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Conjugate covariant derivatives on vector bundles and duality.

Introduction

In a series of papers, namely [7]-[9], we have studied the geometry of pairs of linear connections on a manifold M which are conjugated with respect to a tensor field of (1,1)-type T [member of] [[TAU].sub.1.sup.1](M) satisfying the reduced quadratic equation: [T.sup.2] = [epsilon]I. Here I is the usual Kronecker tensor field I = ([[delta].sup.i.sub.j]) and [epsilon] [member of] {0, -1, +1}. Recently, an unified approach for the non-degenerate case [epsilon] = [+ or -]1 was presented in [2]. Related to the almost product case is the study [6] concerned with golden structures and [10] dealing with metallic structures.

Let us remark that all the previous studies involve only the tangent bundle TM of M. The aim of the present work is to extend these objects to an arbitrary vector bundle E endowed with a non-degenerate quadratic endomorphism [lambda] and a given covariant derivative [nabla]. An important remark is that the main property mentioned above, namely the duality of conjugate linear connections ([nabla], [[nabla].sup.[lambda]]), continues to hold in this general setting. Two new features of the present paper are: i) the use of local expressions for all the involved objects, which leads to a better picture; for example in Section 2, for the structural and virtual tensor fields of a pair (linear connection, endomorphism), ii) a special attention is given to the mean covariant derivative [[nabla].sup.0] which parallelizes the given [lambda]. More generally, we use [[nabla].sup.[lambda]] and the first associated Obata operator in order to determine the whole class C([lambda]) of covariant derivatives with respect to which [lambda] is parallel. The second Obata operator is used to express the variation of the curvature tensor field. We finish the first section with the study of anchored, particularly Lie algebroid, covariant derivatives on E.

The above computations are applied in the cases of two particular geometries involving again the tangent bundle. The first concerns the so-called Finsler bundle TM [x.sub.M] TM while the second is a proper Finsler geometry provided by a 1-homogeneous function F: TM [right arrow] R+ with non-degenerate square. Let us remark that the class of almost complex and almost product connections in vector bundles endowed with such endomorphisms are discussed also in [19] but our study follows a different path: we unify the treatment of these geometries and in this way we firstly determine the mean covariant derivative from an arbitrary pair ([nabla], [[nabla].sup.[lambda]]) and secondly we derive the set C([lambda]). A strong motivation for an unified treatment of almost complex and almost product geometries comes from the relationship (1.20) of curvatures of the conjugate covariant derivatives; the same relation holds for both geometries i.e. it does not depends on [epsilon]. For the usual case of the tangent bundle we associate to [nabla] two tensor fields of (1,2)-type called the structural and the virtual tensor field of the pair ([lambda], [nabla]) respectively. In the last section we compute these tensor fields in the case of a Finsler connection.

1 Conjugate covariant derivatives for [epsilon]-endomorphisms

Let [pi]: E [right arrow] N be a vector bundle of (paracompact) base [N.sup.n] and fibre [R.sup.k]. As usual, X(N) = [GAMMA](TN) and [GAMMA]([pi]) denote the [C.sup.[infinity]](N)-module of sections for the vector bundle [[tau].sub.N]: TN [right arrow] N and [pi], respectively; in fact we mainly follow the notations of [21]. A classical notion for this setting is:

Definition 1.1. ([21, p. 277]) A covariant derivative operator on [pi] is an R-bilinear map [nabla]: X(N) x r([pi]) [right arrow] [GAMMA]([pi]), (X, s) [right arrow] [[nabla].sub.X]s, such that:

i) [nabla] is tensorial in its first variable: [[nabla].sub.fx]s = f[[nabla].sub.X]s,

ii) [nabla] is a derivation in its second variable: [[nabla].sub.X](f s) = X(f) * s + f [[nabla].sub.X]s,

for all f [member of] [C.sup.[infinity]] (N), X [member of] X(N) and s [member of] [GAMMA]([pi]).

Let now [epsilon] [member of] {[+ or -]1} and [lambda] [member of] [GAMMA](End([pi])) [congruent to] End([GAMMA]([pi])). Recall from [21, p. 284] that [nabla] induces a covariant derivative operator [??] on the bundle End(E) through the relation:

([[??].sub.X][lambda])s:= [[nabla].sub.X]([lambda](s)) - [lambda]([V.sub.X]s), (1.1)

for all X [member of] X(N) and s [member of] [GAMMA](n).

For the given [lambda] the natural problem is to obtain the class of all [nabla] such that [??][lambda] = 0; let us denote by C([lambda]) the set of these covariant derivatives. In the following we restrict to a particular remarkable type of such endomorphisms:

Definition 1.2. [lambda] is an e-endomorphism if: [[lambda].sup.2] = [[epsilon]1.sub.[GAMMA](E)].

