### Article

Kyungpook Mathematical Journal 2021; 61(1): 191-203

**Published online** March 31, 2021

Copyright © Kyungpook Mathematical Journal.

### Second Order Parallel Tensor on Almost Kenmotsu Manifolds

Venkatesha Venkatesha^{*}, Devaraja Mallesha Naik, Aysel-Turgut Vanli

Department of Mathematics, Kuvempu University, Shivamogga 577-451, India

e-mail : vensmath@gmail.com

Department of Mathematics, Kuvempu University, Shivamogga 577-451, India Department of Mathematics, CHRIST (Deemed to be University), Bengaluru 560029, Karnataka, India

e-mail : devaraja.mallesha@christuniversity.in

Department of Mathematics, Gazi University, Ankara, Turkey

e-mail : avanli@gazi.edu.tr

**Received**: November 14, 2019; **Revised**: July 18, 2020; **Accepted**: July 21, 2020

Let *M* be an almost Kenmotsu manifold of dimension *2n+1* having non-vanishing ξ-sectional curvature such that *M* is a constant multiple of the associated metric tensor and obtained some consequences of this. Vector fields keeping curvature tensor invariant are characterized on *M*.

**Keywords**: almost Kenmotsu manifold, second order parallel tensor, nullity distribution, homothetic vector field.

In 1923, Eisenhart [13] proved that if a positive definite Riemannian manifold

In contact geometry, Kenmotsu manifolds, introduced by Kenmotsu in [17], are one of the important classes of manifolds. Such manifolds were observed to be normal. Let

### Theorem 1.1.

Let α be a second order symmetric parallel tensor, and

(i)

$A\xi =\alpha (\xi ,\xi )\xi $ , ifM has non-vanishing ξ-sectional curvature;

(ii)

$trA=\alpha (\xi ,\xi )-tr(A({h}^{2}-2\phi h-\phi ({\nabla}_{\xi}h)))+\alpha (\xi ,\xi )S(\xi ,\xi )$ ;

(iii)

$tr(A\mathcal{l})=\alpha (\xi ,\xi )S(\xi ,\xi ),$

Since the Ricci tensor

### Corollary 1.1.

If an almost Kenmotsu manifold

(i)

$Q\xi =-(2n+tr{h}^{2})\xi $ ;

(ii) the scalar curvature

$r=\Vert Q\xi {\Vert}^{2}-2n-tr({h}^{2})-tr(Q({h}^{2}-2\phi h-\phi ({\nabla}_{\xi}h)));$

(iii)

$tr(Q\mathcal{l})=\Vert Q\xi {\Vert}^{2},$

where

### Theorem 1.2.

Let

### Corollary 1.2.

If

Since

### Corollary 1.3.

### Corollary 1.4.

An affine Killing vector field on

Blair, Koufogiorgos and Papantoniou [3] introduced

for any

### Corollary 1.5.

A second order parallel tensor on an almost Kenmotsu manifold with

The above corollary has been proved by Wang and Liu in [34]. Recently, Dileo-Pastore [12] introduced

for any

### Lemma 1.1.

([12, Proposition 4.1]) Let

### Lemma 1.2.

([11, Theorem 6]) Let

### Lemma 1.3.

([12, Corollary 4.2]) Let

### Lemma 1.4.

([12, Proposition 4.3]) Let

where

We use these to prove:

### Theorem 1.3.

Let

In [34], Wang and Liu proved the above theorem in another way. Now waving the hypothesis non-vanishing ξ-sectional curvature in Theorem 1.2 by

### Theorem 1.4.

Let

In [23] Naik et al. proved that every vector field which leaves the curvature tensor invariant are Killing in a

### Theorem 1.5.

Let

The

where

### Theorem 1.6.

Let

A

where

### Theorem 1.7.

Let

An

We easily obtain from (2.1) that

A manifold

If the 1-form

The two tensor fields

Further, one has the following formulas:

### 3. Proof of Theorems

Now we prove the results stated in Section 1.

Suppose that α is a symmetric (0, 2)-tensor and

Therefore

Thus we get

Setting

For

Since the ξ-sectional curvature

This proves (i). Applying

If

This gives (ii). Now plugging

Using this in (ii) yields (iii). This finishes the proof.

Now (i) follows from (2.8). Note that from (3.4), we have

Hence (ii) and (iii) follows directly from (ii) and (iii) of Theorem 1.1.

