International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 161523 | https://doi.org/10.5402/2011/161523

Venkatesha, C. S. Bagewadi, K. T. Pradeep Kumar, "Some Results on Lorentzian Para-Sasakian Manifolds", International Scholarly Research Notices, vol. 2011, Article ID 161523, 9 pages, 2011. https://doi.org/10.5402/2011/161523

Some Results on Lorentzian Para-Sasakian Manifolds

Academic Editor: M. Dunajski
Received03 Jun 2011
Accepted03 Jul 2011
Published15 Aug 2011

Abstract

The object of the present paper is to study Lorentzian para-Sasakian (briefly LP-Sasakian) manifolds satisfying certain conditions on the 𝑊2-curvature tensor.

1. Introduction

In 1989, Matsumoto [1] introduced the notion of Lorentzian para-Sasakian manifold. Then, Mihai and Roşca [2] introduced the same notion independently and they obtained several results on this manifold. LP-Sasakian manifolds have also been studied by Matsumoto and Mihai [3], Mihai et al. [4], and Venkatesha and Bagewadi [5].

On the other hand, Pokhariyal and Mishra [6] have introduced new curvature tensor called 𝑊2-curvature tensor in a Riemannian manifold and studied their properties. Further, Pokhariyal [7] has studied some properties of this curvature tensor in a Sasakian manifold. Matsumoto et al. [8], and Yìldìz and De [9] have studied 𝑊2-curvature tensor in P-Sasakian and Kenmotsu manifolds, respectively.

In the present paper, we study some curvature conditions on LP-Sasakian manifolds. Firstly, we study LP-Sasakian manifolds satisfying 𝑊2=0 and 𝑊2-semisymmetric LP-Sasakian manifolds. Further, we study LP-Sasakian manifolds which satisfy 𝑃⋅𝑊2=0, 𝑀⋅𝑊2=0 and 𝐶⋅𝑊2=0, where 𝑃 is the projective curvature tensor, 𝑀 is the 𝑀-projective curvature tensor, and 𝐶 is the conformal curvature tensor.

2. Preliminaries

An 𝑛-dimensional differentiable manifold 𝑀 is called an LP-Sasakian manifold [1, 2] if it admits a (1,1) tensor field 𝜙, a contravariant vector field 𝜉, a 1-form 𝜂, and a Lorentzian metric 𝑔 which satisfy𝜙𝜂(𝜉)=−1,2∇𝑋=𝑋+𝜂(𝑋)𝜉,𝑔(𝜙𝑋,𝜙𝑌)=𝑔(𝑋,𝑌)+𝜂(𝑋)𝜂(𝑌),𝑔(𝑋,𝜉)=𝜂(𝑋),𝑋∇𝜉=𝜙𝑋,𝑋𝜙𝑌=𝑔(𝑋,𝑌)𝜉+𝜂(𝑌)𝑋+2𝜂(𝑋)𝜂(𝑌)𝜉,(2.1) where ∇ denotes the operator of covariant differentiation with respect to the Lorentzian metric 𝑔.

It can be easily seen that, in an LP-Sasakian manifold, the following relations hold:𝜙𝜉=0,𝜂(𝜙𝑋)=0,rank𝜙=𝑛−1.(2.2) If we putΦ(𝑋,𝑌)=𝑔(𝑋,𝜙𝑌),(2.3) for any vector fields 𝑋 and 𝑌, then the tensor field Φ(𝑋,𝑌) is a symmetric (0, 2) tensor field [1]. Also, since the 1-form 𝜂 is closed in an LP-Sasakian manifold, we have [1, 4]∇𝑋𝜂(𝑌)=Φ(𝑋,𝑌),Φ(𝑋,𝜉)=0,(2.4) for any vector fields 𝑋 and 𝑌.