From now on we suppose that the fixed [lambda] is an [epsilon]-endomorphism. In order to study the above problem we follow the method of [2] and [7]-[9] by introducing:

Definition 1.3. The [lambda]-conjugate of [nabla] is [[nabla].sup.[lambda]]: X(N) x [GAMMA]([pi]) [right arrow] [GAMMA]([pi]) given by:

[[nabla].sup.[lambda].sub.X]s:= [epsilon][lambda]([[nabla].sub.X]([lambda]s)). (1.2)

More generally, the [lambda]-conjugate of [nabla] is [[nabla].sup.[lambda]]: X(N) x [GAMMA]([pi]) [right arrow] [GAMMA]([pi]) given by:

[[nabla].sup.[lambda]]:= [[lambda].sup.-1] [??] [nabla] [??] ([1.sub.X(N)],[lambda]), (1.2gen)

for any invertible endomorphism [lambda].

It follows immediately that [[nabla].sup.[lambda]] is also a covariant derivative operator on [pi]. Let us remark that defining the mean covariant derivative of [nabla] and [[nabla].sup.[lambda]]:

[[nabla].sup.0] = 1/2 ([nabla] + [[nabla].sup.[lambda]]) (1.3)

we obtain the desired solution of the problem, namely: [[??].sup.0] [lambda] = 0; see also the Proposition 1.5 below. It follows that a study of [[nabla].sup.[lambda]] is necessary and in the following we provide some properties of this conjugate covariant derivative. A first important property is the duality announced in the title:

Proposition 1.4. [([[nabla].sup.[lambda]]).sup.[lambda]] = [nabla], hence [([[nabla].sup.[lambda]]).sup.0] = [[nabla].sup.0].

Proof. A direct consequence of the definition is:

[[nabla].sup.[lambda].sub.X] ([lambda]s) = [lambda]([[nabla].sub.X] s) (1.4)

and hence:

[mathematical expression not reproducible]

which implies the conclusion.

The next property concerns with the hat-versions of the involved [nabla]'s:

Proposition 1.5. [[??].sup.[lambda]][lambda] = -[??][lambda] and then [nabla] [member of] C([lambda]) if and only if [[nabla].sup.[lambda]] [member of] C([lambda]), that is [[nabla].sup.[lambda]] = [nabla] = [[nabla].sup.0].

Proof. We have directly from the definitions:

[mathematical expression not reproducible]

which is the claimed equation.

In the following we express locally the above objects. Let h = (U, [u.sup.[alpha]]; [alpha] = 1,...,n) be a local chart on N and suppose that [E|.sub.U]:= [[pi].sup.-1] (U) has a trivialization. By the same arguments as in [21, p. 279] it follows that there exists a local frame field [S.sub.U] = {[s.sub.i]; i = 1,...,k} of [pi] over U. So, any local section of [[GAMMA].sub.U]([pi]) can be uniquely written as a [C.sup.[infinity]] (U)-linear combination of elements of [S.sub.U]. In particular:

[mathematical expression not reproducible] (1.5)

for some smooth functions [[GAMMA].sup.j.sub.[alpha]i] [member of] [C.sup.[infinity]](U). Following the cited book we call these functions the Christoffel symbols of [nabla] with respect to the chart h and the local frame field [S.sub.U]. Also we have the local expression of [lambda]:

[lambda]([s.sub.i]) = [[lambda].sup.j.sub.i] [s.sub.j] (1.6)

with [[lambda].sup.j.sub.i] [member of] [C.sup.[infinity]](U). Let us denote by [mathematical expression not reproducible] the Christoffel symbols of [[nabla].sup.[lambda]] ([[nabla].sup.0]) with respect to the same pair (h, Su). A straightforward computation yields the change [GAMMA] [right arrow][[GAMMA].sup.[lambda]]:

[mathematical expression not reproducible] (1.7)

Recall the usual local expression of the [nabla]-covariant derivative of [lambda]:

[mathematical expression not reproducible] (1.8)

Then the conjugate and the mean covariant derivatives can be expressed in the following way:

[mathematical expression not reproducible] (1.9)

The relationship in formulae (1 3) and (1 9) can be represented as follows:

[mathematical expression not reproducible] (1.9 fig)

If [omega] = ([[omega].sup.j.sub.i]) is the connection 1-form of [??] and [omega] is the connection 1-form of [??] then, from [[omega].sup.j.sub.i] = [[GAMMA].sup.j.sub.[alpha]i] [du.sup.[alpha]] it follows that:

[[??].sup.j.sub.i] -[[omega].sup.j.sub.i] = [epsilon][[lambda].sup.j.sub.k][[lambda].sup.k.sub.i|[alpha]] [du.sup.[alpha]] = -[epsilon][[lambda].sup.k.sub.i][lambda][[lambda].sup.j.sub.k|[alpha]] [du.sup.[alpha]]. (1.10)

Recall also the curvature of [nabla]:

R([[partial derivative]/[[partial derivative]u.sup.[alpha]]], [[partial derivative]/[[partial derivative]u.sup.[beta]]]) [s.sub.i] = [R.sup.j.sub.[alpha][beta]i] [S.sub.j], (1.11)

where:

[R.sup.j.sub.[alpha][beta]i] = [[[partial derivative][GAMMA].sup.j.sub.[beta]i]/[[partial derivative]u.sup.[alpha]]] - [[[partial derivative][GAMMA].sup.j.sub.[alpha]i]/[[partial derivative]u.sup.[beta]]] + [[GAMMA].sup.k.sub.[beta]i] [[GAMMA].sup.j.sub.[alpha]k] - [[GAMMA].sup.k.sub.[alpha]i] [[GAMMA].sup.j.sub.[beta]k]. (1.12)

