# Every Model Learned by Gradient Descent Is Approximately a Kernel Machine

Pedro Domingos<EMAIL_ADDRESS>
Paul G. Allen School of Computer Science & Engineering
University of Washington
Seattle, WA 98195-2350, USA

###### Abstract

Deep learning’s successes are often attributed to its ability to automatically
discover new representations of the data, rather than relying on handcrafted
features like other learning methods. We show, however, that deep networks
learned by the standard gradient descent algorithm are in fact mathematically
approximately equivalent to kernel machines, a learning method that simply
memorizes the data and uses it directly for prediction via a similarity
function (the kernel). This greatly enhances the interpretability of deep
network weights, by elucidating that they are effectively a superposition of
the training examples. The network architecture incorporates knowledge of the
target function into the kernel. This improved understanding should lead to
better learning algorithms.

Keywords: gradient descent, kernel machines, deep learning, representation
learning, neural tangent kernel

##  1 Introduction

Despite its many successes, deep learning remains poorly understood
(Goodfellow et al., 2016). In contrast, kernel machines are based on a well-
developed mathematical theory, but their empirical performance generally lags
behind that of deep networks (Schölkopf and Smola, 2002). The standard
algorithm for learning deep networks, and many other models, is gradient
descent (Rumelhart et al., 1986). Here we show that every model learned by
this method, regardless of architecture, is approximately equivalent to a
kernel machine with a particular type of kernel. This kernel measures the
similarity of the model at two data points in the neighborhood of the path
taken by the model parameters during learning. Kernel machines store a subset
of the training data points and match them to the query using the kernel. Deep
network weights can thus be seen as a superposition of the training data
points in the kernel’s feature space, enabling their efficient storage and
matching. This contrasts with the standard view of deep learning as a method
for discovering representations from data, with the attendant lack of
interpretability (Bengio et al., 2013). Our result also has significant
implications for boosting algorithms (Freund and Schapire, 1997),
probabilistic graphical models (Koller and Friedman, 2009), and convex
optimization (Boyd and Vandenberghe, 2004).

##  2 Path Kernels

A kernel machine is a model of the form

$y=g\left(\sum_{i}a_{i}K(x,x_{i})+b\right),$

where $x$ is the query data point, the sum is over training data points
$x_{i}$, $g$ is an optional nonlinearity, the $a_{i}$’s and $b$ are learned
parameters, and the kernel $K$ measures the similarity of its arguments
(Schölkopf and Smola, 2002). In supervised learning, $a_{i}$ is typically a
linear function of $y_{i}^{*}$, the known output for $x_{i}$. Kernels may be
predefined or learned (Cortes et al., 2009). Kernel machines, also known as
support vector machines, are one of the most developed and widely used machine
learning methods. In the last decade, however, they have been eclipsed by deep
networks, also known as neural networks and multilayer perceptrons, which are
composed of multiple layers of nonlinear functions. Kernel machines can be
viewed as neural networks with one hidden layer, with the kernel as the
nonlinearity. For example, a Gaussian kernel machine is a radial basis
function network (Poggio and Girosi, 1990). But a deep network would seem to
be irreducible to a kernel machine, since it can represent some functions
exponentially more compactly than a shallow one (Delalleau and Bengio, 2011;
Cohen et al., 2016).

Whether a representable function is actually learned, however, depends on the
learning algorithm. Most deep networks, and indeed most machine learning
models, are trained using variants of gradient descent (Rumelhart et al.,
1986). Given an initial parameter vector $w_{0}$ and a loss function
$L=\sum_{i}L(y_{i}^{*},y_{i})$, gradient descent repeatedly modifies the
model’s parameters $w$ by subtracting the loss’s gradient from them, scaled by
the learning rate $\epsilon$:

$w_{s+1}=w_{s}-\epsilon\nabla_{w}L(w_{s}).$

The process terminates when the gradient is zero and the loss is therefore at
an optimum (or saddle point). Remarkably, we have found that learning by
gradient descent is a strong enough constraint that the end result is
guaranteed to be approximately a kernel machine, regardless of the number of
layers or other architectural features of the model.

Specifically, the kernel machines that result from gradient descent use what
we term a path kernel. If we take the learning rate to be infinitesimally
small, the path kernel between two data points is simply the integral of the
dot product of the model’s gradients at the two points over the path taken by
the parameters during gradient descent:

$K(x,x^{\prime})=\int_{c(t)}\nabla_{w}y(x)\cdot\nabla_{w}y(x^{\prime})\>dt,$

where $c(t)$ is the path. Intuitively, the path kernel measures how similarly
the model at the two data points varies during learning. The more similar the
variation for $x$ and $x^{\prime}$, the higher the weight of $x^{\prime}$ in
predicting $y$. Fig. 1 illustrates this graphically.

Figure 1: How the path kernel measures similarity between examples. In this
two-dimensional illustration, as the weights follow a path on the plane during
training, the model’s gradients (vectors on the weight plane) for $x$, $x_{1}$
and $x_{2}$ vary along it. The kernel $K(x,x_{1})$ is the integral of the dot
product of the gradients $\nabla_{w}y(x)$ and $\nabla_{w}y(x_{1})$ over the
path, and similarly for $K(x,x_{2})$. Because on average over the weight path
$\nabla_{w}y(x)\cdot\nabla_{w}y(x_{1})$ is greater than
$\nabla_{w}y(x)\cdot\nabla_{w}y(x_{2})$, $y_{1}$ has more influence than
$y_{2}$ in predicting $y$, all else being equal.

Our result builds on the concept of neural tangent kernel, recently introduced
to analyze the behavior of deep networks (Jacot et al., 2018). The neural
tangent kernel is the integrand of the path kernel when the model is a
multilayer perceptron. Because of this, and since a sum of positive definite
kernels is also a positive definite kernel (Schölkopf and Smola, 2002), the
known conditions for positive definiteness of neural tangent kernels extend to
path kernels (Jacot et al., 2018). A positive definite kernel is equivalent to
a dot product in a derived feature space, which greatly simplifies its
analysis (Schölkopf and Smola, 2002).

We now present our main result. For simplicity, in the derivations below we
assume that $y$ is a (real-valued) scalar, but it can be made a vector with
only minor changes. The data points $x_{i}$ can be arbitrary structures.

###### Definition 1

The tangent kernel associated with function $f_{w}(x)$ and parameter vector
$v$ is
$K^{g}_{f,v}(x,x^{\prime})=\nabla_{w}f_{w}(x)\cdot\nabla_{w}f_{w}(x^{\prime})$,
with the gradients taken at $v$.

###### Definition 2

The path kernel associated with function $f_{w}(x)$ and curve $c(t)$ in
parameter space is
$K^{p}_{f,c}(x,x^{\prime})={\displaystyle\int_{c(t)}}K^{g}_{f,w(t)}(x,x^{\prime})\>dt$.

###### Theorem 1

Suppose the model $y=f_{w}(x)$, with $f$ a differentiable function of $w$, is
learned from a training set $\\{(x_{i},y_{i}^{*})\\}_{i=1}^{m}$ by gradient
descent with differentiable loss function $L=\sum_{i}L(y_{i}^{*},y_{i})$ and
learning rate $\epsilon$. Then

$\lim_{\epsilon\rightarrow 0}y=\sum_{i=1}^{m}a_{i}K(x,x_{i})+b,$

where $K(x,x_{i})$ is the path kernel associated with $f_{w}(x)$ and the path
taken by the parameters during gradient descent, $a_{i}$ is the average
$-\partial L/\partial y_{i}$ along the path weighted by the corresponding
tangent kernel, and $b$ is the initial model.

Proof  In the $\epsilon\rightarrow 0$ limit, the gradient descent equation,
which can also be written as

$\frac{w_{s+1}-w_{s}}{\epsilon}=-\nabla_{w}L(w_{s}),$

where $L$ is the loss function, becomes the differential equation

$\frac{dw(t)}{dt}=-\nabla_{w}L(w(t)).$

(This is known as a gradient flow (Ambrosio et al., 2008).) Then for any
differentiable function of the weights $y$,

$\frac{dy}{dt}=\sum_{j=1}^{d}\frac{\partial y}{\partial
w_{j}}\frac{dw_{j}}{dt},$

where $d$ is the number of parameters. Replacing $dw_{j}/dt$ by its gradient
descent expression:

$\frac{dy}{dt}=\sum_{j=1}^{d}\frac{\partial y}{\partial
w_{j}}\left(-\frac{\partial L}{\partial w_{j}}\right).$

Applying the additivity of the loss and the chain rule of differentiation:

$\frac{dy}{dt}=\sum_{j=1}^{d}\frac{\partial y}{\partial
w_{j}}\left(-\sum_{i=1}^{m}\frac{\partial L}{\partial y_{i}}\frac{\partial
y_{i}}{\partial w_{j}}\right).$

Rearranging terms:

$\frac{dy}{dt}=-\sum_{i=1}^{m}\frac{\partial L}{\partial
y_{i}}\sum_{j=1}^{d}\frac{\partial y}{\partial w_{j}}\frac{\partial
y_{i}}{\partial w_{j}}.$

Let $L^{\prime}(y_{i}^{*},y_{i})=\partial L/\partial y_{i}$, the loss
derivative for the $i$th output. Applying this and Definition 1:

$\frac{dy}{dt}=-\sum_{i=1}^{m}L^{\prime}(y_{i}^{*},y_{i})K^{g}_{f,w(t)}(x,x_{i}).$

Let $y_{0}$ be the initial model, prior to gradient descent. Then for the
final model $y$:

$\lim_{\epsilon\rightarrow
0}y=y_{0}-\int_{c(t)}\sum_{i=1}^{m}L^{\prime}(y_{i}^{*},y_{i})K^{g}_{f,w(t)}(x,x_{i})\>dt,$

where $c(t)$ is the path taken by the parameters during gradient descent.
Multiplying and dividing by $\int_{c(t)}K^{g}_{f,w(t)}(x,x_{i})\>dt$:

$\lim_{\epsilon\rightarrow
0}y=y_{0}-\sum_{i=1}^{m}\left(\frac{\int_{c(t)}K^{g}_{f,w(t)}(x,x_{i})L^{\prime}(y_{i}^{*},y_{i})\>dt}{\int_{c(t)}K^{g}_{f,w(t)}(x,x_{i})\>dt}\right)\int_{c(t)}K^{g}_{f,w(t)}(x,x_{i})\>dt.$

Let
$\overline{L^{\prime}}(y_{i}^{*},y_{i})=\int_{c(t)}K^{g}_{f,w(t)}(x,x_{i})L^{\prime}(y_{i}^{*},y_{i})\>dt/\int_{c(t)}K^{g}_{f,w(t)}(x,x_{i})\>dt$,
the average loss derivative weighted by similarity to $x$. Applying this and
Definition 2:

$\lim_{\epsilon\rightarrow
0}y=y_{0}-\sum_{i=1}^{m}\overline{L^{\prime}}(y_{i}^{*},y_{i})K^{p}_{f,c}(x,x_{i}).$

Thus

$\lim_{\epsilon\rightarrow 0}y=\sum_{i=1}^{m}a_{i}K(x,x_{i})+b,$

with $K(x,x_{i})=K^{p}_{f,c}(x,x_{i})$,
$a_{i}=-\overline{L^{\prime}}(y_{i}^{*},y_{i})$, and $b=y_{0}$.

