1 Introduction

In 1968, Yano and Sawaki [1] introduced the notion of quasi-conformal curvature tensor in the context of Riemannian geometry. Recently, in tune with Yano and Sawaki [1], the present author [2] introduced and studied generalized quasi-conformal curvature tensor in the context of \(N(k,\mu )\)-contact metric manifold. The beauty of generalized quasi-conformal curvature tensor lies in the fact that it has the flavour of Riemann curvature tensor R, conformal curvature tensor C [3], conharmonic curvature tensor \(\hat{C}\) [4], concircular curvature tensor E [5, p. 84], projective curvature tensor P [5, p. 84] and m-projective curvature tensor H[6] as particular cases. Hereafter in this paper, generalized quasi-conformal curvature tensor \(\mathcal {W}\) will be called quasi-conformal like curvature tensor.

The components of quasi-conformal like curvature tensor \(\mathcal {W} \) in a Riemannian manifold \((M^{2n+1},g)(n>1),\) are given by

$$\begin{aligned} \mathcal {W} (X,Y)Z= & {} \frac{2{n}-1}{2{n}+1}\left[ (1+2na-b)-\{1+2n(a+b)\}c\right] C(X,Y)Z\ \nonumber \\&+\left( 1-b+2{n}a\right) E(X,Y)Z+2n(b-a)P(X,Y)Z \nonumber \\&+\frac{2{n}-1}{2{n}+1}(c-1)\left[ 1+2{n(}a+b) \right] \hat{C}(X,Y)Z \end{aligned}$$
(1.1)

where a, b and c are real constants. The foregoing equation can also be written as

$$\begin{aligned} {\mathcal {W}}(X,Y)Z= & {} R(X,Y)Z+a\left[ S(Y,Z)X-S(X,Z)Y\right] \nonumber \\&+\,b\left[ g(Y,Z)QX-g(X,Z)QY\right] \nonumber \\&-\,\frac{cr}{2n+1}\left( \frac{1}{2n}+a+b\right) \left[ g(Y,Z)X-g(X,Z)Y \right] {,} \end{aligned}$$
(1.2)

R,  S, Q and r being Riemannian curvature tensor, Ricci tensor, Ricci operator and scalar curvature respectively. In particular, the quasi conformal like curvature tensor \({\mathcal {W}}\) induced to (1) Riemann curvature tensor R if \(a=b=c=0\); (2) conformal curvature tensor C if \(a=b=-\frac{1}{2{n}-1}\) and \(c=1\); (3) conharmonic curvature tensor \(\hat{C},\) if \(a=b=-\frac{1}{2{n}-1}\) and \(c=0\); (4) concircular curvature tensor E,  if \(a=b=0\) and \(c=1\); (5) projective curvature tensor P if \(a=-\frac{1}{2{n}},\) \(b=0\) and \(\ c=0\) and (6) m-projective curvature tensor H if \(a=b= -\frac{1 }{4{n}}\) and \(c=0\).

The study of the Ricci solitons in contact geometry has begun with the work of Sharma [7, 8]. Ricci solitons are introduced as triples (M, g, V),  where (M, g) is a Riemannian manifold and V is a potential vector field so that the following equation is satisfied:

$$\begin{aligned} \frac{1}{2}\pounds _{_{V}}g+S+\lambda g=0 \end{aligned}$$
(1.3)

where \(\pounds \) denotes the Lie derivative, S is the Ricci tensor and \( \lambda \in \mathbb {R} \). A Ricci soliton is said to be shrinking, steady or expanding according to \( \lambda \) negative, zero, and positive respectively. During the last two decades, the geometry of Ricci solitons has become a subject of growing interest for many mathematicians. For details we refer to [9,10,11,12,13,14,15,16,17,18,19,20] and the references therein. It has become more important after Grigory Perelman applied Ricci solitons to solve the long standing Poincar é conjecture posed in 1904.

Our work is structured as follows. Section 2 is a very brief review of Sasakian manifolds. In Sect. 3, we investigate Ricci solitons in a Sasakian manifold admitting \({\omega } (\xi ,X){ \cdot }\mathcal {W}=0\) where \(\omega \) and \({\mathcal {W}}\) stand for quasi conformal like curvature tensor with the associated scalar triples \((\bar{a},\bar{b},\bar{c})\) and (a, b, c) respectively (two distinct notions have been used in order to study the nature of 36 semi-symmetric type curvature condition [25] as shown in the table by taking the permutation and combination of the scalar triples into account), the dot means that \({\mathcal {\omega }}(X,Y)\) acts as a derivation on \(\mathcal {W}\). It is observed that that Ricci soliton \( (M,g,\xi )\) in a Sasakian manifold satisfying \({\omega }(\xi ,X )\cdot \mathcal {W}=0\) is sometime expanding and some other time remain shrinking. In Sect. 4, we have pointed out that Ricci soliton in a Sasakian manifold satisfying \(\mathcal {W}(\xi ,X)\cdot S=0\) is always shrinking. Remark that this type of conditions was discussed in [21, 22].