Example 1.6. A) Suppose that E is the tangent bundle TN, hence [lambda] is a tensor field of (1,1)-type on N and n = k while [alpha] [member of] {1,....,n}.

i) The case [epsilon] = 1, corresponding to almost complex geometry, was studied in [7]; n must be an even positive integer.

ii) The case [epsilon] = +1, corresponding to almost product geometry, was studied in [8].

iii) From the expression (1.9) we derive also a relation between the torsions of [nabla] and [[nabla].sup.[lambda]], [[nabla].sup.0]:

[[??].sup.j.sub.[alpha]i]= [T.sup.j.sub.[alpha]i] + [epsilon][[lambda].sup.j.sub.k] ([[lambda].sup.k.sub.i|[alpha]]-[[lambda].sup.k.sub.[alpha]|i]), [[??].sup.j.sub.[alpha]i] = [T.sup.j.sub.[alpha]i] + [[epsilon]/2][[lambda].sup.j.sub.k]([[lambda].sup.k.sub.i|[alpha]]-[[lambda].sup.k.sub.[alpha]|i]). (1.13)

iv) In the setting of G-structures suppose that [lambda] is integrable. Then there exists an atlas on N such that the components of [lambda] are constant and hence:

[[??].sup.j.sub.[alpha]i]= [epsilon][[lambda].sup.j.sub.k][[GAMMA].sup.k.sub.[alpha]l] [[lambda].sup.l.sub.i], [[??].sup.j.sub.[alpha]i]= [[epsilon]/2] [[lambda].sup.j.sub.k] [[GAMMA].sup.k.sub.[alpha]l][[lambda].sup.l.sub.i] + [1/2][[GAMMA].sup.j.sub.[alpha]l] (1.14)

which yields:

[[??].sup.j.sub.[alpha][beta]i] = [epsilon][[lambda].sup.j.sub.k][R.sup.k.sub.[alpha][beta]l][[lambda].sup.l.sub.i], [[??].sup.j.sub.[alpha][beta]i] = [[epsilon]/2][[lambda].sup.j.sub.k][R.sup.k.sub.[alpha][beta]l][[lambda].sup.l.sub.i] + [1/2][R.sup.j.sub.[alpha][beta]i] (1.15)

We recover the result of Proposition 2.1 of [2], namely [nabla] is flat if and only if [[nabla].sup.[lambda]] is also flat; this result holds even if [lambda] is not-integrable as well as for [[nabla].sup.0]. The relationships between the torsions of [nabla] and [[nabla].sup.[lambda]], [[nabla].sup.0] are:

[[??].sup.j.sub.[alpha]i]= [epsilon][[lambda].sup.j.sub.k][[GAMMA].sup.k.sub.[alpha]l] [[lambda].sup.l.sub.i], [[??].sup.j.sub.[alpha]i]= [1/2][T.sup.j.sub.[alpha]i]+[[epsilon]/2][[lambda].sup.j.sub.k][T.sup.k.sub.[alpha]l] [[lambda].sup.l.sub.i] (1.16)

B) Almost complex structures on the vertical bundle associated to the tangent bundle are studied in [4], while almost product structures on the same bundle are studied in [17].

Returning to the general case we describe now the [C.sup.[infinity]] (N)-affine module C([lambda]). The Kronecker endomorphism I [member of] [GAMMA](End([pi])) = End([GAMMA]([pi])) acts locally as:

I ([s.sub.i]) = [[delta].sup.j.sub.i] [s.sub.j] (1.17)

and [lambda] has two associated (2, 2)-tensor fields, called Obata operators:

[[ohm].sup.hk.sub.ij] = [1/2] ([[delta].sup.h.sub.i][[delta].sup.k.sub.j] + [epsilon][[lambda].sup.h.sub.i][[lambda].sup.k.sub.j]), [[PSI].sup.hk.sub.ij] = [1/2] ([[delta].sup.h.sub.i][[delta].sup.k.sub.j] - [epsilon][[lambda].sup.h.sub.i][[lambda].sup.k.sub.j]). (1.18)

A straightforward computation yields:

Proposition 1.7. 1. The generic element [[nabla].sup.g] of C([lambda]) has the expression:

[[??]j.sub.[alpha]i,]=[[??]j.sub.[alpha]i,] +[[ohm].sup.lj.sub.i[alpha]][X.sup.[alpha].sub.[alpha]l], (1.19)

with arbitrary X = ([X.sup.[alpha].sub.[alpha]l]).

