#  Robustness-preserving Lifelong Learning via Dataset Condensation

###### Abstract

Lifelong learning (LL) aims to improve a predictive model as the data source
evolves continuously. Most work in this learning paradigm has focused on
resolving the problem of ‘catastrophic forgetting,’ which refers to a
notorious dilemma between improving model accuracy over new data and retaining
accuracy over previous data. Yet, it is also known that machine learning (ML)
models can be vulnerable in the sense that even tiny, adversarial input
perturbations can deceive the models into producing erroneous predictions.
This motivates the research objective of this paper – specification of a new
LL framework that can salvage model robustness (against adversarial attacks)
from catastrophic forgetting. Specifically, we propose a new memory-replay LL
strategy that leverages modern bi-level optimization techniques to determine
the ‘coreset’ of the current data (i.e., a small amount of data to be
memorized) for ease of preserving adversarial robustness over time. We term
the resulting LL framework ‘Data-Efficient Robustness-Preserving LL’ (DERPLL).
The effectiveness of DERPLL is evaluated for class-incremental image
classification using ResNet-18 over the CIFAR-10 dataset. Experimental results
show that DERPLL outperforms the conventional coreset-guided LL baseline and
achieves a substantial improvement in both standard accuracy and robust
accuracy.

Index Terms—  lifelong learning, coreset selection, adversarial robustness,
class-incremental learning

##  1 Introduction

While ML technologies have achieved significant advances for many use cases
[1, 2, 3], the problem of continual learning remains largely open. In
continual, lifelong learning, ML models keep updated as new data come in or as
the prediction task evolves over time. For example, the number of classes
evolves over time in the class-incremental learning (CIL) setting [4, 5, 6],
which is the setting considered in this paper. The so-called catastrophic
forgetting phenomenon [7, 8] occurs when past knowledge is dominated by newly
acquired information, and it is a major obstacle to accurate continual
learning. This paper develops a strategy to mitigate the forgetting effects
and maintain high performance over tasks in the LL setting.

To alleviate catastrophic forgetting, numerous methods have been proposed and
utilized in LL or CIL [9, 10, 11, 12, 6, 13, 14, 15]; these methods
principally rely on two types of techniques, namely, _memory replay_ and
_regularization_. Firstly, memory replay [9, 10, 11, 12, 6] stores a small
dataset – either by extracting a subset of the past training set [9, 11] or by
generating a synthetic summary [10] – to retain useful knowledge for the
previous tasks, and then reuses it to ‘remind’ the model of the old tasks when
the model is being updated for new tasks. Secondly, regularization techniques
[13, 14, 15] have been successful in enforcing the newly updated model to
prevent deviations far from the old model. In practice, relay and
regularization techniques are used in combination.

There has been a recent surge in interest in _adversarial robustness_ of ML
models to overcome their well-known fragility, especially those having deep
architectures [16, 17]; ML models can be deceived by a tiny amount of
(adversarial) perturbations on the input induced by an adversarial attacker.
While the past decade has witnessed significant progress in adversarially
robust ML in the static learning setting [18, 19], adversarial robustness for
LL is largely unexplored. Chen et at. [6] attempted to achieve adversarially
robust CIL by applying adversarial training techniques to the streaming data
setting. However, extra side information (i.e., queried unlabeled data) have
to be assumed for addressing the problem of ‘catastrophic forgetting’ [6].
This may cause a new problem in data efficiency.

In this work, we propose a novel approach to sustain adversarial robustness
for models learned in the CIL setting in a data-efficient manner.
Specifically, our strategy combines a memory-replay method based on coreset
selection and a regularization technique to retain both prediction accuracy
and adversarial robustness. We find that a carefully curated data summary is
substantially more effective than a random subset for maintaining accuracy and
adversarial robustness.

Contributions. We summarize our contributions below.

(1) We identify the outstanding challenges in attaining adversarially robust
CIL.

(2) We propose the DERPLL strategy to address ‘robustness forgetting’ based on
bi-level optimization implementation of coreset selection.

(3) We demonstrate the effectiveness of DERPLL for CIL with deep models
trained on real-world image datasets.