2 Sasakian manifolds

Let \(M^{2n+1}(\phi ,\xi ,\eta ,g)\) be a Sasakian manifold. Then the following relation hold [23, 24]

$$\begin{aligned} \eta (X)= & {} g(X,\xi ),\text { }\phi ^{2}=-I+\eta \circ \xi ,\eta (\xi )=1, \quad \eta \circ \phi =0, \end{aligned}$$
(2.1)
$$\begin{aligned} \nabla _{X}\xi= & {} -\phi X,g(\phi X,\phi Y)=g(X,Y)-\eta (X)\eta (Y), \end{aligned}$$
(2.2)
$$\begin{aligned} \eta (R(X,Y)Z)= & {} [g(Y,Z)\eta (X)-g(X,Z)\eta (Y)], \end{aligned}$$
(2.3)
$$\begin{aligned} S(X,\xi )= & {} 2n\eta (X), \end{aligned}$$
(2.4)

for any vector fields X,  Y, Z on M, where \(\nabla \) denotes the operator of covariant differentiation with respect to g.

Let \((g,V,\lambda )\) be a Ricci soliton in an \((2n+1)\)-dimensional Sasakian manifold M. From (2.3), we have

$$\begin{aligned} (\pounds _{\xi }g)(X,Y)=0 \end{aligned}$$
(2.5)

From (1.3) and (2.5), we find that

$$\begin{aligned} S(X,Y)=-\lambda g(X,Y). \end{aligned}$$
(2.6)

Thus, in a Sasaskian manifold with Ricci soliton, the quasi conformal like curvature tensor \(\mathcal {W}\) takes the form

$$\begin{aligned} {\mathcal {W}}(X,Y)Z= & {} R(X,Y)Z-\lambda \left[ (a+b)+\frac{cr}{2n+1} \left( \frac{1}{2n}+a+b\right) \right] \nonumber \\&\left[ g(Y,Z)X-g(X,Z)Y\right] . \end{aligned}$$
(2.7)

where Q and r are respectively the Ricci operator and the scalar curvature on M.

3 Ricci solitons in Sasakian manifolds satisfying \({\mathcal {\omega }}({X,Y} ){\cdot }{\mathcal {W}}{=0}\)

Let us consider a \((2n+1)\)-dimensional Sasakian manifold M, satisfying the condition

$$\begin{aligned} ({\mathcal {\omega }}(X,Y)\cdot {\mathcal {W}})(Z,U)V=0, \end{aligned}$$
(3.1)

for any vector fields X,  Y on the manifold and \({\mathcal {\omega } }(X,Y)\) acts on \({\mathcal {W}}\) as derivation, where \(\omega \) and \({\mathcal {W}}\) stand for quasi conformal like curvature tensor with the associated scalar triples \(( \bar{a},\bar{b},\bar{c})\) and (a, b, c) respectively (two distinct notions have been used in order to study the nature of 36 semi-symmetric type curvature condition [25] as shown in the table by taking the permutation and combination of the scalar triples into account). It is equivalent to

$$\begin{aligned}&{g}(\mathcal {\omega }(\xi ,X) {\mathcal {W}} (Y,Z)U,\xi )-{g}(\mathcal {W}(\mathcal {\omega } (\xi ,X )Y,Z)U,\xi ) \nonumber \\&\quad -{g}({\mathcal {W}}(Y,{\mathcal {\omega }} (\xi ,X) Z)U,\xi )-{g}(\mathcal {W}(Y,Z){\mathcal {\omega }}(\xi ,X)U,\xi )=0. \end{aligned}$$
(3.2)

Putting \(X=Y=e_{i}\) in (3.2) where \(\{e_{1}, e_{2}, e_{3},\ldots ,e_{2{n}},\) \(e_{2{n}+1}=\xi \}\) is an orthonormal basis of the tangent space at each point of the manifold M and taking the summation over i,  \(1\le i\le 2{n}+1,\) we get

$$\begin{aligned}&\sum _{i=1}^{2n+1}[g(\omega (\xi ,e_{i}){\mathcal {W}} (e_{i},Z)U,\xi )-g({\mathcal {W}}(\omega (\xi ,e_{i})e_{i},Z)U,\xi ) \nonumber \\&\quad -g({\mathcal {W}}(e_{i},\omega (\xi ,e_{i})Z)U,\xi )-g({\mathcal {W}}(e_{i},Z)\omega (\xi ,e_{i})U,\xi )]=0. \end{aligned}$$
(3.3)

