lagrangian mechanics, lead
us to consider an arbitrary (1, 1)−tensor field F on a differentiable manifold. By using it, we deal here
with a notion which generalizes both the classical equations of geodesics and magnetic curves, namely the
F−geodesics on manifolds, with the purpose to unify these classes of curves on one side, and to provide a
geometrical model for some physical particles, satisfying certain differential equations, on the other side.
The notion of F−geodesic is slightly different from F−planar curve (see [18] and the references therein).
We provide several classes of F−geodesics, which highlight trajectories in Lagrangian mechanics, mag-
netic curves (described by particles moving under the influence of the Lorentz force), and curves on the
total space of the tangent bundle (obtained by using different types of lifts).
We obtain some characterizations of F−geodesics w.r.t. special connections, namely Vranceanu connec-
tions (see [6], [30]) on foliated manifolds and adapted connections (see [15]) on almost contact manifolds
(see [7]). We give a necessary and sufficient condition for a pair of symmetric connections to have the
same system of F−geodesics. Moreover, we use here the notion of F−planar diffeomorphism (see [13],
[17], and [18]), which extends the classical projective transformation (see [29]), the holomorphic-projective
2010 Mathematics Subject Classification. Primary 53B05, 53C22, 53C12, Secondary 53C56, 53C80, 53D10.
Keywords. geodesic, magnetic curve, projective transformation, holomorphically projective transformation, C-projective transfor-
mation, foliated manifold, almost contact manifold, Vranceanu connection.
Received: 20 August 2014; Accepted: 03 November 2014
Communicated by Ljubica S. Velimirovic
Email addresses: bejanliv@yahoo.com (Cornelia-Livia Bejan), simonadruta@yahoo.com (Simona-Luiza Drut¸
˘
a-Romaniuc)
C.L. Bejan, S.L. Drut¸˘a-Romaniuc / Filomat 29:10 (2015), 2367–2379 2368
(H−projective) transformation from both the complex (see [31], [26]) and locally product (para-complex)
context (see [24], [25]), as well as the C−projective transformation from the almost contact case (see [15],
[22]).
On a 1−codimensional foliation, we construct a tensor field of Weyl type which is invariant under any
F−planar diffeomorphism.
Throughout this note, all geometric objects are assumed to be smooth, the Einstein convention summa-
tion is used, and the derivative
˙
γ(t) with respect to t of a curve γ(t) on a manifold denotes the speed vector
field, while the derivative of a function f is denoted by f
0
.
2. F−geodesics
The main ingredients used in the present note are provided in the following:
Notations 1: By a couple (M, F) (resp. a triple (M, F, ∇)) we mean a manifold M endowed with a
(1,1)-tensor field F (resp. a couple as above, with a linear connection ∇).
The following notion is slightly different from the notion of F−planar curve (see [13], [17], and [18]), it
generalizes the geodesics, and it is followed by some examples.
Definition 2.1. We say that a smooth curve γ : I → M on a manifold (M, F, ∇) is an F−geodesic if γ(u) satisfies:
∇
˙
γ(u)
˙
γ(u) = F
˙
γ(u). (1)
Note that the above notion is completely different from that of Φ−geodesic (see [28]), which means a
classical geodesic on a Sasakian manifold, whose velocity vector field is horizontal.
Remark 2.2. (a) If t is another parameter for the same curve γ(u) then the relation (1) becomes:
∇
˙
γ(t)
˙
γ(t) = α(t)
˙
γ(t) + β(t)F
˙
γ(t), (2)
where α and β are some functions on the curve γ(t).
(b) A curve γ(t) satisfying the relation (2) describes an F−geodesic up to a reparameterization.
(c) From geometrical point of view, an F−geodesic (up to a reparameterization) is defined as a curve γ(t) such that
the parallel transport along the curve preserves the tangent subspace (of dimension 1 or 2) spanned by
˙
γ(t) and F
˙
γ(t).
(d) F−geodesics are a special case of F−planar curves. Not every F−planar curve is an F−geodesic, because
generally, a transformation to a canonical parameter in equation (2), with a given tensor field F does not necessarily
lead to the form (1), but to a form
∇
˙
γ(u)
˙
γ(u) = f (t)F
˙
γ(u),
with a function f of parameter t.
e) The variational problem of F−planar curves was solved in [14] (see [16]).