2. If [nabla] [member of] C([lambda]) then [omega] and [PSI] are also covariant constant with respect to [nabla].

The second Obata operator is useful to express globally the first equation in (1.15) through:

[??] = R - 2[PSI](R), (1.20)

a relation obtained in [11] for E = TN and conjugate connections with respect to non-degenerate (0, 2)-tensor fields; see also [12]. In conclusion R= R - [PSI](R). Let us remark that from [[nabla].sup.g] [member of] C([lambda]) it results that R commutes with [lambda]:

[??] (*,*) [??] [lambda] = [lambda][??] [??] (*,*). (1.21)

Another approach for the pair ([[nabla].sup.[lambda]], [[nabla].sup.0]) is expressed in terms of quasi-covariant derivatives, more precisely [lambda]-covariant derivatives, which are maps D as in Definition 1.1 with the second condition replaced by: ii[lambda]) [D.sub.X] (fs) = f [D.sub.X]s + X(f) * [lambda](s).

It is easy to see that any covariant derivative [nabla] yields a [lambda]-covariant derivative [D.sup.[nabla]] through:

[D.sup.[nabla].sub.X]:= [[nabla].sub.x] [??] [lambda] (1.22)

and then:

[[nabla].sup.[lambda]] = [epsilon][lambda] [??] [D.sup.[nabla]], [[nabla].sup.0.sub.x] = [[epsilon]/2]([lambda] [??] [D.sup.[lambda].sub.X] + [D.sup.[lambda]] [??] [lambda]) (1.23)

for every X [member of] X(N). Also, it follows that: [V.sub.X] = [epsilon][D.sup.[nabla].sub.X] [??] [lambda].

We finish this section with a slight generalization concerning algebroid covariant derivatives. Namely, let (A, N, [tau]: A [right arrow] N, [rho]) be an anchored vector bundle over N of rank r, i.e. [rho]: A [right arrow] TN is a morphism of vector bundles over the identity of N, called anchor, [13, p. 7]. Following the cited book we consider:

Definition 1.8. An A-covariant derivative on n is an R-bilinear map

D: [GAMMA]([tau]) x [GAMMA]([pi]) [right arrow] [GAMMA]([pi]), ([xi], s) [right arrow] [D.sub.[xi]]s satisfying:

i) D is tensorial in the first variable: D([f.sub.[xi]])s = f[D.sub.[xi]]s,

ii) D is a derivation in the second variable: [D.sub.[xi]] (fs) = f[D.sub.[xi]]s + ([rho] [??] [xi])(f)s, for all

f [member of] [C.sup.[infinity]] (N), [xi] [member of] [GAMMA]([tau]) and s [member of] [GAMMA]([pi]).

We note that this notion also occurs in the infinite dimensional setting of Banach vector bundles, but under another name, in [3]. We can consider the same problem for [lambda] as above in the setting of anchored covariant derivatives and all the definitions and results 1.3-1.5 hold with X replaced by [xi]. Let us express locally these objects. Let h = (U, [u.sup.[alpha]]), [alpha] = 1,...,n be a local chart on N and the trivialization of [tau] and [pi] respectively:

i) [A|.sub.U]:= [[tau].sup.-1](U) has a trivialization [S.sup.A.sub.U] = {[e.sub.A], A = 1,...,r},

ii) [E|.sub.U] has a trivialization as above.

The trivialization [S.sup.A.sub.U] yields:

1) the linear fibre coordinates [z.sup.A] i.e. [A|.sup.U] has the local coordinates ([u.sup.[alpha]], [z.sup.A]),

2) the smooth functions ([13, p. 7]) [rho]A:= [[??].sup.[alpha]] [??] [rho] [??] [e.sub.A] [member of] [C.sup.[infinity]](U). Hence:

[rho]([e.sub.A]) = [[rho].sup.[alpha].sub.A][[partial derivative]/[[partial derivative]u.sup.[alpha]]] (.)

A fixed A-covariant derivative D has the local expression: [D.sup.U.sub.eA][s.sub.i]:= [[GAMMA].sup.j.sub.Ai][s.sub.j] with [[GAMMA].sup.j.sub.Ai] [member of] [C.sup.[infinity]] (U). Then its conjugate A-covariant derivative [D.sup.[lambda]] is the generalization of (1.7):

[[??].sup.j.sub.Ai] = [epsilon][[lambda].sup.j.sub.i] ([[rho].sup.[alpha].sub.A][[partial derivative][[lambda].sup.k.sub.i]/[[partial derivative]u.sup.[alpha]]]+[[GAMMA].sup.k.sub.Al][[lambda].sup.l.sub.i]). (1.24)

Also, we consider the generalization of (1.8):

[[lambda].sup.k.sub.i|A]:= [[rho].sup.[alpha].sub.A][[partial derivative][[lambda].sup.k.sub.i]/[[partial derivative]u.sup.[alpha]]]+[[GAMMA].sup.k.sub.Al][[lambda].sup.l.sub.i] - [[GAMMA].sup.l.sub.Ai][[lambda].sup.k.sub.l] (1.25)

and hence (1.9) generalizes to:

[mathematical expression not reproducible]. (1.26)

We remark that if A is the usual tangent bundle of N then (1.24) - (1.26) reduce to the formulae (1.7) - (1.9) since [rho] is the Kronecker endomorphism (1.17).

