The notion of recurrent manifold was introduced by Walker ^{1}. A non-flat Riemannian manifold is said to be recurrent if the curvature tensor

where ^{2}.

Lichnerowics ^{3} introduced the 2-recurrent Riemannian manifold which is defined as: A non-flat Riemannian manifold is called 2-recurrent Riemannian manifold if the Riemannian curvature tensor

where ^{ }

where ^{5, 6, 7}. In particular if ^{3. }

It is known that the quasi-conformal curvature tensor ^{ }

where ^{9} as

In tuned with the definition given by Ray^{ }

where

where ^{ }

where ^{n}) → TP(M^{n})

An Einstein manifold is defined by

for which scalar curvature

In this paper, section 1 is the introduction of recurrent and generalized 2-recurrent Riemannian manifold. In section 2, we study generalized 2-recurrent Riemannian manifold with Einstein manifold. In next section 3, we first obtain a necessary and sufficient condition for an Einstein

Riemannian manifold is a part of differentiable manifold which we study by index free notation and tensor notation. Different type of differentiable manifold and their standard results used to characterize the recurrent manifolds. The fundamental theorem of Riemannian geometry, Ricci Identity, Bianchi first and second Identity, Contraction method and Levi-Civita connection are used to finding the results in this paper. Standard techniques and methods in the field of differential geometry developed by investigators such as Jaiswal and Ojha ^{11}, De and Sarkar ^{12}, Bagewadi and Ingalahalli ^{13} and their recent work are used and extended in this paper.

We study

Let us assume that the

where

From (13) it follows that

So using Bianchi’s identity we find that

Covariant differentiation of (14) gives

By virtue of (6) and (14), it follows from (15) that

From (16) on contraction we get

From (13) and (8), we get

Taking

On using (17) and (18) in equation (15), we get

i.e.

Hence from (15) we get

That is, the manifold is of constant curvature. Hence we have the following theorem:

Now, we assume that an Einstein

where

where

Since the manifold is Einstein so from (20), we have

i.e.

which is equivalent to an Einstein manifold to

Conversely, suppose that an Einstein

Since in an Einstein manifold scalar curvature

Thus we have the following theorem:

Next, if the Ricci tensor i.e.

which is

In this article we study generalized 2-recurrent Riemannian manifold with quasi-conformal curvature tensor which is denoted by

Presented in 4^{th} Mizoram Science Congress (MSC 2022) during 20^{th }& 21^{st} October 2022, organized by Mizoram Science, Technology and Innovation Council (MISTIC), Directorate of Science and Technology (DST) Mizoram, Govt. of Mizoram in collaboration with science NGOs in Mizoram such as Mizo Academy of Sciences (MAS), Mizoram Science Society (MSS), Science Teachers’ Association, Mizoram (STAM), Geological Society of Mizoram (GSM), Mizoram Mathematics Society (MMS), Biodiversity and Nature ConservationNetwork (BIOCONE) and Mizoram Information & Technology Society (MITS). The Organizers claim the peer review responsibility.