###### Remark 1

This differs from typical kernel machines in that the $a_{i}$’s and $b$ depend
on $x$. Nevertheless, the $a_{i}$’s play a role similar to the example weights
in ordinary SVMs and the perceptron algorithm: examples that the loss is more
sensitive to during learning have a higher weight. $b$ is simply the prior
model, and the final model is thus the sum of the prior model and the model
learned by gradient descent, with the query point entering the latter only
through kernels. Since Theorem 1 applies to every $y_{i}$ as a query
throughout gradient descent, the training data points also enter the model
only through kernels (initial model aside).

###### Remark 2

Theorem 1 can equally well be proved using the loss-weighted path kernel
$K^{lp}_{f,c,L}=\int_{c(t)}L^{\prime}(y_{i}^{*},y_{i})K^{g}_{f,w(t)}(x,x_{i})\>dt$,
in which case $a_{i}=-1$ for all $i$.

###### Remark 3

In least-squares regression, $L^{\prime}(y_{i}^{*},y_{i})=y_{i}-y_{i}^{*}$.
When learning a classifier by minimizing cross-entropy, the standard practice
in deep learning, the function to be estimated is the conditional probability
of the class, $p_{i}$, the loss is $-\sum_{i=1}^{m}\ln p_{i}$, and the loss
derivative for the $i$th output is $-1/p_{i}$. Similar expressions hold for
modeling a joint distribution by minimizing negative log likelihood, with
$p_{i}$ as the probability of the data point.

###### Remark 4

Adding a regularization term $R(w)$ to the loss function simply adds
$-\int_{c(t)}\sum_{j=1}^{d}(\partial y/\partial w_{j})(\partial R/\partial
w_{j})$ to $b$.

###### Remark 5

The proof above is for batch gradient descent, which uses all training data
points at each step. To extend it to stochastic gradient descent, which uses a
subsample, it suffices to multiply each term in the summation over data points
by an indicator function $I_{i}(t)$ that is 1 if the $i$th data point is
included in the subsample at time $t$ and 0 otherwise. The only change this
causes in the result is that the path kernel and average loss derivative for a
data point are now stochastic integrals. Based on previous results (Scieur et
al., 2017), Theorem 1 or a similar result seems likely to also apply to
further variants of gradient descent, but proving this remains an open
problem.

For linear models, the path kernel reduces to the dot product of the data
points. It is well known that a single-layer perceptron is a kernel machine,
with the dot product as the kernel (Aizerman et al., 1964). Our result can be
viewed as a generalization of this to multilayer perceptrons and other models.
It is also related to Lippmann et al.’s proof that Hopfield networks, a
predecessor of many current deep architectures, are equivalent to the nearest-
neighbor algorithm, a predecessor of kernel machines, with Hamming distance as
the comparison function (Lippmann et al., 1987).

The result assumes that the learning rate is sufficiently small for the
trajectory of the weights during gradient descent to be well approximated by a
smooth curve. This is standard in the analysis of gradient descent, and is
also generally a good approximation in practice, since the learning rate has
to be quite small in order to avoid divergence (e.g., $\epsilon=10^{-3}$)
(Goodfellow et al., 2016). Nevertheless, it remains an open question to what
extent models learned by gradient descent can still be approximated by kernel
machines outside of this regime.

##  3 Discussion

A notable disadvantage of deep networks is their lack of interpretability
(Zhang and Zhu, 2018). Knowing that they are effectively path kernel machines
greatly ameliorates this. In particular, the weights of a deep network have a
straightforward interpretation as a superposition of the training examples in
gradient space, where each example is represented by the corresponding
gradient of the model. Fig. 2 illustrates this. One well-studied approach to
interpreting the output of deep networks involves looking for training
instances that are close to the query in Euclidean or some other simple space
(Ribeiro et al., 2016). Path kernels tell us what the exact space for these
comparisons should be, and how it relates to the model’s predictions.

Figure 2: Deep network weights as superpositions of training examples.
Applying the learned model to a query example is equivalent to simultaneously
matching the query with each stored example using the path kernel and
outputting a weighted sum of the results.

Experimentally, deep networks and kernel machines often perform more similarly
than would be expected based on their mathematical formulation (Brendel and
Bethge, 2019). Even when they generalize well, deep networks often appear to
memorize and replay whole training instances (Zhang et al., 2017; Devlin et
al., 2015). The fact that deep networks are in fact kernel machines helps
explain both of these observations. It also sheds light on the surprising
brittleness of deep models, whose performance can degrade rapidly as the query
point moves away from the nearest training instance (Szegedy et al., 2014),
since this is what is expected of kernel estimators in high-dimensional spaces
(Hardle et al., 2004).

Perhaps the most significant implication of our result for deep learning is
that it casts doubt on the common view that it works by automatically
discovering new representations of the data, in contrast with other machine
learning methods, which rely on predefined features (Bengio et al., 2013). As
it turns out, deep learning also relies on such features, namely the gradients
of a predefined function, and uses them for prediction via dot products in
feature space, like other kernel machines. All that gradient descent does is
select features from this space for use in the kernel. If gradient descent is
limited in its ability to learn representations, better methods for this
purpose are a key research direction. Current nonlinear alternatives include
predicate invention (Muggleton and Buntine, 1988) and latent variable
discovery in graphical models (Elidan et al., 2000). Techniques like structure
mapping (Gentner, 1983), crossover (Holland, 1975) and predictive coding (Rao
and Ballard, 1999) may also be relevant. Ultimately, however, we may need
entirely new approaches to solve this crucial but extremely difficult problem.

Our result also has significant consequences on the kernel machine side. Path
kernels provide a new and very flexible way to incorporate knowledge of the
target function into the kernel. Previously, it was only possible to do so in
a weak sense, via generic notions of what makes two data points similar. The
extensive knowledge that has been encoded into deep architectures by applied
researchers, and is crucial to the success of deep learning, can now be ported
directly to kernel machines. For example, kernels with translation invariance
or selective attention are directly obtainable from the architecture of,
respectively, convolutional neural networks (LeCun et al., 1998) or
transformers (Vaswani et al., 2017).

A key property of path kernels is that they combat the curse of dimensionality
by incorporating derivatives into the kernel: two data points are similar if
the candidate function’s derivatives at them are similar, rather than if they
are close in the input space. This can greatly improve kernel machines’
ability to approximate highly variable functions (Bengio et al., 2005). It
also means that points that are far in Euclidean space can be close in
gradient space, potentially improving the ability to model complex functions.
(For example, the maxima of a sine wave are all close in gradient space, even
though they can be arbitrarily far apart in the input space.)

Most significantly, however, learning path kernel machines via gradient
descent largely overcomes the scalability bottlenecks that have long limited
the applicability of kernel methods to large data sets. Computing and storing
the Gram matrix at learning time, with its quadratic cost in the number of
examples, is no longer required. (The Gram matrix is the matrix of
applications of the kernel to all pairs of training examples.) Separately
storing and matching (a subset of) the training examples at query time is also
no longer necessary, since they are effectively all stored and matched
simultaneously via their superposition in the model parameters. The storage
space and matching time are independent of the number of examples.
(Interestingly, superposition has been hypothesized to play a key role in
combatting the combinatorial explosion in visual cognition (Arathorn, 2002),
and is also essential to the efficiency of quantum computing (Nielsen and
Chuang, 2000) and radio communication (Carlson and Grilly, 2009).) Further,
the same specialized hardware that has given deep learning a decisive edge in
scaling up to large data (Raina et al., 2009) can now be used for kernel
machines as well.

The significance of our result extends beyond deep networks and kernel
machines. In its light, gradient descent can be viewed as a boosting
algorithm, with tangent kernel machines as the weak learner and path kernel
machines as the strong learner obtained by boosting it (Freund and Schapire,
1997). In each round of boosting, the examples are weighted by the
corresponding loss derivatives. It is easily seen that each round (gradient
descent step) decreases the loss, as required. The weight of the model at a
given round is the learning rate for that step, which can be constant or the
result of a line search (Boyd and Vandenberghe, 2004). In the latter case
gradient descent is similar to gradient boosting (Mason et al., 1999).

Another consequence of our result is that every probabilistic model learned by
gradient descent, including Bayesian networks (Koller and Friedman, 2009), is
a form of kernel density estimation (Parzen, 1962). The result also implies
that the solution of every convex learning problem is a kernel machine,
irrespective of the optimization method used, since, being unique, it is
necessarily the solution obtained by gradient descent. It is an open question
whether the result can be extended to nonconvex models learned by non-
gradient-based techniques, including constrained (Bertsekas, 1982) and
combinatorial optimization (Papadimitriou and Steiglitz, 1982).

The results in this paper suggest a number of research directions. For
example, viewing gradient descent as a method for learning path kernel
machines may provide new paths for improving it. Conversely, gradient descent
is not necessarily the only way to form superpositions of examples that are
useful for prediction. The key question is how to optimize the tradeoff
between accurately capturing the target function and minimizing the
computational cost of storing and matching the examples in the superposition.

Acknowledgments

This research was partly funded by ONR grant N00014-18-1-2826. Thanks to Léon
Bottou and Simon Du for feedback on a draft of this paper.