In order to introduce the curvature we suppose that, in addition, A is a Lie algebroid i.e. [GAMMA]([tau]) is endowed with a Lie bracket [*, *]A following [13, p. 7]. Then the curvature tensor field of D is ([13, p. 48]):

R([xi], [eta])s:= [D.sub.[xi]][D.sub.[eta]]s - [D.sub.[eta]][D.sub.[xi]]s - [D.sub.[[xi],[eta]]A]s (1.27)

with local coefficients:

R([e.sub.A],[e.sub.B])[s.sup.i]:= [R.sup.j.sub.ABi][s.sub.j]., [R.sup.j.sub.ABi] [member of] [C.sup.[infinity]] (U). (1.28)

A straightforward computation yields:

[mathematical expression not reproducible] (1.29)

where the coefficients [THETA] are provided by the A-Lie bracket:

[[THETA].sup.C.sub.AB]:= [z.sup.C] [??] [[e.sub.A],[e.sub.B]]A. (1.30)

In the particular case [tau] = [pi] we can also define the torsion of D, ([13, p. 48]):

T([xi], [eta]): = [D.sub.[xi]][eta] - [D.sub.[eta]][xi] - [[[xi], [eta]].sub.A] (1.31)

and hence:

T([e.sub.A],[e.sub.B]):= [T.sup.C.sub.[AB.sup.e]C],:= [T.sup.C.sub.AB]:= [[GAMMA].sup.C.sub.AB] - [[GAMMA].sup.C.sub.BA] - [[THETA].sup.C.sub.AB]. (1.32)

2 The case of a Finsler bundle endowed with an [epsilon]-endomorphism

Let now M be a smooth manifold of dimension m and [tau] = [[tau].sub.M]: TM [right arrow] M be its tangent bundle. A local chart [h.sub.M] = (U, [x.sup.i]; i = 1,...,m) on M induces a local chart ([[tau].sup.-1](U),[x.sup.1][x.sup.2]) on T[M|.sub.U], where u [member of] T[M|.sup.U] is expressed as u = [y.sup.i][[partial derivative]/[[partial derivative]x.sup.i]].

The vector bundle in the preceding section is [pi]: E = TM [x.sub.M] TM [right arrow] N = TM with [21, p. 179]:

TM [x.sub.M] TM = {(u, v) [member of] TM x TM; [tau](u) = [tau]([nu])} (2.1)

and [pi] (u, v) = u. The fiber of n over u e TM is [[pi].sup.-1] (u) = {u} x [T.sub.[tau](u)]M and hence k = m and n = 2m. Usually, [pi] is called the Finsler bundle of M and its sections are called Finsler vector fields. The [C.sup.[infinity]] (TM)-module of Finsler vector fields is canonically isomorphic with the [C.sup.[infinity]] (TM)-module of sections of [[tau].sub.M] along [[tau].sub.M]:

[[GAMMA].sub.[tau]M](TM):= {[X.bar] [member of] [C.sup.[infinity]] (TM, TM); [[tau].sub.M] [??] [X.bar] = [[tau].sub.M]}. (2.2)

Hence the Finsler vector fields are of the form:

[??]: u [member of] TM [right arrow] [??](u) = (u, [X.bar](u)) [member of] TM [x.sub.M] TM (2.3)

or briefly [??] = ([1.sub.TM], [X.bar]). A remarkable Finsler vector field C corresponds to [X.bar] = [1.sub.TM] [member of] [[GAMMA].sub.[tau]M] (TM) and then:

C = ([1.sub.TM], [1.sub.TM]): u [member of] TM [right arrow] (u, u) [member of] TM [x.sub.m] TM. (2.4)

In a compact form we have the Liouville vector field C = [y.sup.i][[partial derivative]/[[partial derivative]y.sup.i]] [member of] X(TM).

The local chart [h.sub.M] of M determines for u [member of] [[tau].sup.1](U) [subset] T[M|.sub.U] the basis {[bar.[partial derivative].sub.i](u)} of the fibre [[pi].sup.-1] (u) given by:

[mathematical expression not reproducible]. (2.5)

Hence [S|.sub.U] = {[s.sub.i] = [bar.[partial derivative].sub.i]; i = 1,n} is a local frame field of [pi] over U.

Fix now an [epsilon]-endomorphism [lambda] of this [pi]; hence [lambda] is a particular case of a Finsler tensor field of (1,1)-type. The equations:

[lambda]([bar.[partial derivative].sub.i]i) = [[lambda].sup.j.sub.i] [bar.[partial derivative].sub.j] (2.6)

give its components [[lambda].sup.j.sub.i] [member of] [C.sup.[infinity]] ([[tau].sup.-1] (U)):

[[lambda].sup.j.sub.i] = [[lambda].sup.j.sub.i] ([u.sup.[alpha]]) = [[lambda].sup.j.sub.i] ([x.sup.1],...,[x.sup.m], [y.sup.1],...,[y.sup.m]). (2.7)

Fix now the covariant differential operator [nabla] for [pi]. The local expression of [nabla]:

[mathematical expression not reproducible] (2.8)

generates the pair of Christoffel symbols: i) horizontal: [[GAMMA].sup.k.sub.ij] = [[GAMMA].sup.k.sub.ij](x,y), ii) vertical: [C.sup.k.sub.ij] = [C.sup.k.sub.ij] (x,y). The conjugate [[nabla].sup.[lambda]] has the pair ([[??].sup.k.sub.ij], [[??].sup.k.sub.ij]):