##  2 Background & Related Work

Class-Incremental Learning (CIL). In the LL paradigm, this paper will focus on
CIL, which refers to the setting where a model is learned to solve a time-
evolving task using a stream of data from a non-stationary distribution [4,
20]. CIL has received increasing attention in LL [9, 10, 11, 12, 6, 13, 14,
15], and various approaches have been proposed to alleviate the _catastrophic
forgetting_ phenomenon (i.e., the interference to the previously learned
knowledge caused by acquiring new information) [7, 8]. The CIL methods devised
to mitigate the forgetting effects largely rely on two types of techniques,
namely, _memory replay_ and _regularization_. The _memory-replay_ -type
approaches maintain a small number of data points to ‘memorize’ the old tasks
and replay them during the learning phase for new tasks. To construct a
summarizing dataset, some of these methods directly extract a subset from the
past training dataset [9, 11], while others generate an artificial summary for
the old tasks using generative models [10], unlabeled auxiliary data [12, 6],
or knowledge distillation [15, 6]. Another category of widely used CIL methods
utilizes regularization techniques to prevent the newly trained model from
deviating too far from the previously trained models [13, 14]. Learning
without forgetting (_LwF_), coined by Li and Hoiem [14], is the epitome of
this type of approach, which utilizes ideas from knowledge distillation to
restrain forgetting. LwF and its variants form a popular and successful family
of regularization methods for CIL [14, 15, 6].

Coreset Selection. The coreset selection problem has a long history in
computational geometry [21], and has appeared in various contexts as a tool
for dimensionality reduction [22]. Applications of coreset selection in ML
include $k$-means clustering [23], logistic regression [24], Gaussian mixture
model learning [25], active learning [26], and Bayesian posterior inference
[27] and relational data analysis [28]. Recently, Borsos _et al._ [5] proposed
a coreset-selection-based approach to maintain the high accuracy of the models
in the task of continual learning. They reformulated the coreset selection
problem as a cardinality-constrained bi-level optimization problem and
proposed a greedy approach. However, this greedy method is difficult to scale
due to its large computational overhead.

Adversarial Training (AT). Recognition of the fragility of ML models has
promoted a flurry of research efforts to devise effective defenses against
adversarial attacks [18, 29, 30]. In particular, adversarial training (AT)
[18], which trains a model by minimizing the worst-case (maximum) training
loss, has been recognized as one of the most effective defense methods [31].
Although adversarial robustness has been extensively studied via AT or its
variants [29, 30], these efforts are mostly restricted to robustifying ML
models in the static learning paradigm. By contrast, the adversarial
robustness of LL is little explored, except [6] to the best of our knowledge.
In [6], the problem of _robustness forgetting_ (i.e., catastrophic forgetting
of of adversarial robustness over time-evolving tasks) was identified in CIL.
The authors attempted to tackle this challenge using robustified
regularization techniques, e.g., TRADES [29], with the assistance of auxiliary
unlabeled data. Compared with [6], we revisit the problem of adversarially
robust CIL and propose a data-efficient robustness-preserving CIL framework,
which does not require additional side information to mitigate the
catastrophic forgetting of adversarial robustness.

##  3 Problem: Adversarially Robust CIL

In this section, we formally define the problem setting we consider, and
identify the outstanding challenges in sustaining the adversarial robustness
of models for CIL. To this end, we start from a basic formulation for learning
problems, and gradually introduce additional layers of complexity.

Consider a random vector
$(\mathsf{x},\mathsf{y})\in\mathcal{X}\times\mathcal{Y}$, which is drawn from
an unknown joint distribution $\mathbf{P}$. We refer to
$\mathsf{x},\mathsf{y}$ as input, and response (or, label), respectively; we
commonly have $\mathcal{X}=\mathbb{R}^{d}$; and $\mathcal{Y}=\\{1,\dots,K\\}$
(classification). We have access to a finite-sized sample
$\mathcal{D}=\\{(\mathbf{x}_{i},y_{i}):i=1,\dots,n\\}$ drawn from
$\mathbf{P}$. Let $\mathcal{F}$ denote the set of functions we consider, e.g.,
those implementable by a DNN (deep neural network), parameterized by model
parameters $\bm{\theta}$. The goal of ML is to find a function
$f_{\bm{\theta}}\in\mathcal{F}$ that approximates well the conditional
expectation of $\mathsf{y}$ given $\mathsf{x}$, i.e.,
$f_{\bm{\theta}}(\mathbf{x})\approx\mathbb{E}_{\mathbf{P}}[\mathsf{y}|\mathsf{x}=\mathbf{x}]$,
using the finite dataset $\mathcal{D}$.