Recall from the Riemannian context, the existence and uniqueness of the solution of a second order
differential equation with initial data, which gives the existence and uniqueness of a geodesic passing
through a given point p ∈ M, with a given velocity X
p
∈ T
p
M. These properties are extended in [3] to
magnetic curves corresponding to an arbitrary magnetic field. The first question arising on a triple (M, F, ∇)
is about the existence of the F−geodesics. The theory of differential systems with Cauchy condition leads
to the following generalization of the mentioned result.
Lemma 2.3. Let (M, F, ∇) be as in Notations 1. Then, for any p ∈ M and X
p
∈ T
p
M, there exists a unique maximal
F−geodesic passing through p and having the velocity X
p
.
C.L. Bejan, S.L. Drut¸˘a-Romaniuc / Filomat 29:10 (2015), 2367–2379 2369
Examples of F−geodesics
(i) If F is identically zero, then an F−geodesic becomes a classical geodesic, and moreover an F−geodesic up to
a reparameterization becomes a geodesic up to a reparameterization.
(ii) When F is the identity endomorphism up to a multiplicative function, then an F−1eodesic is a geodesic
up to a reparameterization.
(iii) In the context of Lagrangian mechanics, the Euler-Lagrange equations of systems with frictions, i.e.
with non-conservative forces F
i
(not of gradient type) take the form
d
dt
∂L
∂
˙
q
i
−
∂L
∂q
i
= F
i
, i = 1, k, (3)
where L(t, q,
˙
q) denotes a Lagrangian function, depending on the coordinates (q
i
) (and on their derivatives
˙
q
i
) of a submanifold M, which are given by x
i
= x
i
(q
i
, . . . , q
k
), i = 1, n, in R
n
, with the cartesian coordinates
(x
1
, . . . , x
n
). For the general theory of a Lagrange space (M, L) we refer to [19].
We assume here that the dissipative forces F
i
are expressed by:
F
i
= −
k
X
j=1
f
ij
˙
q
j
, i = 1, k, (4)
where ( f
ij
)
i,j=1,k
are the function coefficients.
To focus on the classical example in mechanics, we take in particular the function L(q,
˙
q) to be the kinetic
energy T for a particle of mass m:
T =
m
2
k
X
i=1
(
˙
x
i
)
2
=
1
2
k
X
i,j=1
1
ij
(q)
˙
q
i
˙
q
j
, (5)
where (1
ij
)
i,j=1,k
is a Riemannian metric on M. Then, in view of Definition 2.1, the trajectory of a particle
described by (3) (with (4) and (5)) is an F−geodesic.
(iv) In the 3-dimensional Riemannian case, let F be the Lorentz force setting as:
FX = B × X, ∀X ∈ Γ(TM), (6)
where B is the magnetic induction.
Then the notion known in literature as a normal magnetic curve (see [1]-[3], [8], [11], [21], [27]), can be
redefined in view of Definition 2.1, as being an F−geodesic γ(s), parameterized by its arc length, where F is
the Lorentz force.
To extend the above statement to the 3-dimensional pseudo-Riemannian case, we take into account that
a lightlike curve (see e.g. [12]) cannot be parameterized by its arc length. Therefore, the statement remains
true for any spacelike or timelike arc length parameterized curve γ(s), only.
(v) In higher dimensions, F may be the electro-magnetic tensor field, whose action on particle trajectories
was studied e.g. in [23].
(vi) We provide now another example of F−geodesics, by using the Lorentz force defined on a (pseudo)
Riemannian manifold of arbitrary dimension.
To do this, we recall the following notions for which we quote e.g. [2]:
Definition 2.4. On a (pseudo) Riemannian manifold (M, 1), a closed 2-form Ω is called a magnetic field if
it is associated by the following relation to the Lorentz force Φ, defined as a skew symmetric (w.r.t. 1)
endomorphism field on M:
1(Φ(X), Y) = Ω(X, Y), ∀X, Y ∈ Γ(TM). (7)
C.L. Bejan, S.L. Drut¸˘a-Romaniuc / Filomat 29:10 (2015), 2367–2379 2370
The Lorentz force Φ is a divergence free (1,1)-tensor field (i.e. div Φ = 0).