[mathematical expression not reproducible] (2.9)

Then [nabla] yields two covariant derivatives: one horizontal | and one vertical |. For [lambda] we have:

[[lambda].sup.k.sub.j|i]:= [[partial derivative][[lambda].sup.k.sub.j]/[[partial derivative]x.sup.i]] + [[GAMMA].sup.k.sub.il][[lambda].sup.l.sub.j] - [[GAMMA].sup.l.sub.ij][[lambda].sup.k.sub.l], [[lambda].sup.k.sub.j/i]:= [[partial derivative][[lambda].sup.k.sub.j]/[[partial derivative]y.sup.i]] + [C.sup.k.sub.il] [[lambda].sup.l.sub.j]-[C.sup.l.sub.ij] [[lambda].sup.k.sub.l] (2.10)

and then:

[mathematical expression not reproducible] (2.11)

Also, the mean covariant derivative [[nabla].sub.0] has the pair ([[??].sup.k.sub.ij], [[??].sup.k.sub.ij]):

[mathematical expression not reproducible] (2.12)

Example 2.1. Suppose that [lambda] is a tensor field on the base M; then [lambda] = [lambda](x). It follows that:

[mathematical expression not reproducible] (2.13)

and then, the vertical part of both [[nabla].sup.0] and [[nabla].sup.X] follows the path of formula (1.14). In particular, if the vertical part of [nabla] vanishes then also the vertical parts of [[nabla].sup.[lambda]] and [[nabla].sup.0] vanish; in particular this is the case discussed in [2] when [GAMMA] lives also on the base M i.e. [GAMMA] = [GAMMA](x).

Now, with m = 2 and following the example 1.6 part iv) we consider:

[mathematical expression not reproducible]. (2.14)

The first equations (2.13) become:

[mathematical expression not reproducible] (2.15)

or, in more details:

[mathematical expression not reproducible] (2.16)

The corresponding result of Proposition 1.7 for this setting is:

Proposition 2.2. The generic element of C([lambda]) has the expression:

[mathematical expression not reproducible] (2.17)

with arbitrary X = ([X.sup.a.sub.il]) and Y = ([Y.sup.a.sub.il]).

Let us finish this section with the simple case of the tangent bundle [[tau].sub.m]; then [lambda] [member of] [[TAU].sup.1.sub.1](M). Two tensor fields of (1,2)-type are associated in [2] to a given linear connection [nabla]: the structural and the virtual tensor field, denoted respectively by C[??] and B[??]. After a short computation of their initial expression we write them in more useful form as: 1) the structural tensor field:

C[??](X, Y):= [1/2]{([[nabla].sub.[lambda]x][lambda])Y - [lambda][([[nabla].sub.[lambda]x])(Y)]}, (2.18)

2) the virtual tensor field:

B[??](X, Y):= 2{([[nabla].sub.[lambda]X][lambda])Y + [lambda][([[nabla].sub.[lambda]][lambda])(Y)]}. (2.19)

Their utility is provided by the relation (30) in [2]:

[[nabla].sub.X] = [nabla] + [epsilon](B[??] - C[??]) [??] [[nabla].sub.0] = [nabla] + 2(B[??] - C[??]). (2.20)

Suppose that locally we write:

[mathematical expression not reproducible] (2.21)

Since the equation (1.8) is expressed by:

[mathematical expression not reproducible] (2.22)

we derive:

[mathematical expression not reproducible] (2.23)

3 Finsler geometry endowed with an [epsilon]-endomorphism

Recall from [5] that a Finsler fundamental function on M is a map F: TM [right arrow] R+ with the following properties:

F1) F is smooth on the slit tangent bundle [T.sub.0]M:= TM \ O and continuous on the null section O of [[tau].sub.M],

F2) F is positive homogeneous of degree 1: F(x, [lambda]y) = [lambda]F(x,y) for every [lambda] > 0,

F3) the matrix ([g.sub.ij]) = ([1/2][[[partial derivative].sup.2][F.sup2]/[[partial derivative]y.sup.i][[partial derivative]y.sup.i]]) is invertible and its associated quadratic form is positive definite. The tensor field g = {[g.sub.ij] (x,y); 1 [less than or equal to] i, j [less than or equal to] m} is called the Finsler metric and the homogeneity of F implies:

[F.sup.2] (x,y)= [g.sub.ij] [y.sup.i][y.sup.j] = [y.sub.i][y.sup.i], (3.1)

where [y.sub.i] = [g.sub.ij][y.sup.j]. The pair (M, F) is called Finsler manifold. In particular, if g does not depend on y, we recover the Riemannian geometry.