Class-incremental Learning (CIL). Observe from above that the unknown
underlying distribution $\mathbf{P}$ along with a dataset $\mathcal{D}$
defines the target ‘task’ of interest. CIL refers to the setting where
$\mathbf{P}$ evolves over time, e.g., the number of class labels increases. We
model this CIL setup using a discrete-time task sequence
$(\mathcal{T}^{(t)}:t=1,2,\dots)$, where the task at time $t$ is defined as a
pair $\mathcal{T}^{(t)}=(\mathcal{D}^{(t)},\mathbf{P}^{(t)})$ of dataset and
(unknown) distribution. When faced with a stream of tasks as above, the goal
of CIL is to learn a sequence of models $(\bm{\theta}^{(t)}:t=1,2,\dots)$ such
that $\bm{\theta}^{(t)}$ solves task $\mathcal{T}^{(t)}$, and at the same
time, retains good performance for all previous tasks
$\mathcal{T}^{(1)},\dots,\mathcal{T}^{(t-1)}$.

Adversarially Robust Learning. The goal of adversarially robust learning is to
acquire a robust model $\bm{\theta}$ against adversarial input perturbations.
For instance, adversarial training (AT) attempts to learn $\bm{\theta}$ by
solving [18]

$\displaystyle\begin{array}[]{l}\displaystyle\min_{\bm{\theta}}\leavevmode\nobreak\
\ell^{\epsilon}_{\mathrm{AT}}(\bm{\theta};\mathcal{D})\mathrel{\mathop{:}}=\mathbb{E}_{(\mathbf{x},y)\in\mathcal{D}}\Big{\\{}\max_{\|\bm{\delta}\|_{\infty}\leq\epsilon}\ell(\bm{\theta};\mathbf{x}+\bm{\delta},y)\Big{\\}},\end{array}$
(AT)

where $\bm{\delta}$ is a variable representing the input perturbation that
falls in the $\ell_{\infty}$-norm ball of radius $\epsilon$, and $\ell$ is a
loss function in use for training. That is, AT ‘robustifies’ the learned model
by modifying the loss used in the standard empirical risk minimization
formulation to reflect the worst-case behavior;
$\ell(\bm{\theta};\mathbf{x},y)\mapsto\max_{\|\bm{\delta}\|_{\infty}\leq\epsilon}\ell(\bm{\theta};\mathbf{x}+\bm{\delta},y)$.

Main Problem: Challenges in Adversarially Robust CIL. Both CIL and adversarial
robustness capture important features in practical ML problems. However, these
two topics have been studied separately in the literature. In this work, we
aim to address adversarial robustness in the CIL setting. Specifically, we
focus on two challenges that arise while we attempt to naïvely extend AT to
CIL:

(C1) Lack of principles for memory replay: While memory replay is a popular
approach for CIL, it is unclear what data should be stored to sustain both
accuracy and robustness.

(C2) Forgetting of robustness: CIL may fail to retain adversarial robustness
over a sequence of time-evolving tasks.

Inspired by these challenges, we ask the question:

(P): How to design a new CIL paradigm that can optimize data efficiency and
model robustness simultaneously?

##  4 Methods

In this section, we describe our DERPLL approach to tackle (P), which combines
a memory-replay technique based on coreset selection and a LwF-type
regularization technique.

###  4.1 Coreset Selection for Memory Replay

Coreset selection aims at ‘compressing’ the dataset
$\mathcal{D}:=\bigcup_{k=1}^{t-1}\mathcal{D}^{(k)}$ into a small summary
$\mathcal{S}^{(t)}$ that contains sufficient information of previous tasks
$\mathcal{T}_{1},\dots,\mathcal{T}_{t-1}$ for training $\bm{\theta}^{(t)}$.
The coreset selection problem can be formulated as the following bi-level
optimization (BLO) problem [5]:

$\displaystyle\begin{array}[]{cl}\displaystyle\operatorname*{\text{minimize}}_{\mathbf{w}\in\Omega}&\displaystyle\frac{1}{N}\sum_{i=1}^{N}\ell\big{(}\bm{\theta}^{*}(\mathbf{w});\mathbf{x}_{i},y_{i}\big{)}\\\
\operatorname*{\text{subject
to}}&\bm{\theta}^{*}(\mathbf{w})=\displaystyle\operatorname*{arg\,min}_{\bm{\theta}}\frac{1}{N}\sum_{i=1}^{N}w_{i}\cdot\ell(\bm{\theta};\mathbf{x}_{i},y_{i}),\end{array}$
(4)

where $\mathbf{w}=[w_{1},\ldots,w_{N}]^{T}\in\\{0,1\\}^{N}$ is the selection
variable,
$\Omega=\\{\mathbf{w}\in\\{0,1\\}^{N}\,|\,\mathbf{1}^{T}\mathbf{w}=n\\}$
encodes the cardinality constraint (i.e., the size of the coreset), and $\ell$
is a model training loss (e.g., cross entropy loss). Note that problem (4)
involves two levels of entangled optimization sub-problems: (i) the lower-
level problem (w.r.t. $\bm{\theta}$) finds the best model given a coreset
given the upper-level variables $\mathbf{w}$; (ii) the upper-level problem
finds the best coreset (i.e., the best $\bf w$) by assessing the performance
of the model $\bm{\theta}^{*}(\mathbf{w})$ over the entire dataset.

Computational Challenge in Solving (4). While (4) provides a concise
formulation, it can be difficult to solve for two reasons: (i) the cardinality
constraint on the Boolean variable $\bf w$; and (ii) the need of hard implicit
gradient computation $\frac{d\bm{\theta}^{*}(\mathbf{w})}{d\mathbf{w}}$ to
obtain a closed-form expression for the upper-level gradient. To circumvent
these computational challenges, Borsos _et al._ [5] developed a greedy
algorithm that constructs a coreset by incrementally adding a training sample
with the maximum influence score (IS) – i.e., the sensitivity of the upper-
level objective with respect to the change in $w_{i}$. While it is possible to
compute the closed form of IS using the influence function theory [32, 33], it
involves the inverse Hessian of the inner objective w.r.t. $\bm{\theta}$. This
makes the use of IS impractical for deep models such as ResNet-18 [34] used in
our experiments.

Our Approaches to Solve (4). We take an alternative approach to numerically
solve (4). To be precise, we adopt two techniques: (i) we relax the Boolean
constraint $\mathbf{w}\in\\{0,1\\}^{N}$ to a more amenable box constraint
$\mathbf{w}\in[0,1]^{N}$, enabling the use of standard BLO solvers; and (ii)
we leverage a straight-through (ST) gradient estimator [35], which retains the
discrete function value evaluation in the forward pass of DL but the continual
derivative evaluation in the backward pass. Concretely, we reformulate (4) as
the following:

$\displaystyle\begin{array}[]{cl}\displaystyle\operatorname*{\text{minimize}}_{\mathbf{w}\in{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Omega^{\prime}}}&\displaystyle
f(\mathbf{w})\mathrel{\mathop{:}}=\frac{1}{N}\sum_{i=1}^{N}\ell\big{(}\bm{\theta}^{*}(\mathbf{w});\mathbf{x}_{i},y_{i}\big{)}\\\
\operatorname*{\text{subject
to}}&\bm{\theta}^{*}(\mathbf{w})=\displaystyle\operatorname*{arg\,min}_{\bm{\theta}}\frac{1}{N}\sum_{i=1}^{N}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{B}_{i}(\mathbf{w})}\cdot\ell(\bm{\theta};\mathbf{x}_{i},y_{i}),\end{array}$
(7)

where
$\Omega^{\prime}=\\{\mathbf{w}\in[0,1]^{N}\,|\,\mathbf{1}^{T}\mathbf{w}=n\\}$,
and $\mathcal{B}_{i}:\mathbb{R}^{N}\to\\{0,1\\}$ is a thresholding function
such that $\mathcal{B}_{i}(\mathbf{w})=1$ if and only if $w_{i}\geq w_{[n]}$
where $w_{[n]}$ is the $n$-th largest element of $\mathbf{w}$.

The reformulated problem (7) bypasses the two challenges in (4). The upper-
level problem is continuous, and the projected gradient descent (PGD) method
is applicable. In addition, the derivative of thresholding functions in the ST
gradient estimator is omitted and thus makes the computation of implicit
gradient $\frac{d\bm{\theta}^{*}(\mathbf{w})}{d\mathbf{w}}$ tractable in the
continuous domain. Thus, we can utilize standard DL toolkits to compute the
implicit gradient, e.g., gradient unrolling [36], which uses automatic
differentiation to compute the implicit gradient evaluated at a finite-step
gradient-descent approximation of $\bm{\theta}^{*}(\mathbf{w})$.