## References

  * Aizerman et al. (1964) M. A. Aizerman, E. M. Braverman, and L. I. Rozonoer.  Theoretical foundations of the potential function method in pattern recognition learning.  _Autom. & Remote Contr._, 25:821–837, 1964.
  * Ambrosio et al. (2008) Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré.  _Gradient Flows: In Metric Spaces and in the Space of Probability Measures_.  Birkhäuser, Basel, 2nd edition, 2008.
  * Arathorn (2002) D. W. Arathorn.  _Map-Seeking Circuits in Visual Cognition: A Computational Mechanism for Biological and Machine Vision_.  Stanford Univ. Press, Stanford, CA, 2002.
  * Bengio et al. (2005) Y. Bengio, O. Delalleau, and N. L. Roux.  The curse of highly variable functions for local kernel machines.  _Adv. Neural Inf. Proc. Sys._ , 18:107–114, 2005.
  * Bengio et al. (2013) Y. Bengio, A. Courville, and P. Vincent.  Representation learning: A review and new perspectives.  _IEEE Trans. Patt. An. & Mach. Intell._, 35:1798–1828, 2013.
  * Bertsekas (1982) P. Bertsekas.  _Constrained Optimization and Lagrange Multiplier Methods_.  Academic Press, Cambridge, MA, 1982.
  * Boyd and Vandenberghe (2004) S. Boyd and L. Vandenberghe.  _Convex Optimization_.  Cambridge University Press, Cambridge, UK, 2004.
  * Brendel and Bethge (2019) W. Brendel and M. Bethge.  Approximating CNNs with bag-of-local-features models works surprisingly well on ImageNet.  In _Proc. Int. Conf. Learn. Repr._ , 2019.
  * Carlson and Grilly (2009) A. B. Carlson and P. B. Grilly.  _Communication Systems: An Introduction to Signals and Noise in Electrical Communication_.  McGraw-Hill, New York, 5th edition, 2009.
  * Cohen et al. (2016) N. Cohen, O. Sharir, and A. Shashua.  On the expressive power of deep learning: A tensor analysis.  In _Proc. Ann. Conf. Learn. Th._ , pages 698–728, 2016.
  * Cortes et al. (2009) C. Cortes, M. Mohri, and A. Rostamizadeh.  Learning non-linear combinations of kernels.  _Adv. Neural Inf. Proc. Sys._ , 22:396–404, 2009.
  * Delalleau and Bengio (2011) O. Delalleau and Y. Bengio.  Shallow vs. deep sum-product networks.  _Adv. Neural Inf. Proc. Sys._ , 24:666–674, 2011.
  * Devlin et al. (2015) J. Devlin, H. Cheng, H. Fang, S. Gupta, L. Deng, X. He, G. Zweig, and M. Mitchell.  Language models for image captioning: The quirks and what works.  In _Proc. Ann. Meet. Assoc. Comp. Ling._ , pages 100–105, 2015.
  * Elidan et al. (2000) G. Elidan, N. Lotner, N. Friedman, and D. Koller.  Discovering hidden variables: A structure-based approach.  _Adv. Neural Inf. Proc. Sys._ , 13:479–485, 2000.
  * Freund and Schapire (1997) Y. Freund and R. Schapire.  A decision-theoretic generalization of on-line learning and an application to boosting.  _J. Comp. & Sys. Sci._, 55:119–139, 1997.
  * Gentner (1983) D. Gentner.  Structure-mapping: A theoretical framework for analogy.  _Cog. Sci._ , 7:155–170, 1983.
  * Goodfellow et al. (2016) I. Goodfellow, Y. Bengio, and A. Courville.  _Deep Learning_.  MIT Press, Cambridge, MA, 2016.
  * Hardle et al. (2004) W. Hardle, M. Müller, and A. Werwatz.  _Nonparametric and Semiparametric Models_.  Springer, Berlin, 2004.
  * Holland (1975) J. H. Holland.  _Adaptation in Natural and Artificial Systems_.  Univ. Michigan Press, Ann Arbor, MI, 1975.
  * Jacot et al. (2018) A. Jacot, F. Gabriel, and C. Hongler.  Neural tangent kernel: Convergence and generalization in neural networks.  _Adv. Neural Inf. Proc. Sys._ , 31:8571–8580, 2018.
  * Koller and Friedman (2009) D. Koller and N. Friedman.  _Probabilistic Graphical Models: Principles and Techniques_.  MIT Press, Cambridge, MA, 2009.
  * LeCun et al. (1998) Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner.  Gradient-based learning applied to document recognition.  _Proc. IEEE_ , 86:2278–2324, 1998.
  * Lippmann et al. (1987) R. Lippmann, B. Gold, and M. Malpass.  A comparison of Hamming and Hopfield neural networks for pattern classification.  Technical Report 769, MIT Lincoln Lab, Lexington, MA, 1987.
  * Mason et al. (1999) L. Mason, J. Baxter, P. L. Bartlett, and M. R. Frean.  Boosting algorithms as gradient descent.  _Adv. Neural Inf. Proc. Sys,_ , 12:512–518, 1999.
  * Muggleton and Buntine (1988) S. Muggleton and W. Buntine.  Machine invention of first-order predicates by inverting resolution.  In _Proc. Int. Conf. Mach. Learn._ , pages 339–352, 1988.
  * Nielsen and Chuang (2000) M. A. Nielsen and I. L. Chuang.  _Quantum Computation and Quantum Information_.  Cambridge Univ. Press, Cambridge, UK, 2000.
  * Papadimitriou and Steiglitz (1982) C. Papadimitriou and K. Steiglitz.  _Combinatorial Optimization: Algorithms and Complexity_.  Prentice-Hall, Upper Saddle River, NJ, 1982.
  * Parzen (1962) E. Parzen.  On estimation of a probability density function and mode.  _Ann. Math. Stat._ , 33:1065–1076, 1962.
  * Poggio and Girosi (1990) T. Poggio and F. Girosi.  Regularization algorithms for learning that are equivalent to multilayer networks.  _Science_ , 247:978–982, 1990.
  * Raina et al. (2009) R. Raina, A. Madhavan, and A. Y. Ng.  Large-scale deep unsupervised learning using graphics processors.  In _Proc. Int. Conf. Mach. Learn._ , pages 873–880, 2009.
  * Rao and Ballard (1999) R. P. N. Rao and D. H. Ballard.  Predictive coding in the visual cortex: A functional interpretation of some extra-classical receptive field effects.  _Nature Neurosci._ , 2:79–87, 1999.
  * Ribeiro et al. (2016) M. T. Ribeiro, S. Singh, and C. Guestrin.  “Why should I trust you?”: Explaining the predictions of any classifier.  In _Proc. Int. Conf. Knowl. Disc. & Data Mining_, pages 1135–1144, 2016.
  * Rumelhart et al. (1986) D. E. Rumelhart, G. E. Hinton, and R. J. Williams.  Learning representations by back-propagating errors.  _Nature_ , 323:533–536, 1986.
  * Schölkopf and Smola (2002) B. Schölkopf and A. J. Smola.  _Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond_.  MIT Press, Cambridge, MA, 2002.
  * Scieur et al. (2017) D. Scieur, V. Roulet, F. Bach, and A. d’Aspremont.  Integration methods and optimization algorithms.  _Adv. Neural Inf. Proc. Sys._ , 30:1109–1118, 2017.
  * Szegedy et al. (2014) C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. Goodfellow, and R. Fergus.  Intriguing properties of neural networks.  In _Proc. Int. Conf. Learn. Repr._ , 2014.
  * Vaswani et al. (2017) A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, N. Aidan, L. Kaiser, and I. Polosukhin.  Attention is all you need.  _Adv. Neural Inf. Proc. Sys._ , 30:5998–6008, 2017.
  * Zhang et al. (2017) C. Zhang, S. Bengio, M. Hardt, B. Recht, and O. Vinyals.  Understanding deep learning requires rethinking generalization.  In _Proc. Int. Conf. Learn. Repr._ , 2017.
  * Zhang and Zhu (2018) Q. Zhang and S.-C. Zhu.  Visual interpretability for deep learning: A survey.  _Front. Inf. Tech. & Elec. Eng._, 19:27–39, 2018.


#  Generalized Quasi-Einstein Manifolds in Contact Geometry

İnan Ünal
Department of Computer Engineering
Munzur University, Tunceli, Turkey
<EMAIL_ADDRESS>
###### Abstract

In this study, we investigate generalized quasi-Einstein structure for normal
metric contact pair manifolds. Firstly, we deal with elementary properties and
examine, existence, and characterizations of generalized quasi-Einstein normal
metric contact pair manifold. Secondly, the generalized quasi-constant
curvature of normal metric contact pair manifolds are studied and it is proven
that a normal metric contact pair manifold with generalized quasi-constant
curvature is a generalized quasi-Einstein manifold. Normal metric contact pair
manifolds satisfying cyclic Ricci tensor and the Codazzi type of Ricci tensor
are considered and its shown that a generalized quasi-Einstein normal metric
contact pair manifold does not satisfy Codazzi type of Ricci tensor. Finally,
we work on normal metric contact pair manifolds satisfying certain curvature
conditions related to $\mathcal{M}-$projective, conformal, and concircular
curvature tensors. We show that a normal metric contact pair manifold with
generalized quasi-constant curvature is locally isometric to the Hopf manifold
$S^{2n+1}(1)\times S^{1}$.

######  Keywords.

generalized quasi-Einstein, Contact pairs, generalized quasi-constant
curvature,
$\mathcal{M}-$projevtice curvature tensor

##  1 Introduction

An Einstein manifold, which comes from the solution of Einstein fields
equation, is defined as Ricci tensor $Ric$ of type $(0,2)$ is non-zero and
proportional to the metric tensor. Since Einstein manifolds have important
differential geometric properties as well as their physical applications, they
are studied by geometers in a wide perspective. A generalization of Einstein
manifolds is quasi-Einstein manifolds. A Riemannian manifold $M$ is called
quasi-Einstein if the Ricci curvature tensor has the following form:

$Ric(X_{1},X_{2})=\lambda g(X,Y)+\beta\omega(X_{1})\omega(X_{2})$

for all $X_{1},X_{2}\in\Gamma(TM)$, where $\lambda,\beta$ are scalars and
$\omega$ is a $1-$form. In contact geometry $\eta-$Einstein manifolds could be
seen as a special case of quasi-Einstein manifolds. Quasi-Einstein manifolds
arose during the study of exact solutions of the Einstein field equations as
well as during considerations of quasi-umbilical hypersurfaces. For instance,
the Robertson-Walker space-times are quasi-Einstein manifolds. For more
details about quasi-Einstein manifolds see [15, 22, 30].

The generalization of quasi-Einstein manifolds has been presented in different
perspectives. One of them was given by Chaki in [11] and another was presented
by Catino [9] with to generalize the concepts of the Ricci soliton, the Ricci
almost soliton and quasi-Einstein manifolds. The third definition of
generalized quasi-Einstein manifolds were given by De and Ghosh [16] as
follow,

$Ric(X_{1},X_{2})=\lambda
g(X,Y)+\beta\omega(X_{1})\omega(X_{2})+\mu\eta(X_{1})\eta(X_{2})$

where $\omega,\mu$ are two non-zero $1-$ forms and $\lambda,\beta$ are certain
non-zero scalars [10]. The unit and orthogonal vector fields $\xi_{1}$ and
$\xi_{2}$ corresponding to the $1-$forms $\omega$ and $\eta$ are defined by
$g(X_{1},\xi_{1})=\omega(X_{1}),\ g(X_{2},\xi_{2})=\eta(X_{1})$, respectively
[16]. The geometric properties of generalized quasi-Einstein manifolds have
been studied in [16, 29, 17, 27, 18]. A generalized quasi-Einstein manifold ,
in addition to its geometrical features, it also has physical applications in
general relativity [24, 23, 19]. Complex $\eta-$Einstein manifolds are a
special case of generalized quasi-Einstein manifolds (see [33]).