On N:= [T.sub.0]M we have two distributions:

i) V ([T.sub.0]M):= ker [pi]*, called the vertical distribution; does not depends of F. It is integrable and has the basis {[[partial derivative]/[[partial derivative]y.sup.i]]; 1 [less than or equal to] i [less than or equal to] m}. A remarkable section of it is the Liouville vector field C = [y.sup.1][[partial derivative]/[[partial derivative]y.sub.i]].

ii) H([T.sub.0]M) with the basis {[[delta]/[[delta]x.sup.i]]:= [[partial derivative]/[[partial derivative]y.sub.i]] - [N.sup.j.sub.i][[partial derivative]/[[partial derivative]y.sub.j]]}, where:

[N.sup.i.sub.j] = [1/2] [[partial derivative][gamma].sup.i.sub.00]/[[partial derivative]y.sup.i] (3.2)

and [[gamma].sup.i.sub.00] = [[gamma].sup.i.sub.jk][y.sup.j][y.sup.k] is built from the usual Christoffel symbols:

[mathematical expression not reproducible] (3.3)

H([T.sub.0]M) is often called the Cartan (or canonical) nonlinear connection of the geometry (M, F) and a remarkable section of it is the geodesic spray:

[S.sub.F]=[y.sup.i] [[delta]/[[delta]x.sup.i]] (3.4)

The Finslerian connections are triples [GAMMA] = ([N.sup.k.sub.i], [F.sup.k.sub.ij](x,y),[C.sup.k.sub.ij]-(x,y)) where [??] behave like the coefficients of a linear connection on M and C is a d-tensor field on [T.sup.0]M. Such a Finslerian connection yields the covariant derivative [GAMMA][DELTA] on [T.sub.0]M given by:

[mathematical expression not reproducible] (3.5)

There are four remarkable Finslerian connections, [5, p. 227]:

-Cartan C[alpha] = ([N.sup.k.sub.i], [F.sup.k.sub.ij], [C.sup.k.sub.ij]),

-Chern-Rund CR = ([N.sup.k.sub.i], [F.sup.k.sub.ij], 0),

-Berwald B = ([N.sup.k.sub.i],[G.sup.k.sub.ij]., 0),

-Hashiguchi H = ([N.sup.k.sub.i], [G.sup.k.sub.ij], [C.sup.k.sub.ij]),

where [G.sup.k] = [N.sup.i.sub.j][y.sup.j] and:

[mathematical expression not reproducible] (3.6)

Let now [lambda] be an e-endomorphism given locally by [lambda] = ([[lambda].sup.j.sub.i](x,y)) as in (2.7). [GAMMA], or equivalently [GAMMA][DELTA], yields two covariant derivatives: one horizontal | and one vertical | which for [lambda] are:

[mathematical expression not reproducible] (3.7)

Then 1 has:

1) a conjugate Finsler connection [[GAMMA].sup.[lambda]] = ([N.sub.i.sup.k], [[[lambda].sup.k].[F.sub.ij]] (x, y), [[[lambda].sup.k].[C.sub.ij]] (x, y)) with coefficients as in (2.11),

2) a mean Finsler connection [[GAMMA].sup.0] = ([N.sub.i.sup.k], [[0.sup.k].[F.sub.ij]] (x, y), [[0.sup.k].[C.sub.ij]] (x, y)) with coefficients as in (2.12). [lambda] is covariant constant with respect to this Finsler connection:

[[[lambda].sup.k.sub.0] over j|i] = [[[lambda].sup.k.sub.0] over j|i] = 0. (3.8)

Example 3.1. The almost complex case ([epsilon] = -1) of [[GAMMA].sup.0] appears in [20, p. 15] while the almost product case ([epsilon] = +1) in [18, p. 61], see also [1]. An important remark of these works is that the set C([lambda]) derived from (2.12) is a commutative group.

Returning to the general case let us remark that the Finsler connection [GAMMA] defines the pairs: ([h.C.sup.[lambda].sub.[GAMMA]], [h.B.sup.[lambda].sub.[GAMMA]]), ([v.C.sup.[lambda].sub.[GAMMA]], [v.B.sup.[lambda].sub.[GAMMA]]) with:

[mathematical expression not reproducible] (3.9)

[mathematical expression not reproducible] (3.10)

[mathematical expression not reproducible] (3.11)

[mathematical expression not reproducible]. (3.12)

Hence, for the case [GAMMA] [member of] {Chern - Rund, Berwald} we have:

[mathematical expression not reproducible] (3.13)

and these tensor fields are zero if [lambda] is a basic endomorphism, i.e. [lambda] [member of] [T.sub.1.sup.1](M), or an integrable one, i.e. with constant components in a preferential atlas on M.

Let (d[x.sup.i], [delta][y.sup.i]:= d[y.sup.i] + [N.sub.j.sup.i]d[x.sup.j]) be the dual of the Berwald basis ([delta]/[delta][x.sup.i], [partial derivative]/[partial derivative][y.sup.i]). A final remark is that the given [lambda] yields four endomorphisms of N:

[A.sub.[+ or -]]([lambda]):= [[lambda].sub.i.sup.j][delta]/[delta][x.sup.j] [cross product] d[x.sup.i] [+ or -] [[lambda].sub.i.sup.j] [partial derivative]/[partial derivative][y.sup.j] [cross product] [delta][y.sup.i], [B.sub.[+ or -]]([lambda]):= [[lambda].sub.i.sup.j][delta]/[delta][x.sup.j] [cross product] [delta][y.sup.i] [+ or -] [[lambda].sub.i.sup.j] [partial derivative]/[partial derivative][y.sup.j] [cross product] d[x.sup.i]. (3.14)

We note that [A.sub.[+ or -]]([lambda]) and [B.sub.+]([lambda]) are exactly [epsilon]-endomorphisms on N while [B.sub.-]([lambda]) is an (-[epsilon])-endomorphism on N.