###  4.2 DERPLL: Robustness-aware LL with Coresets

Next, we illustrate how to integrate the selected coreset $\mathcal{S}^{(t)}$
to achieve data-efficient robustness-preserving lifelong learning (DERPLL). As
described in Sect. 3, we aim to sustain good performance of the ML models
$\bm{\theta}^{(t)}$ over continually evolving tasks $\mathcal{T}^{(t)}$. We
propose to train $\bm{\theta}^{(t)}$ using $\mathcal{D}^{(t)}$ and
$\mathcal{S}^{(t)}$, with a training objective function that is designed to
meet the desiderata – i.e., simultaneous adaptation and preservation.

To this end, we design an objective function that consists of two components.
First, we utilize the usual AT loss, i.e.,
$\ell^{\epsilon}_{\mathrm{AT}}(\bm{\theta}^{(t)};\mathcal{D}^{(t)})$ to ensure
robustness at the current time $t$. Second, we introduce an LwF loss term to
retain accuracy/robustness on the past tasks, which is implemented by
minimizing the prediction disagreement between the old model
$\bm{\theta}^{(t-1)}$ and the new model $\bm{\theta}^{(t)}$ on the archived
coreset, $\mathcal{S}^{(t)}$:

$\displaystyle\ell_{\mathrm{LwF}}(\bm{\theta}^{(t)};\bm{\theta}^{(t-1)},\mathcal{S}^{(t)})=\mathbb{E}_{\mathbf{x}\in\mathcal{S}^{(t)}}\big{\\{}d(\bm{\theta}^{(t)},\bm{\theta}^{(t-1)};\mathbf{x})\big{\\}}$
$\displaystyle\hskip
48.36967pt+\mathbb{E}_{\mathbf{x}\in\mathcal{S}^{(t)}}\Big{\\{}\max_{\|\bm{\delta}\|_{\infty}\leq\epsilon}d(\bm{\theta}^{(t)},\bm{\theta}^{(t-1)};\mathbf{x}+\bm{\delta})\Big{\\}},$

where $d(\bm{\theta}_{1},\bm{\theta}_{2};\mathbf{x})$ is a disparity measure
(e.g., KL divergence [6] used in our experiments) of predictions from two
models $\bm{\theta}_{1}$ and $\bm{\theta}_{2}$ at input $\mathbf{x}$; the two
terms are designed to penalize forgetting of accuracy and robustness,
respectively.

Overall, we propose the DERPLL framework that learns $\bm{\theta}^{(t)}$ by
minimizing the composite objective function, i.e.,
$\ell^{\epsilon}_{\mathrm{AT}}$ regularized with $\ell_{\mathrm{LwF}}$:

$\displaystyle\displaystyle\operatorname*{\text{minimize}}_{\bm{\theta}^{(t)}}\leavevmode\nobreak\
\ell^{\epsilon}_{\mathrm{AT}}(\bm{\theta}^{(t)};\mathcal{D}^{(t)})+\gamma\ell_{\mathrm{LwF}}(\bm{\theta}^{(t)};\bm{\theta}^{(t-1)},\mathcal{S}^{(t)}),$
(8)

where $\gamma>0$ is a tunable parameter. After acquiring $\bm{\theta}^{(t)}$,
the coreset $\mathcal{S}^{(t)}$ is updated to $\mathcal{S}^{(t+1)}$ by
augmenting a newly selected subset from $\mathcal{D}^{(t)}$; cf. (7) in Sect.
4.1.

##  5 Experiments

In this section, we evaluate the effectiveness of our proposal in the task of
class-incremental image classification.

Datasets & Model Architectures. We consider the CIFAR-10 [37] dataset using
the neural network model ResNet-18 [34]. In the context of CIL, CIFAR-10 with
10 classes is randomly divided into 5 tasks, each with 2 classes. The model is
trained with a time-evolving task stream and will update its classification
ability when a new task arrives.

Implementation Details. For each new task, the model will be adversarially
trained with both the complete new training data and the selected old data
from the memory bank stored from previous tasks. The coreset selection for the
new data will take place after the model is updated at the current task. We
set the size of the memory bank to 100 images per class if not otherwise
specified. We leverage the widely adopted 10-step PGD attack [18] to generate
adversarial examples with the attack strength of $\epsilon=8/255$, where
$\epsilon$ was defined in (AT). To solve problem (8), we use the SGD optimizer
as the model training recipe. And we tune the regularization parameter
$\gamma$ in $[10^{-3},1]$, and find that $\gamma=0.1$ consistently provides
the best robustness and accuracy performance over time.