Also in an LP-Sasakian manifold, the following relations hold [3, 4]:𝑔(𝑅(𝑋,𝑌)𝑍,𝜉)=𝜂(𝑅(𝑋,𝑌)𝑍)=𝑔(𝑌,𝑍)𝜂(𝑋)−𝑔(𝑋,𝑍)𝜂(𝑌),(2.5)𝑅(𝜉,𝑋)𝑌=𝑔(𝑋,𝑌)𝜉−𝜂(𝑌)𝑋,(2.6)𝑅(𝑋,𝑌)𝜉=𝜂(𝑌)𝑋−𝜂(𝑋)𝑌,(2.7)𝑆(𝑋,𝜉)=(𝑛−1)𝜂(𝑋),(2.8)𝑆(𝜙𝑋,𝜙𝑌)=𝑆(𝑋,𝑌)+(𝑛−1)𝜂(𝑋)𝜂(𝑌),(2.9) for any vector fields 𝑋,𝑌, and 𝑍, where 𝑅 is the Riemannian curvature tensor and 𝑆 is the Ricci tensor of 𝑀.

An LP-Sasakian manifold 𝑀 is said to be Einstein if its Ricci tensor 𝑆 is of the form𝑆(𝑋,𝑌)=ğ‘Žğ‘”(𝑋,𝑌),(2.10) for any vector fields 𝑋 and 𝑌, where ğ‘Ž is a function on 𝑀.

In [6], Pokhariyal and Mishra have defined the curvature tensor 𝑊2, given by𝑊21(𝑋,𝑌,𝑈,𝑉)=𝑅(𝑋,𝑌,𝑈,𝑉)+[],𝑛−1𝑔(𝑋,𝑈)𝑆(𝑌,𝑉)−𝑔(𝑌,𝑈)𝑆(𝑋,𝑉)(2.11) where 𝑆 is a Ricci tensor of type (0, 2).

Now, consider an LP-Sasakian manifold satisfying 𝑊2=0; then, (2.11) becomes1𝑅(𝑋,𝑌,𝑈,𝑉)=[].𝑛−1𝑔(𝑌,𝑈)𝑆(𝑋,𝑉)−𝑔(𝑋,𝑈)𝑆(𝑌,𝑉)(2.12) Taking 𝑋=𝑈=𝜉 in (2.12) and using (2.7) and (2.8), we have𝑆(𝑌,𝑉)=(𝑛−1)𝑔(𝑌,𝑉).(2.13) Therefore, 𝑀 is an Einstein manifold.

Again using (2.13) in (2.12), we get[].𝑅(𝑋,𝑌,𝑈,𝑉)=𝑔(𝑌,𝑈)𝑔(𝑋,𝑉)−𝑔(𝑋,𝑈)𝑔(𝑌,𝑉)(2.14)

Corollary 2.1. An LP-Sasakian manifold satisfying 𝑊2=0 is a space of constant curvature −1, that is, it is locally isometric to the hyperbolic space.

Definition 2.2. An LP-Sasakian manifold is called 𝑊2-semisymmetric if it satisfies 𝑅(𝑋,𝑌)⋅𝑊2=0,(2.15) where 𝑅(𝑋,𝑌) is to be considered as a derivation of the tensor algebra at each point of the manifold for tangent vectors 𝑋 and 𝑌.
It can be easily shown that in an LP-Sasakian manifold the 𝑊2-curvature tensor satisfies the condition 𝜂𝑊2(𝑋,𝑌)𝑍=0.(2.16)

Theorem 2.3. A 𝑊2-semisymmetric LP-Sasakian manifold 𝑀 is an Einstein manifold.