In [2] Bande and Hadjar defined a new contact structure on a differentiable
$(m=2p+2q+2)-$dimensional manifold $M$ with two $1-$forms
$\alpha_{1},\alpha_{2}$. This structure was firstly studied by Blair, Ludden
and Yano [8] as the name of bicontact manifolds. Bande and Hadjar considered a
special type of $f-$structure with complementary frames related to these
contact forms and they obtained associated metric. A differentiable manifold
with this structure is called metric contact pair (MCP) manifold . Riemannian
geometry of MCP manifolds is given in [4, 3].

This paper is on applications of generalized quasi-Einstein manifolds in
contact geometry. We consider the normal metric contact pair manifold admits
generalized quasi-Einstein structure and call this type of manifold as
generalized quasi-Einstein normal metric contact pair manifolds. After
presenting definitions and basic properties, we examine the existence of such
manifolds. Also, we present a characterization of generalized quasi-Einstein
normal metric contact pair manifolds. Moreover, we consider the notion of
generalized quasi-constant curvature for normal metric contact pair manifolds
and we obtain some results on the sectional curvature. We investigate a
generalized quasi-Einstein normal metric contact pair manifold under some
conditions for Ricci tensor. Finally, we study on $M-$projective curvature
tensor, conformal curvature tensor and concircular curvature tensor on a
normal metric contact pair manifold and we gave some results under flatness
and symmetry conditions.

##  2 Preliminaries

Contact pairs were defined by Bande and Hadjar [2] in 2005, for details see
[2, 3, 4]. In this section we give some fundamental facts about contact metric
pair manifolds.

######  Definition 2.1.

A pair of two $1-$forms, $(\alpha_{1},\alpha_{2})$ on an $(m=2p+2q+2)$
-dimensional differentiable manifold $M$, is called contact pair of type
$(p,q)$ if we have

  * •

$\alpha_{1}\wedge(d\alpha_{1})^{p}\wedge\alpha_{2}\wedge(d\alpha_{2})^{q}\neq
0$,

  * •

$(d\alpha_{1})^{p+1}=0$ and $(d\alpha_{2})^{q+1}=0,$

where $p,q$ are positive integers [2].

For $1-$forms $\alpha_{1}$ and $\alpha_{2}$, we have two integrable subbundle
of $TM$; $\mathcal{D}_{1}=\\{X:\alpha_{1}(X)=0,X\in\Gamma(TM)\\}$ and
$\mathcal{D}_{2}=\\{X:\alpha_{2}(X)=0,X\in\Gamma(TM)\\}$. Then we have two
characteristic foliations of $M$, denoted $\mathcal{F}_{1}=\mathcal{D}_{1}\cap
kerd\alpha_{1}$ and $\mathcal{F}_{2}=\mathcal{D}_{2}\cap kerd\alpha_{2}$
respectively. $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are $(2p+1)$ and
$(2q+1)-$dimensional contact manifolds with contact form induced by
$\alpha_{2}$ and $\alpha_{1}$. Thus we can define $(2p+2q)-$dimensional
horizontal subbundle $\mathcal{H}=ker\alpha_{1}\cap ker\alpha_{2}$. To a
contact pair $(\alpha_{1},\alpha_{2})$ of type $(p,q)$ there are associated
two commuting vector fields $Z_{1}$ and $Z_{2}$, called Reeb vector fields of
the pair, which are uniquely determined by the following equations:

$\displaystyle\alpha_{1}(Z_{1})=\alpha_{2}(Z_{2})=1,\
\alpha_{1}(Z_{2})=\alpha_{2}(Z_{1})=0$ $\displaystyle
i_{Z_{1}}d\alpha_{1}=i_{Z_{1}}d\alpha_{2}=i_{Z_{2}}d\alpha_{2}=0$

where $i_{X}$ is the contraction with the vector field X. In particular, since
the Reeb vector fields commute, they determine a locally free
$\mathbb{R}^{2}$-action, called the Reeb action.

The tangent bundle of $(M,(\alpha_{1},\alpha_{2}))$ can be split into as
follow. For the two subbundle of $TM$

$\displaystyle T\mathcal{G}_{i}=kerd\alpha_{i}\cap ker\alpha_{1}\cap
ker\alpha_{2},\ i=1,2$

and we can write

$T\mathcal{F}_{i}=T\mathcal{G}_{i}\oplus\mathbb{R}Z_{1}.$

Therefore we get $TM=T\mathcal{G}_{1}\oplus
T\mathcal{G}_{2}\oplus\mathbb{R}Z_{1}\oplus\mathbb{R}Z_{2}$. The horizontal
subbundle can be written as $\mathcal{H}=T\mathcal{G}_{1}\oplus
T\mathcal{G}_{2}$. Also, we write
$\mathcal{V}=\mathbb{R}Z_{1}\oplus\mathbb{R}Z_{2}$, we call $\mathcal{V}$ is
vertical subbundle of $TM.$

Let $X$ be an arbitrary vector field on $M$. We can write
$X=X^{\mathcal{H}}+X^{\mathcal{V}}$, where $X^{\mathcal{H}},X^{\mathcal{V}}$
horizontal and vertical component of $X$ respectively. We call a vector field
$X$ as horizontal vector field if $X\in\Gamma(\mathcal{H})$, and vertical
vector field if $X\in\Gamma(\mathcal{V})$.

An almost contact structure is related to contact form and it is defined with
tensorial viewpoint. Many geometric properties of contact manifolds examined
by this contact structure. Similarly an almost contact pair structure has been
defined as follow by Bande and Hadjar[3].

######  Definition 2.2.

An almost contact pair structure on a $(m=2p+2q+2)-$dimensional differentiable
manifold $M$ is a triple $\alpha_{1},\alpha_{2},\phi$, where
$(\alpha_{1},\alpha_{2})$ is a contact pair and $\phi$ is a $(1,1)$ tensor
field such that:

$\displaystyle\phi^{2}=-I+\alpha_{1}\otimes Z_{1}+\alpha_{2}\otimes Z_{2},\ \
\phi Z_{1}=\phi Z_{2}=0.$ (1)

The rank of $\phi$ is $(2p+2q)$ and $\alpha_{i}(\phi)=0$ for $i=1,2$ [3].

The endomorphism $\phi$ is said to be decomposable, i.e
$\phi=\phi_{1}+\phi_{2}$, if $T\mathcal{F}_{i}$ is invariant under $\phi$. If
$\phi$ is decomposable then $(\alpha_{i},Z_{i},\phi)$ induce an almost contact
structure on $\mathcal{F}_{j}$ for $i\neq j$ [2]. The decomposability of
$\phi$ do not satisfy for every almost contact pair structure. An example was
given in [2], has an almost contact pair structure but $\phi$ is not
decomposable. In this study we assume that $\phi$ is decomposable.

######  Definition 2.3.

Let $(\alpha_{1},\alpha_{2},Z_{1},Z_{2},\phi)$ be an almost contact pair
structure on a manifold $M$. A Riemannian metric $g$ is called

  1. $\bullet$

compatible if $g(\phi X_{1},\phi
X_{2})=g(X_{1},X_{2})-\alpha_{1}(X_{1})\alpha_{1}(X_{2})-\alpha_{2}(X_{1})\alpha_{2}(X_{2})$
for all $X_{1},X_{2}\in TM$,

  2. $\bullet$

associated if $g(X_{1},\phi X_{2})=(d\alpha_{1}+d\alpha_{2})(X_{1},X_{2})$ and
$g(X_{1},Z_{i})=\alpha_{i}(X_{1}),$ for $i=1,2$ and for all
$X_{1},X_{2}\in\Gamma(TM)$

$4-$tuple $(\alpha_{1},\alpha_{2},\phi,g)$ is called a metric almost contact
pair on a manifold $M$ and g is an associated metric with respect to contact
pair structure $(\alpha_{1},\alpha_{2},\phi)$. We recall
$(M,\phi,Z_{1},Z_{2},\alpha_{1},\alpha_{2},g)$ is a metric almost contact pair
manifold.

We have following properties for a metric almost contact pair manifold $M$
[2]:

$\displaystyle g(Z_{i},X)=\alpha_{i}(X),\ \ g(Z_{i},Z_{j})=\delta_{ij}$ (2)
$\displaystyle\nabla_{Z_{i}}Z_{j}=0,\ \nabla_{Z_{i}}\phi=0$ (3)

and for every $X$ tangent to $\mathcal{F}_{i}\,\ i=1,2$ we have

$\nabla_{X}Z_{1}=-\phi_{1}X,\ \ \nabla_{X}Z_{2}=-\phi_{2}X$

where $\phi=\phi_{1}+\phi_{2}$. Normality of metric almost contact pair
manifold is given by Bande and Hadjar [4]. For a metric almost contact pair
manifold, we have two almost complex structure:

$J=\phi-\alpha_{2}\otimes Z_{1}+\alpha_{1}\otimes Z_{2},\ \
T=\phi+\alpha_{2}\otimes Z_{1}-\alpha_{1}\otimes Z_{2}.$

######  Definition 2.4.

A metric almost contact pair manifold is said to be normal if $J$ and $T$ are
integrable [4].

######  Theorem 2.1.

Let $(M,\phi,Z_{1},Z_{2},\alpha_{1},\alpha_{2},g)$ be a normal metric contact
pair manifold then we have

$g((\nabla_{X_{1}}\phi)X_{2},X_{3})=\sum_{i=1}^{2}(d\alpha_{i}(\phi
X_{2},X_{1})\alpha_{i}(X_{3})-d\alpha_{i}(\phi X_{3},X_{1})\alpha_{i}(X_{2}))$
(4)

where $X_{1},X_{2},X_{3}$ are arbitrary vector fields on $M$ [3].