But by hypothesis,

Let α be a (0,2)-tensor such that

First, suppose that α is symmetric. Then from item

To show

Now differentiating (3.5) along

Replacing

Now putting

and

Multiplying

Similarly, we can find

Since

for any

The skew-symmetry of α then gives

For

From (3.10), the hypothesis

One can easily verify the constancy of

Next, we take

If

Now, as before we put

for all

First suppose that

That

and

Since

for any

and

Thus, we obtain

Proceeding the similar manner as in the Theorem 1.2 one gets

Now suppose that

Then Lemma shows

Putting

Note that 2.6 takes the form

Using (3.15) in (3.14), and putting

and

from which we obtain

and

Since

for any

Then using (3.15) in above, and putting _{i}

Now the condition

Let

Taking

Using (3.15) in above, and putting

and

from which we obtain

and

Since _{i}

and

So that, we have

that is,

for some function

Since

Since

which for _{i}

Consequently, we obtain

- C. L. Bejan and M. Crasmareanu. Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry, Ann Global Anal. Geom. 46 (2014), 117-127.
- D. E. Blair. Riemannian geometry of contact and symplectic manifolds, Mathematics 203, Birkhäuser, New York, 2010.
- D. E. Blair, T. Koufogiorgos, and B. J. Papantoniou. Contact metric manifolds satis-fying a nullity condition, Israel J. Math. 91 (1995), 189-214.
- C. C˘alin and M. Crasmareanu. From the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds, Bull. Malays. Math. Sci. Soc. 33 (2010), 361-368.
- J. Case, Y. Shu, and G. Wei. Rigidity of quasi-Einstein metrics, Differential Geom. Appl. 29 (2011), 93-100.
- M. Crasmareanu. Parallel tensors and Ricci solitons in N(κ)-quasi Einstein mani-folds, Indian J. Pure Appl. Math. 43 (2012), 359-369.
- U. C. De. Second order parallel tensors on P-Sasakian manifolds, Publ. Math. Debrecen 49 (1996), 33-37.
- U. C. De, S. K. Chaubey, and Y. J. Suh. A note on almost co-Kahler manifolds, Int. J. Geom. Methods Mod. Phys. 17 (2020), 2050153. 14 pp.
- U. C. De and K. Mandal. Ricci solitons and gradient Ricci solitons on N(κ)-paracontact manifolds, Zh. Mat. Fiz. Anal. Geom. 15 (2019), 307-320.
- U. C. De, Y. J. Suh, and P. Majhi. Ricci solitons on η-Einstein contact manifolds, Filomat 32 (2018), 4679-4687.
- G. Dileo and A. M. Pastore. Almost Kenmotsu manifolds and local symmetry, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), 343-354.
- G. Dileo and A. M. Pastore. Almost Kenmotsu manifolds and nullity distributions, J. Geom. 93 (2009), 46-61.
- L. P. Eisenhart. Symmetric tensors of the second order whose first covariant deriva-tives are zero, Trans. Amer. Math. Soc. 25 (1923), 297-306.
- A. Ghosh and R. Sharma. Some results on contact metric manifolds, Ann. Global Anal. Geom. 15 (1997), 497-507.
- C. He, P. Petersen, and W. Wylie. On the classification of warped product Einstein metrics, Comm. Anal. Geom. 20 (2012), 271-311.
- D. Janssens and L. Vanhecke. Almost contact structures and curvature tensors, Kodai Math. J. 4 (1981), 1-27.
- K. Kenmotsu. A class of almost contact Riemannian manifolds, Tohoku Math. J. 24 (1972), 93-103.
- H. Levy. Symmetric tensors of the second order whose covariant derivatives vanish, Ann. of Math. 27 (1925), 91-98.
- A. K. Mondal, U. C. De, and C. Özgür. Second order parallel tensors on (κ,µ)-contact metric manifolds, An. St. Univ. Ovidius Const. Ser. Mat. 18 (2011), 229-238.
- D. M. Naik and V. Venkatesha. η-Ricci solitons and almost η-Ricci solitons on para-Sasakian manifolds, Int. J. Geom. Methods Mod. Phys. 16 (2019), 1950134. 18 pp.
- D. M. Naik, V. Venkatesha, and D. G. Prakasha. Certain results on Kenmotsu pseudo-metric manifolds, Miskolc Math. Notes 20 (2019), 1083-1099.
- D. M. Naik, V. Venkatesha, and H. A. Kumara. Ricci solitons and certain related metrics on almost co-Kaehler manifolds, J. Math. Phys. Anal. Geom. 16 (2020), 402-417.
- D. M. Naik, V. Venkatesha, and H. A. Kumara. Some results on almost Kenmotsu manifolds, Note Mat. 40 (2020), 87-100.
- R. Sharma. Second order parallel tensor in real and complex space forms, Internat. J. Math. Math. Sci. 12 (1989), 787-790.
- R. Sharma. Second order parallel tensors on contact manifolds. I, Algebras Groups Geom. 7 (1990), 145-152.
- R. Sharma. Second order parallel tensors on contact manifolds. II, C. R. Math. Rep. Acad. Sci. Canada 13 (1991), 259-264.
- R. Sharma. On the curvature of contact metric manifolds, J. Geom. 53 (1995), 179-190.
- Y. J. Suh and U. C. De. Yamabe solitons and Ricci solitons on almost co-Kähler manifolds, Canad. Math. Bull. 62(3) (2019), 653-661.
- D. Tarafdar and U. C. De. Second order parallel tensors on P-Sasakian manifolds, Northeast. Math. J. 11 (1995), 260-262.
- M. Tarafdar and A. Mayra. On nearly Sasakian manifold, An. Stiint. Univ. “Al. I. Cuza Iasi. Mat. 45 (1999), 291-294.
- V. Venkatesha, H. A. Kumara, and D. M. Naik. Almost *-Ricci soliton on paraKen-motsu manifolds, Arab. J. Math. 9 (2020), 715-726.
- V. Venkatesha and D. M. Naik. On 3-dimensional normal almost contact pseudometric manifolds, Afr. Mat. (2020). (accepted).
- V. Venkatesha, D. M. Naik, and H. A. Kumara. *-Ricci solitons and gradient almost *-Ricci solitons on Kenmotsu manifolds, Math. Slovaca 69 (2019), 1447-1458.
- Y. Wang and X. Liu. Second order parallel tensors on an almost Kenmotsu manifolds satisfying the nullity distributions, Filomat 28 (2014), 839-847.