Proof. Since 𝑅(𝑋,𝑌)⋅𝑊2=0, we have 𝑅(𝑋,𝑌)𝑊2(𝑈,𝑉)𝑍−𝑊2(𝑅(𝑋,𝑌)𝑈,𝑉)𝑍−𝑊2(𝑈,𝑅(𝑋,𝑌)𝑉)𝑍−𝑊2(𝑈,𝑉)𝑅(𝑋,𝑌)𝑍=0.(2.17) Putting 𝑋=𝜉 in (2.17) and then taking the inner product with 𝜉, we obtain 𝑔𝑅(𝜉,𝑌)𝑊2𝑊(𝑈,𝑉)𝑍,𝜉−𝑔2𝑊(𝑅(𝜉,𝑌)𝑈,𝑉)𝑍,𝜉−𝑔2(𝑊𝑈,𝑅(𝜉,𝑌)𝑉)𝑍,𝜉−𝑔2(𝑈,𝑉)𝑅(𝜉,𝑌)𝑍,𝜉=0.(2.18) Using (2.6) in (2.18), we obtain −𝑔𝑌,𝑊2𝑊(𝑈,𝑉)𝑍−𝜂2𝜂𝑊(𝑈,𝑉)𝑍(𝑌)−𝑔(𝑌,𝑈)𝜂2𝑊(𝜉,𝑉)𝑍−𝑔(𝑌,𝑉)𝜂2(𝑊𝑈,𝜉)𝑍−𝑔(𝑌,𝑍)𝜂2(𝑊𝑈,𝑉)𝜉+𝜂(𝑈)𝜂2(𝑊𝑌,𝑉)𝑍+𝜂(𝑉)𝜂2𝑊(𝑈,𝑌)𝑍+𝜂(𝑍)𝜂2(𝑈,𝑉)𝑌=0.(2.19) By using (2.16) in (2.19), we get 𝑊2(𝑈,𝑉,𝑍,𝑌)=0.(2.20) In view of (2.11) and (2.20), it follows that 1𝑅(𝑈,𝑉,𝑍,𝑌)=[].𝑛−1𝑔(𝑉,𝑍)𝑆(𝑈,𝑌)−𝑔(𝑈,𝑍)𝑆(𝑉,𝑌)(2.21) Contracting (2.21), we have 𝑆(𝑉,𝑍)=(𝑛−1)𝑔(𝑉,𝑍).(2.22) Therefore, 𝑀 is an Einstein manifold.

Again using (2.22) in (2.12), we get[].𝑅(𝑈,𝑉,𝑍,𝑌)=𝑔(𝑉,𝑍)𝑔(𝑈,𝑌)−𝑔(𝑈,𝑍)𝑔(𝑉,𝑌)(2.23)

Corollary 2.4. A 𝑊2-semisymmetric LP-Sasakian manifold is a space of constant curvature −1, that is, it is locally isometric to the hyperbolic space.

3. LP-Sasakian Manifolds Satisfying 𝑃(𝑋,𝑌)⋅𝑊2=0

The projective curvature tensor 𝑃 is defined as [10]1𝑃(𝑋,𝑌)𝑍=𝑅(𝑋,𝑌)𝑍−[].𝑛−1𝑆(𝑌,𝑍)𝑋−𝑆(𝑋,𝑍)𝑌(3.1) Using (2.6) and (2.8), (3.1) reduces to1𝑃(𝜉,𝑌)𝑍=𝑔(𝑌,𝑍)𝜉−𝑛−1𝑆(𝑌,𝑍)𝜉.(3.2) Let us suppose that in an LP-Sasakian manifold 𝑃(𝑋,𝑌)⋅𝑊2=0.(3.3) This condition implies that𝑃(𝑋,𝑌)𝑊2(𝑈,𝑉)𝑍−𝑊2(𝑃(𝑋,𝑌)𝑈,𝑉)𝑍−𝑊2(𝑈,𝑃(𝑋,𝑌)𝑉)𝑍−𝑊2(𝑈,𝑉)𝑃(𝑋,𝑌)𝑍=0.(3.4) Putting 𝑋=𝜉 in (3.4) and then taking the inner product with 𝜉, we obtain𝑔𝑃(𝜉,𝑌)𝑊2𝑊(𝑈,𝑉)𝑍,𝜉−𝑔2𝑊(𝑃(𝜉,𝑌)𝑈,𝑉)𝑍,𝜉−𝑔2(𝑊𝑈,𝑃(𝜉,𝑌)𝑉)𝑍,𝜉−𝑔2(𝑈,𝑉)𝑃(𝜉,𝑌)𝑍,𝜉=0.(3.5) Using (3.2) in (3.5), we obtain−𝑔𝑌,𝑊2𝑊(𝑈,𝑉)𝑍−𝑔(𝑌,𝑈)𝜂2𝑊(𝜉,𝑉)𝑍−𝑔(𝑌,𝑉)𝜂2𝑊(𝑈,𝜉)𝑍−𝑔(𝑌,𝑍)𝜂2+1(𝑈,𝑉)𝜉𝑆𝑛−1𝑌,𝑊2𝑊(𝑈,𝑉)𝑍+𝑆(𝑌,𝑈)𝜂2𝑊(𝜉,𝑉)𝑍+𝑆(𝑌,𝑉)𝜂2𝑊(𝑈,𝜉)𝑍+𝑆(𝑌,𝑍)𝜂2(𝑈,𝑉)𝜉=0.(3.6) By using (2.16) in (3.6), we get𝑔𝑌,𝑊2−1(𝑈,𝑉)𝑍𝑆𝑛−1𝑌,𝑊2(𝑈,𝑉)𝑍=0.(3.7) Taking 𝑈=𝑍=𝜉 in (3.7) and using (2.11) and (2.6), we have𝑆(𝑄𝑌,𝑉)=2(𝑛−1)𝑆(𝑌,𝑉)−(𝑛−1)2𝑔(𝑌,𝑉).(3.8) This implies that𝑄𝑌=(𝑛−1)𝑌.(3.9) From this, we get𝑆(𝑌,𝑉)=(𝑛−1)𝑔(𝑌,𝑉).(3.10) Thus, we can state the following.