We use the following statements for the Riemann curvature;

$\displaystyle
R(X_{1},X_{2})X_{3}=\nabla_{X_{1}}\nabla_{X_{2}}X_{3}-\nabla_{X_{2}}\nabla_{X_{1}}X_{3}-\nabla_{[X_{1},X_{2}]}X_{3},$
$\displaystyle R(X_{1},X_{2},X_{3},X_{4})=g(R(X_{1},X_{2})X_{3},X_{4}).$

for all $X_{1},X_{2},X_{3},X_{4}\in\Gamma(TM)$. Curvature properties of normal
metric contact pair manifolds were given in [6]. For
$X_{1},X_{2},X_{3}\in\Gamma(M)$ and $Z=Z_{1}+Z_{2}$ Reeb vector field, we have

$\displaystyle R(X_{1},Z)X_{2}$ $\displaystyle=$ $\displaystyle-g(\phi
X_{1},\phi X_{2})Z,$ (5) $\displaystyle R(X_{1},X_{2},Z,X_{3})$
$\displaystyle=$ $\displaystyle d\alpha_{1}(\phi
X_{3},X_{1})\alpha_{1}(X_{2})+d\alpha_{2}(\phi X_{3},X_{1})\alpha_{2}(X_{2})$
$\displaystyle-d\alpha_{1}(\phi X_{3},X_{2})\alpha_{1}(X_{1})-d\alpha_{2}(\phi
X_{3},X_{2})\alpha_{2}(X_{1}),$ $\displaystyle R(X_{1},Z)Z$ $\displaystyle=$
$\displaystyle-\phi^{2}X_{1}.$ (7)

Let take an orthonormal basis of $M$ by

$S=\\{e_{1},e_{2},...,e_{p},\phi e_{1},\phi e_{2},...,\phi
e_{p},e_{p+1},e_{p+2},...,e_{p+q},\phi e_{p+1},\phi e_{p+2},...,\phi
e_{p+q},Z_{1},Z_{2}\\}.$

Then for all $X_{1}\in\Gamma(TM)$ we get the Ricci curvature of $M$ as

$Ric(X_{1},Z)=\sum_{i=1}^{2p+2q}{d\alpha_{1}(\phi
E_{i},E_{i})\alpha_{1}(X)+d\alpha_{2}(\phi E_{i},E_{i})\alpha_{2}(X)}.$

where $E_{i}\in S,\ \ 1\leq i\leq 2p+2q+2$. So, we obtain

$\displaystyle Ric(X_{1},Z)=0,\ \ \text{for }X_{1}\in\Gamma(\mathcal{H}),$ (8)
$\displaystyle Ric(Z,Z)=2p+2q.$ (9) $\displaystyle Ric(Z_{1},Z_{1})=2p,\
Ric(Z_{2},Z_{2})=2q,\ Ric(Z_{1},Z_{2})=0.$ (10)

##  3 Existence and Characterization of Generalized Quasi-Einstein Normal
Metric Contact Pairs

######  Definition 3.1.

Let $M$ be a normal metric contact pair manifold. $M$ is called generalized
quasi-Einstein normal metric contact pair manifold if the Ricci curvature of
$M$ has following form;

$Ric(X_{1},X_{2})=\lambda
g(X_{1},X_{2})+\beta\alpha_{1}(X_{1})\alpha_{1}(X_{2})+\mu\alpha_{2}(X_{1})\alpha_{2}(X_{2})$

for functions $\lambda,\beta,\mu$ on $M$ and all $X_{1},X_{2}\in\Gamma(TM)$.

If we set $X_{1}=X_{2}=Z_{1}$ and $X_{1}=X_{2}=Z_{2}$, respectively, we obtain
$\beta=2p-\lambda$ and $\mu=2q-\lambda$. Thus the Ricci curvature of
generalized quasi-Einstein normal metric contact pair manifold is given by

$Ric(X_{1},X_{2})=\lambda
g(X_{1},X_{2})+(2p-\lambda)\alpha_{1}(X_{1})\alpha_{1}(X_{2})+(2q-\lambda)\alpha_{2}(X_{1})\alpha_{2}(X_{2})$
(11)

for all $X_{1},X_{2}\in\Gamma(TM)$ and the scalar curvature is

$scal=2(\lambda+1)(p+q).$ (12)

Let $X$ be an arbitrary vector field on $M$. We can write
$X=X^{\mathcal{H}}+X^{\mathcal{V}}$. Since the Ricci curvature is a linear
tensor we have

$Ric(X_{1},X_{2})=Ric(X^{\mathcal{H}}_{1},X^{\mathcal{H}}_{2})+Ric(X^{\mathcal{V}}_{1},X^{\mathcal{V}}_{2}).$

With take consider the decomposition of tangent bundle mentioned above (see
[32] for details), we get

$Ric(X^{\mathcal{V}}_{1},X^{\mathcal{V}}_{2})=2p\alpha_{1}(X_{1})\alpha_{1}(X_{2})+2q\alpha_{2}(X_{1})\alpha_{2}(X_{2}).$

Thus, we reach following useful result.

######  Proposition 3.1.

A normal metric contact pair manifold is generalized quasi-Einstein manifold
if and only if the horizontal bundle is Einstein, that is for a function
$\lambda$ on M, we have $Ric(X^{\mathcal{H}}_{1},X^{\mathcal{H}}_{2})=\lambda
g(X^{\mathcal{H}}_{1},X^{\mathcal{H}}_{2})$.

In [16] De and Ghosh gave a theorem for the existence of a generalized quasi-
Einstein Riemannian manifold.

######  Theorem 3.1.

If the Ricci tensor $Ric$ of a Riemannian manifold satisfies the relation

$\displaystyle
Ric(X_{2},X_{3})Ric(X_{1},X_{4})-Ric(X_{1},X_{3})Ric(X_{2},X_{4})$
$\displaystyle=$ $\displaystyle\gamma[g(X_{2},X_{3})g(X_{1},X_{4})$
$\displaystyle-g(X_{1},X_{3})g(X_{2},X_{4})]$

where $\gamma$ is a non-zero scalar, then the manifold is a generalized quasi
Einstein manifold [16].

Assume that (3.1) is satisfied on a normal metric contact pair manifold $M$.
By setting $X_{1}=X_{4}=Z$ and $X_{2},X_{3}\in\Gamma(\mathcal{H})$, and from
(8) we have

$Ric(X_{2},X_{3})=\frac{2\gamma}{m-2}g(X_{2},X_{3}).$

Thus, with consider Proposition 3.1 $M$ is a generalized quasi-Einstein
manifold. Using (12), we get $\gamma=\frac{1}{2}(scal-m+2)$ and finally we
state;

######  Corollary 3.1.

Let $M$ be normal metric contact pair manifold with scalar curvature $scal\neq
m-2$. If we have the relation

$\displaystyle
Ric(X_{2},X_{3})Ric(X_{1},X_{4})-Ric(X_{1},X_{3})Ric(X_{2},X_{4})$
$\displaystyle=$
$\displaystyle\frac{1}{2}(scal-m+2)[g(X_{2},X_{3})g(X_{1},X_{4})$
$\displaystyle-g(X_{1},X_{3})g(X_{2},X_{4})]\ $

on $M$ for all $X_{1},X_{2},X_{3},X_{4}\in\Gamma(TM)$, then $M$ is a
generalized quasi-Einstein manifold.

The sectional curvature of a Riemann manifold gives us many important
geometric properties. The characterizations of Einstein [14, 20], quasi-
Einstein [7] and generalized quasi-Einstein [29, 21] manifolds have been
obtained by using the sectional curvature of subspaces of tangent bundle. We
take into consideration Theorem 2.2. in [29], and thus we obtain following
theorem. Firstly we should to give some fundamental facts.

Let $\pi$ be a plane section in $T_{Q}M$ for any $Q\in M$. The sectional
curvature of $\pi$ is given as $Sec(\pi)=Sec(u\wedge v)$, where $u,v$
orthonormal vector fields . For any $(p+q)$-dimensional subspace
$\mathcal{L}\subset T_{Q}M,\ 2\leq p+q\leq m$, its scalar curvature
$scal(\mathcal{L})$ is denoted by

$scal(\mathcal{L})=\sum_{1\leq i,j\leq p+q}Sec(E_{i}\wedge E_{j})$ (14)

where ${E_{1},...,E_{n}}$ is any orthonormal basis of $\mathcal{L}$ [13]. When
$\mathcal{L}=T_{Q}M$, the scalar curvature is just the scalar curvature
$scal(Q)$ of $M$ at $Q\in M$.

######  Theorem 3.2.

An $(m=2p+2q+2)-$dimensional normal metric contact pair manifold is a
generalized quasi-Einstein manifold if and only if there exist a function
$\lambda$ on $M$ satisfying

$\displaystyle scal(P)+p+q-\lambda$ $\displaystyle=$
$\displaystyle\sec(P^{\bot}),\ \ Z_{1},Z_{2}\in T_{Q}P^{\bot}$ $\displaystyle
scal(N)+p+q$ $\displaystyle=$ $\displaystyle\sec(N^{\bot}),\ \ Z_{1},Z_{2}\in
T_{Q}N^{\bot}$ $\displaystyle scal(R)+q-p$ $\displaystyle=$
$\displaystyle\sec(P^{\bot}),\ \ Z_{1}\in T_{Q}R,Z_{2}\in T_{Q}R^{\bot}$

where $(p+q+1)-$plane sections $P,R$ and $(p+q)-$plane section $N$;
$P^{\bot},N^{\bot}$ and $R^{\bot}$ denote the orthogonal complements of $P,N$
and $R$ in $T_{Q}M$, respectively.

##  4 Generalized Quasi-constant Curvature of Normal Metric Contact Pair
Manifolds

The notion of quasi-constant curvature was defined by Chen and Yano [12]. De
and Ghosh generalized this notion for a Riemannian manifold. Let $M$ be a
normal metric contact pair manifold. $M$ is called a normal metric contact
pair manifold of generalized quasi-constant curvature if the Riemannian
curvature tensor of $M$ satisfying;

$\displaystyle R(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$ $\displaystyle
A[g(X_{2},X_{3})g(X_{1},X_{4})-g(X_{1},X_{3})g(X_{2},X_{4})]$ $\displaystyle
B[g(X_{1},X_{4})\alpha_{1}(X_{2})\alpha_{1}(X_{3})-g(X_{1},X_{3})\alpha_{1}(X_{2})\alpha_{1}(X_{4})$
$\displaystyle+g(X_{2},X_{3})\alpha_{1}(X_{1})\alpha_{1}(X_{4})-g(X_{2},X_{4})\alpha_{1}(X_{1})\alpha_{1}(X_{3})]$
$\displaystyle
C[g(X_{1},X_{4})\alpha_{2}(X_{2})\alpha_{2}(X_{3})-g(X_{1},X_{3})\alpha_{2}(X_{2})\alpha_{2}(X_{4})$
$\displaystyle+g(X_{2},X_{3})\alpha_{2}(X_{1})\alpha_{2}(X_{4})-g(X_{2},X_{4})\alpha_{2}(X_{1})\alpha_{2}(X_{3})]$

for all $X_{1},X_{2},X_{3},X_{4}\in\Gamma(TM)$, where $A,B$ and $C$ are scalar
functions.