From the Eqs. (2.3) and (2.7), we can easily bring out the followings

$$\begin{aligned}&\eta ({\mathcal {W}}(\xi ,U)Z) =\left[ 1-\lambda (a+b){{ +c\lambda }}\left( \frac{1}{2n}+a+b\right) \right] g(\phi Z,\phi U), \end{aligned}$$
(3.4)
$$\begin{aligned}&\sum _{i=1}^{2n+1}{\mathcal {\bar{W}}}\mathcal (e_{i},Z{, }U,e_{i}) =\lambda \left[ (c-1)\left\{ 1+2n(a+b)\right\} \right] g(Z,U). \end{aligned}$$
(3.5)
$$\begin{aligned}&\text {Now, }\sum _{i=1}^{2n+1}{g}(\mathcal {\omega }( \xi ,e_{i}){\mathcal {W}}(e_{i},Z)U,\xi ) = \left[ 1-\lambda (\bar{a}+\bar{b})+\bar{c}\lambda \left( \frac{1}{2n}+\bar{a}+\bar{b}\right) \right] \nonumber \\&\quad \times \,\left\{ { \eta }(\mathcal {W}(\xi ,Z)U)-\sum _{i=1}^{2n+1}\mathcal {\bar{ W}}\mathcal (e_{i},Z,U,e_{i})\right\} \end{aligned}$$
(3.6)

In view of (3.4) and (3.5), (3.6) becomes

$$\begin{aligned}&\sum _{i=1}^{2n+1} {g}{(\mathcal {\omega }(}\xi ,e_{i} ){\mathcal {W}}(e_{i},Z)U,\xi ) =\left[ 1-\lambda (\bar{a}+\bar{b})+\bar{c}\lambda \left( \frac{1}{2n}+ \bar{a}+\bar{b}\right) \right] \nonumber \\&\qquad \times \left[ 1-\lambda (a+b)+c\lambda \left( \frac{1}{2n}{{ +a+b}} \right) \right] [g(Z,U)-\eta (Z)\eta (U)] \nonumber \\&\qquad -\left[ 1-\lambda (\bar{a}+\bar{b})+\bar{c}\lambda \left( \frac{1}{2n}+\bar{a}+ \bar{b}\right) \right] \nonumber \\&\qquad \times \left[ { -}\lambda -2n\left\{ \lambda (a+b)-c{{ \lambda }}\left( \frac{1}{2n}+a+b\right) \right\} \right] g(Z,U). \end{aligned}$$
(3.7)

As a consequence of (3.4)–(3.5), we obtain the followings

$$\begin{aligned}&\sum _{i=1}^{2n+1}{g}{(\mathcal {W}(\mathcal {\omega }} (\xi ,e_{i})e_{i},Z)U,\xi ) =2n\left[ 1-\lambda (\bar{a}+\bar{b})+\bar{c}{ { \lambda }} \left( \frac{1}{2n}+\bar{a}+\bar{b}\right) \right] \nonumber \\&\qquad \times \left[ 1{ { -\lambda (a+b)+}}c{ { \lambda } }\left( \frac{1}{2n}{ { +a+b}}\right) \right] \left[ g(Z,U)-\eta (Z)\eta (U)\right] f{.} \end{aligned}$$
(3.8)
$$\begin{aligned}&\sum _{i=1}^{2n+1} {g}{(\mathcal {W}(}e_{i},{\mathcal {\omega }}(\xi ,e_{i})Z) U,\xi ) =-\left[ 1-\lambda (\bar{a}+\bar{b})+\bar{c}{ { \lambda }} \left( \frac{1}{2n}+\bar{a}+\bar{b}\right) \right] \nonumber \\&\quad \times \left[ 1{ { -\lambda (a+b)+c\lambda }}\left( \frac{1}{2n} { { +a+b}}\right) \right] \left[ g(Z,U)-\eta (Z)\eta (U)\right] \end{aligned}$$
(3.9)
$$\begin{aligned}&\sum _{i=1}^{2n+1}{g}{(\mathcal {W}}{ ( }e_{i},Z){\mathcal {\omega }}(\xi ,e_{i})U,\xi ) =2n\left[ 1-\lambda (\bar{a}+\bar{b})+\bar{c}{ { \lambda }} \left( \frac{1}{2n}+\bar{a}+\bar{b}\right) \right] \nonumber \\&\quad \times \left[ 1{ { -\lambda (a+b)+c\lambda }}\left( \frac{1}{2n} { { +a+b}}\right) \right] \eta (Z)\eta (U). \end{aligned}$$
(3.10)