Example 3.2. Suppose that [epsilon] = +1 and [lambda] = [delta] = ([[delta].sub.i.sup.j]). Then: [A.sub.+]([delta]) = [1.sub.TN] is the Kronecker tensor field of tangent bundle TN; [A.sub.-]([delta]) is (together with the Sasaki-type metric [G.sub.F] on TN induced by ([g.sub.ij])) the almost para-Kahler structure [P.sub.F] from [15, p. 1880] while [B.sub.-]([delta]) is (again together with [G.sub.F]) the almost Kahler structure [[PSI].sub.F] from [16, p. 243]. For the general setting of tangent manifolds, particularly tangent bundles, endowed with nonlinear connections, the almost product structure [A.sub.-]([delta]) appears in [14, p. 14] and it is well-known that [A.sub.-]([delta]) is integrable if and only if the corresponding nonlinear connection is without curvature i.e. flat. Also, for any Finsler connection [GAMMA] we have [v.C.sup.[delta].sub.[GAMMA]]=[v.B.sup.[delta].sub.[GAMMA]]=[h.C.sup.[delta].sub.[GAMMA]]=[h.B.sup.[delta].sub.[GAMMA]]= 0.

References

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[2] C.-L. Bejan, M. Crasmareanu, Conjugate connections with respect to a quadratic endomorphism and duality, Filomat, 30(2) (2016), 2367-2374.

[3] A. Bejancu, h-connexions sur h-fibres vectoriels banachiques, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat., 54 (1973), 68-74.

[4] A. Bejancu, Complex structures on vertical bundle and CR-structures, Tensor, 46 (1987), 361-364.

[5] A. Bejancu, H. R. Farran, Foliations and geometric structures, Mathematics and Its Applications (Springer), 580, Springer, Dordrecht, 2006.

[6] A. M. Blaga, The geometry of Golden conjugate connections, Sarajevo J. Math., 10(23)(2) (2014), 237-245.

[7] A. M. Blaga, M. Crasmareanu, The geometry of complex conjugate connections, Hacet. J. Math. Stat., 41(1) (2012), 119-126.

[8] A. M. Blaga, M. Crasmareanu, The geometry of product conjugate connections, An. Stiint. Univ. Al. I. Cuza Iasi. Mat., 59(1) (2013), 73-84.

[9] A. M. Blaga, M. Crasmareanu, The geometry of tangent conjugate connections, Hacet. J. Math. Stat., 44(4) (2015), 767-774.

[10] A. M. Blaga, C. E. Hretcanu, Metallic conjugate connections, Rev. Union Mat. Argent., 59(1) (2018), 179-192.

[11] A. Bucki, Curvature tensors of conjugate connections on a manifold, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 33 (1979), 13-22 (1981).

[12] A. Bucki, On the existence of a linear connection so as a given tensor field of the type (1, 1) is parallel with respect to this connection, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 33 (1979), 23-28 (1981).

[13] M. Crampin, D. Saunders, Cartan geometries and their symmetries. A Lie algebroid approach, Atlantis Studies in Variational Geometry 4, Amsterdam: Atlantis Press, 2016.

[14] M. Crasmareanu, Nonlinear connections and semisprays on tangent manifolds, Novi Sad J. Math., 33(2) (2003), 11-22.

[15] M. Crasmareanu, L.-I. Piscoran, Para-CR structures of codimension 2 on tangent bundles in Riemann-Finsler geometry, Acta Math. Sin. (Engl. Ser.), 30(11) (2014), 1877-1884.

[16] M. Crasmareanu, L.-I. Piscoran, CR-structures of codimension 2 on tangent bundles in Riemann-Finsler geometry, Period. Math. Hungar., 73(2) (2016), 240-250.

[17] M. Crasmareanu, L.-I. Piscoran, Weak para-CR structures on vertical bundles, Adv. Appl. Clifford Algebr., 26(4) (2016), 1127-1136.

[18] F. C. Klepp, Almost product Finsler structures, An. Stiint. Univ. Al. I. Cuza Iasi Sect. I-a Mat., 28(2, suppl.) (1982), 59-67.

[19] F. C. Klepp, Some remarkable Finsler structures on vector bundles, An. Stiint. Univ. Al. I. Cuza Iasi Sect. I-a, 30(4) (1984), 45-48.

[20] R. Miron, On almost complex Finsler structures, An. Stiint. Univ. Al. I. Cuza Iasi Sect. I a Mat., 28(2) (1982), 13-17.

[21] J. Szilasi, R. L. Lovas, D. Cs. Kertesz, Connections, sprays and Finsler structures, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014.

Mircea Crasmareanu

Faculty of Mathematics, University "Al. I. Cuza", Iasi, 700506, Romania

E-mail: mcrasm@uaic.ro
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