Theorem 3.1. An LP-Sasakian manifold 𝑀 satisfying the condition 𝑃(𝑋,𝑌)⋅𝑊2=0 is an Einstein manifold.

4. LP-Sasakian Manifold Satisfying 𝑀(𝑋,𝑌)⋅𝑊2=0

The 𝑀-projective curvature tensor 𝑀 is defined as [11]1𝑀(𝑋,𝑌)𝑍=𝑅(𝑋,𝑌)𝑍−2[].(𝑛−1)𝑆(𝑌,𝑍)𝑋−𝑆(𝑋,𝑍)𝑌+𝑔(𝑌,𝑍)𝑄𝑋−𝑔(𝑋,𝑍)𝑄𝑌(4.1) Using (2.6) and (2.8), (4.1) reduces to1𝑀(𝜉,𝑌)𝑍=2[]−1𝑔(𝑌,𝑍)𝜉−𝜂(𝑍)𝑌2[].(𝑛−1)𝑆(𝑌,𝑍)𝜉−𝜂(𝑍)𝑄𝑌(4.2) Suppose that in an LP-Sasakian manifold 𝑀(𝑋,𝑌)⋅𝑊2=0.(4.3) This condition implies that𝑀(𝑋,𝑌)𝑊2(𝑈,𝑉)𝑍−𝑊2𝑍𝑀(𝑋,𝑌)𝑈,𝑉−𝑊2𝑈,𝑀(𝑋,𝑌)𝑉𝑍−𝑊2(𝑈,𝑉)𝑀(𝑋,𝑌)𝑍=0.(4.4) Putting 𝑋=𝜉 in (4.4) and then taking the inner product with 𝜉, we obtain𝑔𝑀(𝜉,𝑌)𝑊2𝑊(𝑈,𝑉)𝑍,𝜉−𝑔2𝑊𝑀(𝜉,𝑌)𝑈,𝑉𝑍,𝜉−𝑔2𝑊𝑈,𝑀(𝜉,𝑌)𝑉𝑍,𝜉−𝑔2(𝑈,𝑉)𝑀(𝜉,𝑌)𝑍,𝜉=0.(4.5) Using (4.2) in (4.5), we obtain12−𝑔𝑌,𝑊2𝑊(𝑈,𝑉)𝑍−𝑔(𝑌,𝑈)𝜂2𝑊(𝜉,𝑉)𝑍−𝑔(𝑌,𝑉)𝜂2𝑊(𝑈,𝜉)𝑍−𝑔(𝑌,𝑍)𝜂2𝑊(𝑈,𝑉)𝜉+𝜂(𝑈)𝜂2𝑊(𝑌,𝑉)𝑍+𝜂(𝑉)𝜂2𝑊(𝑈,𝑌)𝑍+𝜂(𝑍)𝜂2+1(𝑈,𝑉)𝑌𝑆2(𝑛−1)𝑌,𝑊2𝑊(𝑈,𝑉)𝑍+𝑆(𝑌,𝑈)𝜂2𝑊(𝜉,𝑉)𝑍+𝑆(𝑌,𝑉)𝜂2𝑊(𝑈,𝜉)𝑍+𝑆(𝑌,𝑍)𝜂2𝑊(𝑈,𝑉)𝜉−𝜂(𝑈)𝜂2𝑊(𝑄𝑌,𝑉)𝑍−𝜂(𝑉)𝜂2𝑊(𝑈,𝑄𝑌)𝑍−𝜂(𝑍)𝜂2(𝑈,𝑉)𝑄𝑌=0.(4.6) By using (2.16) in (4.6), we get12𝑔𝑌,𝑊2−1(𝑈,𝑉)𝑍2𝑆(𝑛−1)𝑌,𝑊2(𝑈,𝑉)𝑍=0.(4.7) Taking 𝑈=𝑍=𝜉 in (4.7) and using (2.11) and (2.6), we have𝑆(𝑄𝑌,𝑉)=2(𝑛−1)𝑆(𝑌,𝑉)−(𝑛−1)2𝑔(𝑌,𝑉).(4.8) This implies that𝑄𝑌=(𝑛−1)𝑌,(4.9) which gives𝑆(𝑌,𝑉)=(𝑛−1)𝑔(𝑌,𝑉).(4.10) Thus, we can state the following.