By virtue of (3.7), (3.8), (3.9) and (3.10), the Eq. (3.3) yields

$$\begin{aligned} \lambda [1+4n(1-c)(a+b)-2c]=2nk\text { or }1=\lambda \left[ (\bar{ c}-1)(\bar{a}+\bar{b})-\frac{\bar{c}}{2n}\right] \nonumber \end{aligned}$$

for \(Z=U=\xi \). From the above equation, by taking permutation and combination of different values of \((\bar{a},\bar{b},\bar{c})\) and (a, b, c) (like 0, \(-\frac{1}{2n-1}\), \(-\frac{1}{2n}\), \(-\frac{1}{4n}\) etc.,) one will get 36 curvature conditions and we found that Ricci soliton \((M,g,\xi )\) in a Sasakian manifold for each curvature restriction is sometime expanding and some other time remain shrinking. This leads to the following:

Theorem 3.1

Ricci soliton \((M,g,\xi )\) in a Sasakian manifold satisfying \({\omega (}\xi ,X)\cdot {\mathcal {W}}=0\) is sometime expanding and some other time remain shrinking.

4 Sasakian manifolds with \({\mathcal {W}}\cdot S=0\)

Let \(M^{2n+1}(\phi ,\xi ,\eta ,g)(n>1),\) be a Sasakian manifold, satisfying the condition

$$\begin{aligned} {\mathcal {W(}}\xi ,X{)\cdot S}= & {} 0. \end{aligned}$$
(4.1)
$$\begin{aligned} i.e.\text { }{S}({\mathcal {W}}\mathcal (\xi ,X)Y,Z)+{S}{{( }}Y,{\mathcal {W}}\mathcal (\xi ,X)Z)= & {} 0. \end{aligned}$$
(4.2)

Putting \(Z=\xi \) in (4.2) and then using (2.4), we get

$$\begin{aligned} 2{n\eta }({\mathcal {W}}\mathcal (\xi ,X )Y)+{S}( { Y},{\mathcal {W}}\mathcal (\xi ,X) \xi )=0. \end{aligned}$$
(4.3)

In view of (2.7), we have

$$\begin{aligned} {{{ \eta }}}(\mathcal {W}(\xi ,X)Y)=\left[ 1-\lambda (a+b)+c\lambda \left( \frac{1}{2n}+a+b\right) \right] \left[ g(X,Y)-\eta (X)\eta (Y)\right] . \end{aligned}$$
(4.4)

Making use of (4.4) in (4.3), we have

$$\begin{aligned} \left[ 1-\lambda (a+b)+c\lambda \left( \frac{1}{2n}+a+b\right) \right] \left[ { 2n}g(X,Y)-S(X,Y)\right] { =0.} \end{aligned}$$
(4.5)

Contracting X over Y in (4.5), we get

$$\begin{aligned} 1=\lambda \left[ (1-c)(a+b)-\frac{c}{2n}\right] \text { { or, }} \lambda { =-2n.} \end{aligned}$$
(4.6)

From, the Eq. (4.6) one can easily bring out the following table

Curvature condition

Value of \(\lambda \)

\({ R(\xi ,X)\cdot S}=0\) (obtained by \(a=b=c=0\))

\(\lambda =-2nk,\)

\({ E(\xi ,X)\cdot S}=0\) (obtained by \(a=b=0,\) \(c=1\))

\(\lambda =-2nk,\)

\({ C(\xi ,X)\cdot S}=0\) (obtained by \(a=b=-\frac{1}{2n-1},c=1\))

\( \lambda =-2nk,\)

\({ \hat{C}(\xi ,X)\cdot S}=0\) (obtained by \(a=b=-\frac{1}{2n-1},c=0\))

\(\lambda =-2nk,-\frac{2n-1}{2}\)

\({ P(\xi ,X)\cdot S}=0\) (obtained by \(a=-\frac{1}{2n},b=c=0\))

\(\lambda =-2nk,\)

\({ H(\xi ,X)\cdot S}=0\) (obtained by \(a=b=-\frac{1}{4n},\) \(c=0\))

\(\lambda =-2nk.\)

From the above table, we can state the following:

Theorem 4.1

Ricci soliton \((M,g,\xi )\) in a Sasakian manifold admitting \(\mathcal { W\cdot }{ S}=0\) is always shrinking.