Let ∇ be the Levi-Civita connection of 1, and let q be the charge of a particle, describing a smooth
trajectory γ on M. Then the curve γ(t) whose speed
˙
γ(t) satisfies the Lorentz equation
∇
˙
γ(t)
˙
γ(t) = qΦ(
˙
γ(t)), (8)
is known in the literature as a magnetic curve of the magnetic field Ω.
According to Definition 2.1, the above Lorentz equation expresses the relation satisfied by an F−geodesic
of M, where F is defined by FX = qΦ(X), ∀X ∈ Γ(TM).
The action of the Lorentz force on particle trajectories in the sense of the present paper was studied e.g.
in [23].
3. Constructions of F−geodesics on TM by using lifts
Here, we use the well known method of lifting some geometric objects from the base manifold M to
the total space of its tangent bundle TM (for which we mention the classical monograph [32]), aiming to
provide some new classes of F−geodesics on TM.
Proposition 3.1. Let L (resp. ∇) be a (1, 1)−tensor field (resp. a linear connection) on a manifold M
n
, and let L
H
(resp. ∇
H
) denote its horizontal lift on TM.
(i) An integral curve of any vector field X on M is an L−geodesic w.r.t. ∇ if and only if the integral curve of X
H
is an L
H
−geodesic w.r.t. ∇
H
.
(ii) The above statement remains true, if ”L−geodesic” and ”L
H
−geodesic”, are replaced by ”L−geodesic up to a
reparameterization” and ”L
H
−geodesic up to a reparameterization”, respectively.
Proof. Let π : TM → M, be the tangent bundle of the manifold (M, ∇), and let (x
1
, . . . , x
n
) (resp. (x
1
, . . . , x
n
,
y
1
, . . . , y
n
)) be the local coordinates on M (resp. on TM). Recall that the horizontal lift of a vector field
X = X
i
∂
∂x
i
∈ Γ(TM) to the total space TM of the tangent bundle has the expression X
H
= X
i
δ
δx
i
, where Γ
h
ki
(x)
are the coefficients of the connection ∇ and
δ
δx
i
=
∂
∂x
i
− Γ
h
ki
y
k
∂
∂y
h
.
The horizontal lift of a vector field X ∈ Γ(TM) has the property:
( f X)
H
= f
V
X
H
, (9)
for every function f on M, where f
V
= f ◦ π.
Let γ be an L−geodesic up to a reparameterization (w.r.t. ∇) on M. Then relation (2) is satisfied.
Considering the horizontal lift in (2), then using (9) and the following properties of the horizontal lifts
of the (1,1)-tensor field L and of the conection ∇:
(LX)
H
= L
H
X
H
, ∇
H
X
H
Y
H
= (∇
X
Y)
H
, ∀X, Y ∈ Γ(TM), (10)
we obtain
∇
H
˙
γ(t)
H
˙
γ(t)
H
− α(t)
V
˙
γ(t)
H
− β(t)
V
L
H
˙
γ(t)
H
= 0. (11)
Since any vector field X ∈ Γ(TM) vanishes if and only if its horizontal lift X
H
vanishes, then the
equivalence between the relations (2) and (11) follows, and hence (ii) is proved.
In the particular case when α(t) = 0 and β(t) = 1, one obtains (i).
Our aim now is to obtain another class of F−geodesics on the total space of the tangent bundle, by using
metrics of natural type.