Theorem 4.1. An LP-Sasakian manifold 𝑀 satisfying the condition 𝑀(𝑋,𝑌)⋅𝑊2=0 is an Einstein manifold.

5. LP-Sasakian Manifolds Satisfying 𝐶(𝑋,𝑌)⋅𝑊2=0

The conformal curvature tensor 𝐶 is defined as [12]1𝐶(𝑋,𝑌)𝑍=𝑅(𝑋,𝑌)𝑍−[]+𝑟𝑛−2𝑆(𝑌,𝑍)𝑋−𝑆(𝑋,𝑍)𝑌+𝑔(𝑌,𝑍)𝑄𝑋−𝑔(𝑋,𝑍)𝑄𝑌([].𝑛−1)(𝑛−2)𝑔(𝑌,𝑍)𝑋−𝑔(𝑋,𝑍)𝑌(5.1) Using (2.6) and (2.8), (5.1) reduces to𝐶(𝜉,𝑌)𝑍=1−𝑛+𝑟[]−1(𝑛−1)(𝑛−2)𝑔(𝑌,𝑍)𝜉−𝜂(𝑍)𝑌[].𝑛−2𝑆(𝑌,𝑍)𝜉−𝜂(𝑍)𝑄𝑌(5.2) Now consider an LP-Sasakian manifold satisfying 𝐶(𝑋,𝑌)⋅𝑊2=0.(5.3) This condition implies that𝐶(𝑋,𝑌)𝑊2(𝑈,𝑉)𝑍−𝑊2(𝐶(𝑋,𝑌)𝑈,𝑉)𝑍−𝑊2(𝑈,𝐶(𝑋,𝑌)𝑉)𝑍−𝑊2(𝑈,𝑉)𝐶(𝑋,𝑌)𝑍=0.(5.4) Putting 𝑋=𝜉 in (5.4) and then taking the inner product with 𝜉, we obtain𝑔𝐶(𝜉,𝑌)𝑊2𝑊(𝑈,𝑉)𝑍,𝜉−𝑔2𝑊(𝐶(𝜉,𝑌)𝑈,𝑉)𝑍,𝜉−𝑔2(𝑊𝑈,𝐶(𝜉,𝑌)𝑉)𝑍,𝜉−𝑔2(𝑈,𝑉)𝐶(𝜉,𝑌)𝑍,𝜉=0.(5.5) Using (5.2) in (5.5), we obtain1−𝑛+𝑟(𝑛−1)(𝑛−2)−𝑔𝑌,𝑊2𝑊(𝑈,𝑉)𝑍−𝑔(𝑌,𝑈)𝜂2𝑊(𝜉,𝑉)𝑍−𝑔(𝑌,𝑉)𝜂2𝑊(𝑈,𝜉)𝑍−𝑔(𝑌,𝑍)𝜂2𝑊(𝑈,𝑉)𝜉−𝜂(𝑌)𝜂2𝑊(𝑈,𝑉)𝑍+𝜂(𝑈)𝜂2𝑊(𝑌,𝑉)𝑍+𝜂(𝑉)𝜂2𝑊(𝑈,𝑌)𝑍+𝜂(𝑍)𝜂2+1(𝑈,𝑉)𝑌𝑆𝑛−2𝑌,𝑊2𝑊(𝑈,𝑉)𝑍+𝑆(𝑌,𝑈)𝜂2𝑊(𝜉,𝑉)𝑍+𝑆(𝑌,𝑉)𝜂2𝑊(𝑈,𝜉)𝑍+𝑆(𝑌,𝑍)𝜂2𝑊(𝑈,𝑉)𝜉+(𝑛−1)𝜂(𝑌)𝜂2𝑊(𝑈,𝑉)𝑍−𝜂(𝑈)𝜂2𝑊(𝑄𝑌,𝑉)𝑍−𝜂(𝑉)𝜂2𝑊(𝑈,𝑄𝑌)𝑍−𝜂(𝑍)𝜂2(𝑈,𝑉)𝑄𝑌=0.(5.6) By using (2.16) in (5.6), we get1−𝑛+𝑟𝑔(𝑛−1)(𝑛−2)𝑌,𝑊2−1(𝑈,𝑉)𝑍𝑆𝑛−2𝑌,𝑊2(𝑈,𝑉)𝑍=0.(5.7) Taking 𝑈=𝑍=𝜉 in (5.7) and then using (2.11) and (2.6), we have𝑆(𝑄𝑌,𝑉)=2+𝑛(𝑛−3)+𝑟𝑛−1𝑆(𝑌,𝑉)−(𝑛−1+𝑟)𝑔(𝑌,𝑉).(5.8) This implies that𝑄𝑌=(𝑛−1)𝑌,(5.9)and it follows that𝑆(𝑌,𝑉)=(𝑛−1)𝑔(𝑌,𝑉).(5.10) Thus, we can state the following.