######  Proposition 4.1.

Let $M$ be a normal metric contact pair manifold of generalized quasi-constant
curvature. Then, we have followings;

  1. $\bullet$

the sectional curvature of horizontal bundle is $A$,

  2. $\bullet$

the sectional curvature of plane section spanned by $X\in\Gamma(\mathcal{H})$
and $Z$ is $2A+B+C$,

  3. $\bullet$

the sectional curvature of plane section spanned by $X\in\Gamma(\mathcal{H})$
and $Z_{1},Z_{2}$ is $A+B$ and $A+C$, respectively.

###### Proof.

Let take $X_{1}=X_{4}=X,X_{2}=X_{4}=X^{\prime}$, where $X,X^{\prime}$ unit and
mutually orthogonal horizontal vector fields. Then from (4) we get

$\displaystyle sec(X,X^{\prime})$ $\displaystyle=$ $\displaystyle
A[g(X^{\prime},X^{\prime})g(X,X)-g(X,X^{\prime})g(X^{\prime},X)]$
$\displaystyle=$ $\displaystyle A$

For $X_{1}=X_{4}=X,X_{2}=X_{4}=Z$ for unit horizontal vector field $X$ we get

$\displaystyle sec(X,Z)$ $\displaystyle=$ $\displaystyle
A[g(Z,Z)g(X,X)-g(X,X^{\prime})g(X^{\prime},X)-g(X,Z)g(Z,X)]$
$\displaystyle+B(g(X,X)\alpha_{1}(Z)\alpha_{1}(Z)+Cg(X,X)\alpha_{2}(Z)\alpha_{2}(Z)$
$\displaystyle=$ $\displaystyle 2A+B+C$

By following same steps one can obtain other results. ∎

From above proposition we get

######  Corollary 4.1.

normal metric contact pair manifold of generalized quasi-constant curvature,
we have

$sec(X,Z)=sec(X,Z_{1})+sec(X,Z_{2})$

for any horizontal and unit vector field $X$.

We can also characterize a generalized quasi-Einstein normal metric contact
pair manifold of generalized quasi-constant curvature. We get following
result.

######  Theorem 4.1.

A normal metric contact pair manifold of generalized quasi-constant curvature
is a generalized quasi-Einstein manifold with coefficients
$\lambda_{1}=A(m-1)+B+C,\ \lambda_{2}=B(m-2)$, and $\lambda_{3}=C(m-2)$.

###### Proof.

Let $M$ be normal metric contact pair manifold of generalized quasi-constant
curvature. Take an orthonormal basis of $M$ as

$\displaystyle
S=\\{e_{1},e_{2},...,e_{p},\phi_{1}e_{1},\phi_{1}e_{2},...,\phi_{1}e_{p},e_{p+1},e_{p+2},...,e_{p+q},\phi_{2}e_{p+1},\phi_{2}e_{p+2},...,\phi_{2}e_{p+q},Z_{1},Z_{2}\\}.$

By taking sum of (4) from $i=1$ to $i=2p+2q+2$ for $X_{2}=X_{3}=E_{i}\in S$,
we obtain

$\displaystyle\sum_{i=1}^{2p+2q+2}R(X_{1},E_{i},E_{i},X_{4})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2p+2q+2}\\{A[g(E_{i},E_{i})g(X_{1},X_{4})-g(X_{1},E_{i})g(E_{i},X_{4})]$
$\displaystyle
B[g(X_{1},X_{4})\alpha_{1}(E_{i})\alpha_{1}(E_{i})-g(X_{1},E_{i})\alpha_{1}(E_{i})\alpha_{1}(X_{4})$
$\displaystyle+g(E_{i},E_{i})\alpha_{1}(X_{1})\alpha_{1}(X_{4})-g(E_{i},X_{4})\alpha_{1}(X_{1})\alpha_{1}(E_{i})]$
$\displaystyle
C[g(X_{1},X_{4})\alpha_{2}(E_{i})\alpha_{2}(E_{i})-g(X_{1},E_{i})\alpha_{2}(E_{i})\alpha_{2}(X_{4})$
$\displaystyle+g(E_{i},E_{i})\alpha_{2}(X_{1})\alpha_{2}(X_{4})-g(E_{i},X_{4})\alpha_{2}(X_{1})\alpha_{2}(E_{i})]\\}.$

For $1\leq i\leq 2p+2q$ since $\alpha_{j}(E_{i})=0$, $j=1,2$ and
$\sum_{i=1}^{2p+q+2}g(X_{1},E_{i})g(E_{i},X_{2})=g(X_{1},X_{2})$ we get

$\displaystyle Ric(X_{1},X_{4})$ $\displaystyle=$
$\displaystyle[A(m-1)+B+C]g(X_{1},X_{4})+B(m-2)\alpha_{1}(X_{1})\alpha_{1}(X_{4})$
$\displaystyle+C(m-2)\alpha_{2}(X_{1})\alpha_{2}(X_{4})$

which completes the proof. ∎

##  5 Normal Metric Contact Pair Manifold Satisfying Cyclic Ricci Tensor

A Riemann manifold satisfies cyclic Ricci tensor if we have

$(\nabla_{X_{1}}Ric)(X_{2},X_{3})+(\nabla_{X_{2}}Ric)(X_{3},X_{1})+(\nabla_{X_{3}}Ric)(X_{1},X_{2})=0$

for all $X_{1},X_{2},X_{3}\in\Gamma(TM)$. De and Mallick proved that a
generalized quasi-Einstein Riemann manifold satisfies cyclic Ricci tensor if
generators of the manifolds are Killing [18].

The characteristic vector fields of a normal metric contact pair manifold
$Z_{1},Z_{2}$ are Killing vector fields [5]. Thus, by easy computations, we
get

$\displaystyle(\nabla_{X_{1}}\alpha_{1})X_{2}+(\nabla_{X_{2}}\alpha_{1})X_{1}=0,\
\ (\nabla_{X_{1}}\alpha_{2})X_{2}+(\nabla_{X_{2}}\alpha_{2})X_{1}=0.$ (16)

Let $M$ be a generalized quasi-Einstein normal metric contact pair manifold.
It is well known that

$(\nabla_{X_{1}}Ric)(X_{2},X_{3})=\nabla_{X_{1}}Ric(X_{2},X_{3})-Ric(\nabla_{X_{1}}X_{2},X_{3})-Ric(X_{2},\nabla_{X_{1}}X_{3}).$

Then, from (11) we obtain

$\displaystyle(\nabla_{X_{1}}Ric)(X_{2},X_{3})$ $\displaystyle=$
$\displaystyle X_{1}[\lambda]g(\phi X_{2},\phi X_{3})$
$\displaystyle+(2p-\lambda)[\left((\nabla_{X_{1}}\alpha_{1})X_{3}\right)\alpha_{1}(X_{3})+\left((\nabla_{X_{1}}\alpha_{1})X_{2}\right)\alpha_{1}(X_{2})]$
$\displaystyle(2q-\lambda)[\left((\nabla_{X_{1}}\alpha_{1})X_{3}\right)\alpha_{2}(X_{2})+\left((\nabla_{X_{1}}\alpha_{1})X_{2}\right)\alpha_{2}(X_{3})]$

where $X_{1}[\lambda]$ is the derivation of $\lambda$ in the direction of
$X_{1}$. Thus from (16) we obtain

$\displaystyle(\nabla_{X_{1}}Ric)(X_{2},X_{3})+(\nabla_{X_{2}}Ric)(X_{3},X_{1})+(\nabla_{X_{3}}Ric)(X_{1},X_{2})$
$\displaystyle=X_{1}[\lambda]g(\phi X_{2},\phi X_{3})+X_{2}[\lambda]g(\phi
X_{3},\phi X_{1})+X_{3}[\lambda]g(\phi X_{1},\phi X_{2}).$

As a consequence , we can state following theorem.

######  Theorem 5.1.

Let $M$ be generalized quasi-Einstein normal metric contact pair manifold. If
$\lambda$ is constant then $M$ satisfies cyclic Ricci tensor.

A Riemann manifold satisfies Codazzi type of Ricci tensor if

$(\nabla_{X_{1}}Ric)(X_{2},X_{3})-(\nabla_{X_{2}}Ric)(X_{1},X_{3})=0$

for all $X_{1}\,X_{2}$ vector fields on $M$. In [18] it has been proven that
if a generalized quasi-Einstein Riemann manifold satisfies Codazzi type of
Ricci tensor, then the associated 1-forms are closed.

Suppose that Ricci tensor $Ric$ of a normal metric contact pair manifold $M$
is Codazzi type. Then, from (16) and (5) we obtain

$\displaystyle(2p-\lambda)[((\nabla_{X_{1}}\alpha_{1})X_{2}-(\nabla_{X_{2}}\alpha_{1})X_{1})\alpha_{1}(X_{3})$
$\displaystyle+((\nabla_{X_{1}}\alpha_{1})X_{3}\alpha_{1}(X_{2})+(\nabla_{X_{2}}\alpha_{1})X_{3}\alpha_{1}(X_{1}))]$
$\displaystyle+(2q-\lambda)[((\nabla_{X_{1}}\alpha_{1})X_{2}-(\nabla_{X_{2}}\alpha_{1})X_{1})\alpha_{2}(X_{3})$
$\displaystyle+((\nabla_{X_{1}}\alpha_{1})X_{3}\alpha_{2}(X_{2})+(\nabla_{X_{2}}\alpha_{1})X_{3}\alpha_{2}(X_{1}))]=0.$

Let take $X_{3}=Z_{1}$, then we get

$(2p-\lambda)((\nabla_{X_{1}}\alpha_{1})X_{2}-(\nabla_{X_{2}}\alpha_{1})X_{1})=0$

which implies $\lambda=2p$ or
$(\nabla_{X_{1}}\alpha_{1})X_{2}-(\nabla_{X_{2}}\alpha_{1})X_{1}=0$. If
$\lambda=2p$ the manifold is not generalized quasi-Einstein, so this case is
not possible. In the other case we obtain

$0=(\nabla_{X_{1}}\alpha_{1})X_{2}-(\nabla_{X_{2}}\alpha_{1})X_{1}=d\alpha_{1}(X_{1},X_{2})=0$

and so $\alpha_{1}$ is closed. Similarly by choosing $X_{3}=Z_{2}$ we obtain
$\alpha_{2}$ is closed. As we know contact pairs $(\alpha_{1},\alpha_{2})$ are
not closed. So our assumption is not valid. Finally we conclude that

######  Theorem 5.2.

A generalized quasi-Einstein normal metric contact pair manifold do not
satisfy Codazzi type of Ricci tensor.

##  6 Normal Metric Contact Pair Manifold Satisfying Certain curvature
Conditions

Curvature tensors give us many geometric properties of contact manifolds. Some
properties of normal metric contact pair manifold satisfying certain
conditions of curvature tensors were given in [6, 32]. In this section we
examine the $\mathcal{M}-$projective curvature tensor $\mathcal{W}$, conformal
curvature tensor $\mathcal{C}$ and concircular curvature tensor $\mathcal{Z}$
on a normal metric contact pair manifold.