C.L. Bejan, S.L. Drut¸˘a-Romaniuc / Filomat 29:10 (2015), 2367–2379 2371
On a Riemannian manifold (M, 1), let ∇ be the Levi-Civita connection of 1. We denote by π : TM → M
the tangent bundle of M, whose total space is endowed with a natural diagonal metric G, i.e. a metric
defined by:
G(X
H
y
, Y
H
y
) = c
1
1
π(y)
(X, Y) + d
1
1
π(y)
(X, y)1
π(y)
(Y, y),
G(X
V
y
, Y
V
y
) = c
2
1
π(y)
(X, Y) + d
2
1
π(y)
(X, y)1
π(y)
(Y, y),
G(X
V
y
, Y
H
y
) = 0,
(12)
for all X, Y ∈ Γ(TM), y ∈ TM, where c
1
, c
2
, d
1
, d
2
are smooth functions depending on the energy density ρ
of y, defined as
ρ =
1
2
1
π(y)
(y, y). (13)
The metric G is positive definite provided that
c
1
, c
2
> 0, c
1
+ 2ρd
1
, c
2
+ 2ρd
2
> 0.
When c
1
= c
2
= 1 and d
1
= d
2
= 0, the metric G reduces to the Sasaki metric 1
S
.
The Levi-Civita connection of G, denoted by
e
∇ has the following expressions on the horizontal and resp.
on the vertical distribution of TTM:
e
∇
X
V
Y
V
=
c
0
2
2c
2
(1(X, y)Y
V
+ 1(Y, y)X
V
)−
−
c
0
2
−2d
2
2(c
2
+2ρd
2
)
1(X, Y)y
V
+
c
2
d
0
2
−2c
0
2
d
2
2c
2
(c
2
+2ρd
2
)
1(X, y)1(Y, y)y
V
,
(14)
e
∇
X
H
Y
H
= (∇
X
Y)
H
−
d
1
2c
1
(1(X, y)Y
V
+ 1(Y, y)X
V
)−
−
c
0
1
2(c
2
+2ρd
2
)
1(X, Y)y
V
−
c
2
d
0
1
−2d
1
d
2
2c
2
(c
2
+2ρd
2
)
1(X, y)1(Y, y)y
V
−
1
2
(R(X, Y)y)
V
,
(15)
for all X, Y ∈ Γ(TM), y ∈ TM, where R is the curvature tensor field on the base manifold M.
By using the expression (15), we provide the following:
Proposition 3.2. Let (M, 1) be a Riemannian manifold endowed with a (1, 1)−tensor field L.
(i) An integral curve of any vector field X ∈ Γ(TM) is an L−geodesic w.r.t. the Levi-Civita connection ∇ of 1
if and only if the integral curve of the horizontal lift X
H
is an L
H
−geodesic w.r.t. the Levi-Civita connection
e
∇ of a
natural diagonal metric G, given by (12), provided that c
2
= const ∈ R and d
2
= 0.
(ii) The above assertion remains true, if instead of an ”L−geodesic” (resp. an ”L
H
−geodesic”) we take an
”L−geodesic up to a reparameterization” (resp. an ”L
H
−geodesic up to a reparameterization”).
Proof. Let γ be an L−geodesic up to a reparameterization (w.r.t. ∇) on M, i.e. γ satisfies
∇
˙
γ
˙
γ = α
˙
γ + βL
˙
γ, (16)
where α and β are some smooth functions on the curve.
For X = Y =
˙
γ, the relation (15) becomes
e
∇
˙
γ
H
˙
γ
H
= (∇
˙
γ
˙
γ)
H
−
d
1
c
1
1(
˙
γ, y)
˙
γ
V
−
c
0
1
2(c
2
+2ρd
2
)
1(
˙
γ,
˙
γ)y
V
−
c
2
d
0
1
−2d
1
d
2
2c
2
(c
2
+2ρd
2
)
(1(
˙
γ, y))
2
y
V
.
Replacing (16) into the above relation, and taking into account (9) and (10), if follows that
e
∇
˙
γ
H
˙
γ
H
= α
V
˙
γ
H
+ β
V
L
H
˙
γ
H
,
if and only if
d
1
c
1
1(
˙
γ, y)
˙
γ
V
+
c
0
1
2(c
2
+ 2ρd
2
)
1(
˙
γ,
˙
γ)y
V
+
c
2
d
0
1
− 2d
1
d
2
2c
2
(c
2
+ 2ρd
2
)
(1(
˙
γ, y))
2
y
V
= 0.
From [10, Lemma 3.2] it follows that the coefficients involved in the above relation vanish, and thus
item (ii) is proved.
If in particular α = 0 and β = 1, it follows that item (i) is also true.