Theorem 5.1. An LP-Sasakian manifold 𝑀 satisfying the condition 𝐶(𝑋,𝑌)⋅𝑊2=0 is an Einstein manifold.

Acknowledgment

The authors express their thanks to DST (Department of Science and Technology), Government of India for providing financial assistance under major research project.

References

  1. K. Matsumoto, “On Lorentzian paracontact manifolds,” Bulletin of the Yamagata University. Natural Science, vol. 12, no. 2, pp. 151–156, 1989. View at: Google Scholar | Zentralblatt MATH
  2. I. Mihai and R. Roşca, “On Lorentzian P-Sasakian manifolds,” in Classical Analysis (Kazimierz Dolny, 1991), pp. 155–169, World Scientific, River Edge, NJ, USA, 1992. View at: Google Scholar
  3. K. Matsumoto and I. Mihai, “On a certain transformation in a Lorentzian para-Sasakian manifold,” Tensor, vol. 47, no. 2, pp. 189–197, 1988. View at: Google Scholar | Zentralblatt MATH
  4. I. Mihai, A. A. Shaikh, and U. C. De, “On Lorentzian para-Sasakian manifolds,” Rendiconti del Seminario Matematico di Messina, no. 3, pp. 149–158, 1999. View at: Google Scholar
  5. Venkatesha and C. S. Bagewadi, “On concircular ϕ-recurrent LP-Sasakian manifolds,” Differential Geometry—Dynamical Systems, vol. 10, pp. 312–319, 2008. View at: Google Scholar
  6. G. P. Pokhariyal and R. S. Mishra, “Curvature tensors' and their relativistics significance,” Yokohama Mathematical Journal, vol. 18, pp. 105–108, 1970. View at: Google Scholar | Zentralblatt MATH
  7. G. P. Pokhariyal, “Study of a new curvature tensor in a Sasakian manifold,” Tensor, vol. 36, no. 2, pp. 222–226, 1982. View at: Google Scholar | Zentralblatt MATH
  8. K. Matsumoto, S. Ianuş, and I. Mihai, “On P-Sasakian manifolds which admit certain tensor fields,” Publicationes Mathematicae Debrecen, vol. 33, no. 3-4, pp. 199–204, 1986. View at: Google Scholar
  9. A. Yíldíz and U. C. De, “On a type of Kenmotsu manifolds,” Differential Geometry—Dynamical Systems, vol. 12, pp. 289–298, 2010. View at: Google Scholar | Zentralblatt MATH
  10. K. Yano and M. Kon, Structures on Manifolds, vol. 3 of Series in Pure Mathematics, World Scientific, Singapore, 1984.
  11. G. P. Pokhariyal and R. S. Mishra, “Curvature tensors and their relativistic significance. II,” Yokohama Mathematical Journal, vol. 19, no. 2, pp. 97–103, 1971. View at: Google Scholar | Zentralblatt MATH
  12. L. P. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, NJ, USA, 1949.

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