Let $M$ be an $(m=2p+2q+2)-$dimensional normal metric contact pair manifold.
$\mathcal{M}-$projective curvature tensor of $M$ is given by ,

$\displaystyle\mathcal{W}(X_{1},X_{2})X_{3}$ $\displaystyle=$ $\displaystyle
R(X_{1},X_{2})X_{3}-\frac{1}{2(m-1)}[Ric(X_{2},X_{3})X_{1}$ $\displaystyle-
Ric(X_{1},X_{3})X_{2}+g(X_{2},X_{3})QX_{1}-g(X_{1},X_{3})QX_{2}]$

for $X_{1},X_{2},X_{3}\in\Gamma(TM)$, where $Q$ is Ricci operator is given by
$Ric(X_{1},X_{2})=g(QX_{1},X_{2})$ [26]. $\mathcal{M}-$projective curvature
tensor on manifolds with different structures studied by many authors [1, 28,
35].

From (6), we have

$\displaystyle\mathcal{W}(X_{1},Z)Z$ $\displaystyle=$
$\displaystyle\frac{m}{2m-1}X_{1}-\frac{1}{m-1}QX_{1}$ (19)
$\displaystyle\mathcal{W}(X_{1},X_{2})Z$ $\displaystyle=$ $\displaystyle
R(X_{1},X_{2})Z$ (20) $\displaystyle\mathcal{W}(X_{1},Z)X_{2}$
$\displaystyle=$
$\displaystyle\left[\frac{(2m-1)(m-2)}{2(m-1)}g(X_{1},X_{2})+\frac{1}{2(m-1)}Ric(X_{1},X_{2})\right]Z$
(21)

for $X_{1},X_{2},X_{3}\in\Gamma(\mathcal{H})$. Also, since
$Ric(X_{1},X_{2})=g(QX_{1},X_{2})$, where $Q$ is the Ricci operator, we have

$\displaystyle\mathcal{W}(X_{1},X_{2},X_{3},X_{4})$
$\displaystyle=R(X_{1},X_{2},X_{3},X_{4})-\frac{1}{2(m-1)}[Ric(X_{2},X_{3})g(X_{1},X_{4})$
$\displaystyle-Ric(X_{1},X_{3})g(X_{2},X_{4})+g(X_{2},X_{3})Ric(X_{1},X_{4})$
$\displaystyle-g(X_{1},X_{3})Ric(X_{2},X_{4})].$

for all $X_{1},X_{2},X_{3}\in\Gamma(M)$. $M$ is called
$\mathcal{M}-$projective flat if $\mathcal{W}$ vanishes identically on $M$.

######  Theorem 6.1.

A generalized quasi-Einstein normal metric contact pair manifold is
$\mathcal{M}-$projective flat if and only if it is of generalized quasi-
constant curvature.

###### Proof.

Suppose that $M$ is a generalized quasi-Einstein manifold. Then, from (11) and
(6) we have

$\displaystyle\mathcal{W}(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$
$\displaystyle R(X_{1},X_{2},X_{3},X_{4})$
$\displaystyle-\frac{\lambda}{m-1}[g(X_{2},X_{3})g(X_{1},X_{4})-g(X_{1},X_{3})g(X_{2},X_{4})]$
$\displaystyle-\frac{2p-\lambda}{2(m-1)}[g(X_{1},X_{4})\alpha_{1}(X_{2})\alpha_{1}(X_{3})-g(X_{1},X_{3})\alpha_{1}(X_{2})\alpha_{1}(X_{4})$
$\displaystyle+g(X_{2},X_{3})\alpha_{1}(X_{1})\alpha_{1}(X_{4})-g(X_{2},X_{4})\alpha_{1}(X_{1})\alpha_{1}(X_{3})]$
$\displaystyle-\frac{2q-\lambda}{2(m-1)}[g(X_{1},X_{4})\alpha_{2}(X_{2})\alpha_{2}(X_{3})-g(X_{1},X_{3})\alpha_{2}(X_{2})\alpha_{2}(X_{4})$
$\displaystyle+g(X_{2},X_{3})\alpha_{2}(X_{1})\alpha_{2}(X_{4})-g(X_{2},X_{4})\alpha_{2}(X_{1})\alpha_{2}(X_{3})].$

Thus it is seen that $M$ is $\mathcal{M}-$projective flat if and only if $M$
is of generalized quasi-constant curvature with coefficients
$A=\frac{\lambda}{m-1},\ B=\frac{2p-\lambda}{2(m-1)}$ and
$C=\frac{2q-\lambda}{2(m-1)}$. ∎

The Riemann manifolds satisfying $R(X_{1},X_{2}).R=0$ are called semi-
symmetric, where $R(X_{1},X_{2})$ acts on $R$ as a derivation. Semi-symmetric
contact manifolds were studied by Perrone [25]. Similarly, if
$\mathcal{W}(X_{1},X_{2}).R=0$ then $M$ is called $\mathcal{M}-$projective
semi-symmetric. $\mathcal{W}(X_{1},X_{2}).R$ is defined as

$\displaystyle(\mathcal{W}(X_{1},X_{2}).R)(X_{3},X_{4})X_{5}$
$\displaystyle=\mathcal{W}(X_{1},X_{2}).R(X_{3},X_{4})X_{5}-R(\mathcal{W}(X_{1},X_{2}),X_{3},X_{4})X_{5}$
$\displaystyle-R(X_{3},\mathcal{W}(X_{1},X_{2})X_{4})X_{5}-R(X_{3},X_{4})\mathcal{W}(X_{1},X_{2})X_{5}$

for all $X_{1},X_{2},X_{3},X_{4},X_{5}\in\Gamma(TM)$. Also, we have

$\displaystyle(\mathcal{W}(X_{1},X_{2}).Ric)(X_{3},X_{4})=-Ric(\mathcal{W}(X_{1},X_{2}),X_{3})-Ric(X_{3},\mathcal{W}(X_{1},X_{2})X_{4}).$
(25)

If $\mathcal{W}(X_{1},X_{2}).Ric=0$ then $M$ is called
$\mathcal{M}-$projective Ricci semi-symmetric.

######  Theorem 6.2.

A normal metric contact pair manifold is $\mathcal{M}-$projective semi-
symmetric if and only if $M$ is generalized quasi-Einstein manifold.

###### Proof.

From (6) and using (19), (20) and (21) we obtain

$\displaystyle(\mathcal{W}(X_{1},Z).R)(X_{3},X_{4})X_{5}$ $\displaystyle=$
$\displaystyle KR(X_{1},X_{3},X_{4},X_{5})Z+LR(X_{3},X_{4},X_{5},QX_{1})Z$
$\displaystyle-(Kg(X_{1},X_{3})+LRic(X_{1},X_{3}))g(X_{4},X_{5})Z$
$\displaystyle+(Kg(X_{1},X_{5})+LRic(X_{1},X_{5}))R(X_{3},X_{4})Z$

where $K=\frac{(2m-1)(m-2)}{2(m-1)}$ and $L=\frac{1}{2(m-1)}$.

Let take $X_{1},X_{3},X_{4},X_{5}$ horizontal vector fields and $X_{4}=Z$,
then from (5), (2), (7), (8) and (9), then, we get

$\displaystyle(\mathcal{W}(X_{1},Z).R)(X_{3},Z)X_{5}=-(Kg(X_{1},X_{5})+LRic(X_{1},X_{5}))X_{3}.$

Thus, we conclude that $(\mathcal{W}(X_{1},Z).R)(X_{3},X_{4})X_{5}=0$ if and
only if horizontal bundle of $M$ is Einstein. Take into consideration of
Proposition 3.1, we obtain

$\displaystyle
Ric(X_{1},X_{5})=-\frac{K}{L}g(X_{1},X_{5})+(2p+\frac{K}{L})\alpha_{1}(X_{1})\alpha_{1}(X_{5})+(2q+\frac{K}{L})\alpha_{2}(X_{1})\alpha_{2}(X_{5})$
(26)

So, the manifold is generalized quasi-Einstein. ∎

######  Theorem 6.3.

An $(m=2p+2q+2)$-dimensional normal metric contact pair manifold admitting an
$\mathcal{M}-$projective curvature tensor and a nonzero Ricci tensor $Ric$
satisfies the equality $\mathcal{W}.Ric=0$ if and only $M$ is generalized
quasi-Einstein manifold.

###### Proof.

For $X_{1},X_{3},X_{4}\in\Gamma(TM)$ from (25) we get

$\displaystyle(\mathcal{W}(X_{1},Z).Ric)(X_{3},X_{4})$ $\displaystyle=$
$\displaystyle-Ric([Kg(X_{1},X_{3})+LRic(X_{1},X_{3})]Z,X_{4})$
$\displaystyle-Ric(X_{3},[Kg(X_{1},X_{4})+LRic(X_{1},X_{4})]Z).$

Let take $X_{1},X_{4}$ horizaontal vector fields and $X_{3}=Z$ from (8), (9),
we obtain

$\displaystyle(\mathcal{W}(X_{1},Z).Ric)(X_{3},X_{4})=-(2p+2q)[Kg(X_{1},X_{4})+LRic(X_{1},X_{4})].$

Therefore, $(\mathcal{W}(X_{1},Z).Ric)(X_{3},X_{4})=0$ if and only if
horizontal bundle is Einstein. From Proposition (3.1) we get (26), which
completes the proof. ∎

Conformal $\mathcal{C}$ and concircular $\mathcal{Z}$ on a
$(m=2p+2q+2)$-dimensional normal contact metric pair manifold are given by:

$\displaystyle\mathcal{C}(X_{1},X_{2})X_{3}$ $\displaystyle=$ $\displaystyle
R(X_{1},X_{2})X_{3}$
$\displaystyle+\frac{scal}{(m-1)(m-2)}\left(g(X_{2},X_{3})X_{1}-g(X_{1},X_{3})X_{2}\right)$
$\displaystyle+\frac{1}{m-2}(g(X_{1},X_{3})QX_{2}-g(X_{2},X_{3})QX_{1}$
$\displaystyle+Ric(X_{1},X_{3})X_{2}-Ric(X_{2},X_{3})X_{1}),$
$\displaystyle\mathcal{Z}(X_{1},X_{2})X_{3}$ $\displaystyle=$ $\displaystyle
R(X_{1},X_{2})X_{3}-\frac{scal}{m(m-1)}[g(X_{2},X_{3})X_{1}-g(X_{1},X_{3})X_{2}]$

where $X_{1},X_{2},X_{3}\in\Gamma(TM)$. Conformal and concircular curvature
tensors on contact manifolds have been studied in [31, 32, 34].

