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 : firstname.lastname@example.org
Department of Mathematics, Kuvempu University, Shivamogga 577-451, India Department of Mathematics, CHRIST (Deemed to be University), Bengaluru 560029, Karnataka, India
e-mail : email@example.com
Department of Mathematics, Gazi University, Ankara, Turkey
e-mail : firstname.lastname@example.org
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
Keywords: almost Kenmotsu manifold, second order parallel tensor, nullity distribution, homothetic vector field.
In 1923, Eisenhart  proved that if a positive definite Riemannian manifold
In contact geometry, Kenmotsu manifolds, introduced by Kenmotsu in , are one of the important classes of manifolds. Such manifolds were observed to be normal. Let
Let α be a second order symmetric parallel tensor, and
, if Mhas non-vanishing ξ-sectional curvature;
Since the Ricci tensor
If an almost Kenmotsu manifold
(ii) the scalar curvature
An affine Killing vector field on
Blair, Koufogiorgos and Papantoniou  introduced
A second order parallel tensor on an almost Kenmotsu manifold with
([12, Proposition 4.1]) Let
([11, Theorem 6]) Let
([12, Corollary 4.2]) Let
([12, Proposition 4.3]) Let
We use these to prove:
In , Wang and Liu proved the above theorem in another way. Now waving the hypothesis non-vanishing ξ-sectional curvature in Theorem 1.2 by
In  Naik et al. proved that every vector field which leaves the curvature tensor invariant are Killing in a
If the 1-form
The two tensor fields
Further, one has the following formulas:
Now we prove the results stated in Section 1.
Suppose that α is a symmetric (0, 2)-tensor and
Thus we get
Since the ξ-sectional curvature
This proves (i). Applying
This gives (ii). Now plugging
Using this in (ii) yields (iii). This finishes the proof.
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
Now differentiating (3.5) along
Similarly, we can find
The skew-symmetry of α then gives
From (3.10), the hypothesis
One can easily verify the constancy of
Next, we take
Now, as before we put
First suppose that
Thus, we obtain
Proceeding the similar manner as in the Theorem 1.2 one gets
Now suppose that
Then Lemma shows
Note that 2.6 takes the form
from which we obtain
Then using (3.15) in above, and putting
Now the condition
Using (3.15) in above, and putting
from which we obtain
So that, we have
for some function
Consequently, we obtain
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