Medical, Pharma, Engineering, Science, Technology and Business

Department of Mathematics, Kuvempu University, Shankaraghatta 577451,
Shimoga, Karnataka, India. **E-mail:** [email protected]

Department of Mathematics, Kuvempu University, Shankaraghatta 577451,
Shimoga, Karnataka, India. **E-mail:** [email protected]

Department of Mathematics, Bangalore University, Central College Campus,
Bangalore 577001, Karnataka, India. **E-mail:** [email protected]

Department of Mathematics, Kuvempu University, Shankaraghatta 577451,
Shimoga, Karnataka, India. **E-mail:** [email protected]

**Received Date:** September 22, 2008; **Revised Date:** January 12, 2009

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

The purpose of the present paper is to consider and study a special form of Rund's h-curvature tensor Ki ljk and Berwald's curvature tensor Hi ljk in an R3-like C-reducible Finsler space. In this paper, we modify the Rund's h-curvature tensor Ki ljk to special form by using some special Finsler spaces like C-reducible, R3-like Finsler spaces.

Let *F ^{n}* = (

We use the following notations from [5,8]:

(1.1)

**Definition 1.1** (see [1,2]). A Finsler space *F ^{n}* (n > 3) is called

(1.2)

where is the tensor.

**Definition 1.2** (see [2,3]). A Finsler space *F ^{n}* is called C-reducible if it satises the equation

(1.3)

where .

**Definition 1.3** (see[6]). A Finsler space *F ^{n}* is called P-reducible if the torsion tensor

(1.4)

where and.

The *v*-covariant derivative of P-reducible Finsler space is given by [9]

(1.5)

**Definition 1.4** (see[7]). A Finsler space is called Landsberg Finsler space if .
We use the following identities from [5,7,9]:

(1.6)

where is the (*v*) h-torsion tensor and the sux `0' means contraction with *y ^{i}*

We use the following lemma in the next section.

**Definition 1.5** (see [4]). *If the equation * *holds in F ^{n}, then we
have*,

Let *F ^{n}* be a Finsler space with Rund's h-curvature tensor of the special form [9]

(2.1)

where are Finsler tensor elds. Consider h-curvature tensor of the form

(2.2)

Using equations (1.2) and (1.3) in (2.2), we get

By using some Finsler identities, the above equation can be written as

After simplication and the rearrange the terms, we get

In simple form, the above equation can be written as a special form of (2.1) as

(2.3)

where

(2.4)

Thus we state the following.

**Theorem 2.1.** *In an R3-like, C-reducible Finsler space, the h-curvature tensor reduces to
special form of Rund's h-curvature tensor (2.3).*

Now we compare the Rund's curvature tensor and h-curvature tensor. Thus, from (2.1) and (2.2), we have

(2.5)

Contracting (2.5) with respect to *y ^{l}*,

(2.6)

Again contracting (2.6) with respect to *i* and *j*, we get

(2.7)

Now, we will nd *D _{00}*. Consider

(2.8)

Substituting (2.8) in (2.7), we have

Thus we state the following.

**Theorem 2.2.*** If the Rund's h-curvature tensor has the special form (2.1), then the scalar
curvature of the space is* 2(*n* – 1)*LF*^{2}.

Let us suppose that *F ^{n}* is

(2.9)

where

Thus we have the following.

**Theorem 2.3.** *In an R3-like C-reducible Finsler space, if the Rund's h-curvature tensor has
the special form (2.3), then the Cartan h-curvature tensor * *has the special form (2.9).*

Using the special form of Rund's h-curvature tensor in the Bianchi identity (1.6d), we get

(2.10)

Due to Lemma 1.1, equation (2.10) can be written as

Thus we have the following.

**Theorem 2.4.** *If the Rund's h-curvature tensor * *is of the special form (2.3), then both
the tensor elds A _{ij} and D_{ij} are symmetric simultaneously.*

It is also known that a Finsler space is Landsberg space with . If *F ^{n}* is
Landsberg, then from (1.6a) and (1.6e), we get

Thus we can propose the following.

**Corollary 2.5.*** If F ^{n} is a Landsberg space and the Rund's h-curvature tensor is of the
form (2.3), then Cartan curvature tensor coincides with the Berwald's curvature tensor.*

Now consider h-curvature tensor (1.6a) of the form

From equation (1.6e), the above equation can be written as

(2.11)

Suppose *F ^{n}* is a P-reducible Finsler space, then by using (1.4), (1.5), (2.3), and (2.11), we
have

(2.12)

where

Thus we have the following.

**Theorem 2.6.** *In a P-reducible Finsler space, and the special form of Rund's h-curvature
tensor has the special form of Berwald's curvature tensor, then is of the form
(2.12).*

Consider the Bianchi identity

(2.13)

Substituting (2.12) in (2.13), we get

(2.14)

Due to Lemma 1.1, equation (2.14) can be written as

Thus we have the following.

**Theorem 2.7.** *If in a Finsler space F ^{n} the h-curvature tensor is of the form (2.12),
then the tensor elds T_{jk} and M_{kj} both are simultaneously symmetric.*

The authors are highly thankful to the referee for valuable comments and suggestions.

- Izumi H, Srivastava TN (1978) On R3-like Finsler spaces. Tensor (N.S.) 32:340-349.
- Kitayama M (1998) Finsler spaces admitting a parallel vector field. Balkan J. Geom. Appl 3: 29-36.
- Matsumoto M (1972) On C-reducible Finsler spaces. Tensor (N.S.) 24: 29-37.
- Matsumoto M (1978) Finsler spaces with h?-curvature tensor Phljk of a special form. Rep. Math. Phys 14:113.
- Matsumoto M (1986) Foundations of Finsler Geometry and Special Finsler Spaces. Kaiseisha Press, Otsu, Saikawa.
- Narasimhamurthy SK, Bagewadi CS (2004) C-conformal special Finsler spaces admitting a parallel vector field. Tensor (N.S.) 65: 162-169.
- Rund H (1959) TheDiffrential Geometry of Finsler Spaces. Springer-Verlag, Berlin.
- Rund H (1959) On Landsberg spaces of scalar curvature. J. Korean Math. Soc 12: 79-100.
- Sinha BB, Dwivedi BB (1983) Finsler space with Rundâ€™s h-curvature tensor Kiljk of a special form. Publ. Inst. Math. (Beograd) (N.S.) 34: 205-209.

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