Blair, Bande and Hadjar [6] studied on conformal flatness of normal metric
contact pair manifolds and they proved following theorem.

######  Theorem 6.4.

A conformal flat normal metric contact pair manifold is locally isometric to
the Hopf manifold $S^{2q+1}(1)\times S^{1}$ [6].

Thus we get following results, for a generalized quasi-Einstein normal metric
contact pair manifold.

######  Theorem 6.5.

Let $M$ be a generalized quasi-Einstein normal metric contact pair manifold.
If $M$ is of generalized quasi-constant curvature with coefficients
$A=-\frac{\lambda m-m+2}{(m-1)(m-2)},\ B=\frac{2p-\lambda}{m-2}$ and
$C=\frac{2q-\lambda}{m-2}$, then it is locally isometric to the Hopf manifold
$S^{2q+1}(1)\times S^{1}$.

###### Proof.

Let $M$ be generalized quasi-Einstein normal metric contact pair manifold.
Then, we have

$\displaystyle\mathcal{C}(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$
$\displaystyle R(X_{1},X_{2},X_{3},X_{4})$
$\displaystyle+\frac{scal}{(m-1)(m-2)}[g(X_{2},X_{3})g(X_{1},X_{4})-g(X_{1},X_{3})g(X_{2},X_{4})]$
$\displaystyle-\frac{2\lambda}{m-2}[g(X_{2},X_{3})g(X_{1},X_{4})-g(X_{1},X_{3})g(X_{2},X_{4})]$
$\displaystyle-\frac{2p-\lambda}{m-2}[g(X_{1},X_{4})\alpha_{1}(X_{2})\alpha_{1}(X_{3})-g(X_{1},X_{3})\alpha_{1}(X_{2})\alpha_{1}(X_{4})$
$\displaystyle+g(X_{2},X_{3})\alpha_{1}(X_{1})\alpha_{1}(X_{4})-g(X_{2},X_{4})\alpha_{1}(X_{1})\alpha_{1}(X_{3})]$
$\displaystyle-\frac{2q-\lambda}{m-2}[g(X_{1},X_{4})\alpha_{2}(X_{2})\alpha_{2}(X_{3})-g(X_{1},X_{3})\alpha_{2}(X_{2})\alpha_{2}(X_{4})$
$\displaystyle-g(X_{2},X_{4})\alpha_{2}(X_{1})\alpha_{2}(X_{3})].$

Suppose that, $M$ is of generalized quasi-constant curvature with coefficients
$A=-\frac{\lambda m-m+2}{(m-1)(m-2)},\ B=\frac{2p-\lambda}{m-2}$ and
$C=\frac{2q-\lambda}{m-2}$ . Then, we get $\mathcal{C}=0$ which means that $M$
is conformal flat. Thus, from Theorem 6.4 $M$ is locally isometric to the Hopf
manifold $S^{2q+1}(1)\times S^{1}$. ∎

By using the definition of $\mathcal{M}-$projective curvature tensor and
conformal curvature tensor, we have

$\displaystyle\mathcal{C}(X_{1},X_{2})X_{3}$ $\displaystyle=$
$\displaystyle\frac{2(m-1)}{m-2}\mathcal{W}(X_{1},X_{2})X_{3}-\frac{m}{m-2}R(X_{1},X_{2},)X_{3}$
$\displaystyle+\frac{scal}{(m-1)(m-2)}[g(X_{2},X_{3})X_{1}-g(X_{1},X_{3})X_{2}].$

If $M$ is a $\mathcal{M}-$projective flat normal metric contact pair manifold.
, then, from (6) , $M$ is conformal flat if and only if

$\displaystyle
R(X_{1},X_{2},)X_{3}=\frac{scal}{m(m-1)}[g(X_{2},X_{3})X_{1}-g(X_{1},X_{3})X_{2}],$

which means $M$ is concircular flat. Finally, we conclude that

######  Theorem 6.6.

Let $M$ be $\mathcal{M}-$projective flat normal metric contact pair manifold.
If $M$ is also concircular flat then it is locally isometric to Hopf manifold
$S^{2q+1}(1)\times S^{1}$.

## References

  * [1] Ayar, G., and Chaubey, S. K. M-projective curvature tensor over cosymplectic manifolds. Differential Geometry-Dynamical Systems 2019; 21: 23-33.
  * [2] Bande, G., Hadjar, A. Contact pairs. Tohoku Mathematical Journal 2005; Second Series:57(2): 247-260. doi:10.2748/tmj/1119888338
  * [3] Bande, G., Hadjar, A. Contact pair structures and associated metrics. Differential Geometry - Proceedings of the $VIII$ International Colloquium 2009; (pp. 266-275).
  * [4] Bande, G., Hadjar, A. On normal contact pairs. International Journal of Mathematics 2010; 21(06): 737-754. doi: 10.1142/S0129167X10006197
  * [5] Bande, G., and Blair, D. E. Symmetry in the geometry of metric contact pairs. Mathematische Nachrichten 2013; 286(17-18): 1701-1709. doi:10.1002/mana.201100300
  * [6] Bande, G., Blair, D. E., Hadjar, A. Bochner and conformal flatness of normal metric contact pairs. Annals of Global Analysis and Geometry 2015; 48(1): 47-56. doi:10.1007/s10455-015-9456-2
  * [7] Bejan, C. L. Characterization of quasi-Einstein manifolds. An. Stiint. Univ. Al. I. Cuza Iasi. Mat.(NS) 2007; 53: 67-72.
  * [8] Blair, D. E., Ludden, G. D., and Yano, K. Geometry of complex manifolds similar to the Calabi-Eckmann manifolds. Journal of Differential Geometry 1974; 9(2),: 263-274.
  * [9] Catino, G. Generalized quasi-Einstein manifolds with harmonic Weyl tensor. Mathematische Zeitschrift 2012; 271(3-4):. 751-756. doi: 10.1007/s00209-011-0888-5
  * [10] Chaki, M. C. On quasi Einstein manifolds. Publicationes Mathematicae Debrecen, 2000; 57:297-306.
  * [11] Chaki, M.C.: On Generalized quasi-Einstein manifold. Publ. Math. Debrecen 2001; 58: 638–691
  * [12] Chen, B. Y., and Yano, K. Hypersurfaces of a conformally flat space. Tensor (NS) 1972; 26: 318-322.
  * [13] Chen, B. Y. Some new obstructions to minimal and Lagrangian isometric immersions. Japanese journal of mathematics. 2000; New series: 26(1): 105-127.
  * [14] Chen, B. Y., Dillen, F., Verstraelen, L., and Vrancken, L. Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces. Proceedings of the American Mathematical Society 2000; 128(2): 589-598.
  * [15] De, U., and Ghosh, G. On quasi Einstein manifolds. Periodica Mathematica Hungarica 2004; 48(1-2); 223-231. Doi: 10.1023/B:MAHU.0000038977.94711.ab
  * [16] De, U. C., and Ghosh, G. C. On generalized quasi Einstein manifolds. Kyungpook mathematical journal 2004; 44(4): 607-607.
  * [17] De, U. C., and Mallick, S. On the existence of generalized quasi-Einstein manifolds. Arch. Math.(Brno) 2011; 47(4): 279-291.
  * [18] De, U. C., and Mallick, S. On generalized quasi-Einstein manifolds admitting certain vector fields. Filomat 2015; 29(3): 599-609. doi:10.2298/FIL1503599D
  * [19] De, U. C., and Shenawy, S. Generalized quasi-Einstein GRW space-times. International Journal of Geometric Methods in Modern Physics 2019; 16(08): 1950124. doi: 10.1142/S021988781950124X
  * [20] Dumitru, D. On Einstein spaces of odd dimension. Bul. Univ. Transilvania Brasov 2007; 14(49): 95-97.
  * [21] Dumitru, D. A characterization of generalized quasi-Einstein manifolds. Novi Sad J. Math, 2012; 42.1: 89-94.
  * [22] Ghosh, G. C., De, U. C., and Binh, T. Q. Certain curvature restrictions on a quasi Einstein manifold. Publicationes Mathematicae-Debrecen 2006; 69(1-2): 209-217.
  * [23] Güler, S., and Demirbağ, S. A. A study of generalized quasi Einstein spacetimes with applications in general relativity. International Journal of Theoretical Physics 2016; 55(1): 548-562. doi: 10.1007/s10773-015-2692-1
  * [24] Mallick, S., and De, U. C. On a Class of Generalized quasi-Einstein Manifolds with Applications to Relativity. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 2016; 55(2): 111-127.
  * [25] Perrone, D. Contact Riemannian manifolds satisfying $R(X,\xi)R=0,$ Yokohama Math. J. 1992; 39 : 141–149.
  * [26] Pokhariyal, G. P. and Mishra, R. S. Curvature tensors and their relativistics significance, Yokohama Mathematical Journal 1970; vol. 18: pp. 105–108.
  * [27] Prakasha, D. G., and Venkatesha, H. Some results on generalized quasi-Einstein manifolds. Chinese Journal of Mathematics; 2014: Article ID 563803 doi: 10.1155/2014/563803
  * [28] Prakasha, D. G., Fernandez, L. M., and Mirji, K. The $\mathcal{M}$-projective curvature tensor field on generalized $(\kappa,\mu)$-paracontact metric manifolds. Georgian Mathematical Journal 2020; 27(1): 141-147. doi:10.1515/gmj-2017-0054.
  * [29] Sular, S., and Özgür, C. Characterizations of generalized quasi-Einstein manifolds. Analele Universitatii” Ovidius” Constanta-Seria Matematica 2012; 20(1): 407-416.
  * [30] Tripathi, M. M., and Kim, J. On $N(k)-$quasi Einstein manifolds. Communications-Korean Mathematical Society 2007; 22(3): 411.
  * [31] Ünal, İ., Sarı, R., and Turgut Vanlı, A. Concircular curvature tensor on generalized kenmotsu manifolds. Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi 2018; 99-105.doi: 10.17714/gumusfenbil.437302
  * [32] Ünal, İ. Some Flatness Conditions on Normal Metric Contact Pairs. arXiv preprint 2019, arXiv:1902.05327.
  * [33] Turgut Vanli, A., and Unal, I. On Complex $\eta$-Einstein Normal Complex Contact Metric Manifolds. Communications In Mathematics And Applicationn 2017, 8(3); 301 - 313. doi:10.26713/cma.v8i3.509
  * [34] Turgut Vanli, A., and Unal, I. Conformal, concircular, quasi-conformal and conharmonic flatness on normal complex contact metric manifolds. International Journal of Geometric Methods in Modern Physics 2017; 14(05): 1750067. doi: 10.1142/S0219887817500670
  * [35] Zengin, F. Ö. On m-projectively flat LP-Sasakian manifolds. Ukrainian Mathematical Journal 2014; 65(11): 1725-1732.