Baselines & Evaluation Metrics. We consider two data selection oracles as our
baselines: ❶ random selection randomly samples a subset of the memory bank
size from the training set; ❷ influence score-based selection (IS) [5] selects
the data with the top influence scores. For evaluation, we evaluate the
performance on all the tasks encountered so far. In particular, we look into
the ① standard accuracy (SA), the test accuracy on the clean test set; and ②
robust accuracy (RA), tested on the perturbed test set. We perform 20-step PGD
attack to generate test-time perturbations for evaluation with the same attack
strength ($\epsilon=8/255$) as training.

Table 1: Performance comparison of different methods for robustness-aware CIL
under (CIFAR-10, ResNet18). Each row shows the standard accuracy (SA) and
robust accuracy (RA) on old tasks and the current task. For example, in the
first row, $\mathcal{T}^{(1)}$ is the old task, and $\mathcal{T}^{(2)}$ is the
current task. The best results are in bold.

Tasks | Methods | CIFAR-10 (SA, RA) |  |
---|---|---|---|---
$\mathcal{T}^{(1)}$ (%) |  $\mathcal{T}^{(2)}$ (%) |  $\mathcal{T}^{(3)}$ (%) |  $\mathcal{T}^{(4)}$ (%) |  $\mathcal{T}^{(5)}$ (%)
SA | RA | SA | RA | SA | RA | SA | RA | SA | RA
$\mathcal{T}^{(1)}\sim\mathcal{T}^{(2)}$ | Random | 83.20 | 65.30 | 95.75 | 83.10 | - | - | - | - | - | -
IS | 84.40 | 66.00 | 97.00 | 83.75 | - | - | - | - | - | -
Ours | 85.05 | 66.25 | 97.55 | 87.20 | - | - | - | - | - | -
$\mathcal{T}^{(1)}\sim\mathcal{T}^{(3)}$ | Random | 80.80 | 62.85 | 93.35 | 79.30 | 94.10 | 84.90 | - | - | - | -
IS | 81.75 | 62.95 | 94.50 | 79.35 | 94.95 | 84.95 | - | - | - | -
Ours | 82.05 | 64.50 | 94.85 | 79.40 | 95.10 | 86.00 | - | - | - | -
$\mathcal{T}^{(1)}\sim\mathcal{T}^{(4)}$ | Random | 75.85 | 59.90 | 82.80 | 63.10 | 89.40 | 76.65 | 75.90 | 62.70 | - | -
IS | 76.20 | 61.35 | 83.30 | 63.30 | 89.90 | 78.80 | 76.15 | 63.20 | - | -
Ours | 76.85 | 61.55 | 87.90 | 69.60 | 89.94 | 81.60 | 77.00 | 63.25 | - | -
$\mathcal{T}^{(1)}\sim\mathcal{T}^{(5)}$ | Random | 71.85 | 55.00 | 77.60 | 60.10 | 84.40 | 72.05 | 74.60 | 55.60 | 74.85 | 60.90
IS | 73.05 | 55.40 | 79.15 | 61.65 | 85.40 | 72.35 | 75.35 | 56.10 | 76.60 | 62.20
Ours | 74.10 | 58.20 | 84.35 | 63.70 | 87.75 | 75.35 | 76.40 | 56.80 | 78.20 | 64.50

Experiment Results. In Tab. 1, we show the overview performance of DERPLL and
baseline methods for robustness-aware CIL on CIFAR-10 under ResNet-18. We
summarize our key observations below. First, our method better helps mitigate
the catastrophic forgetting of both accuracy and robustness. Let us take the
row of ‘$\mathcal{T}^{(1)}\sim\mathcal{T}^{(4)}$’ as an example, where
$\mathcal{T}^{(4)}$ is the current task and $\mathcal{T}^{(i)}$ for
$i\in\\{1,2,3\\}$ are old tasks. We observe that our proposed method (DERPLL)
outperform baselines in model performance preservation on old tasks, as
evidence by a substantial SA/RA improvement over all the baselines, e.g.,
$4\%$ on SA and over $6\%$ on RA evaluated at $\mathcal{T}^{(2)}$ in the row
of ‘$\mathcal{T}^{(1)}\sim\mathcal{T}^{(4)}$’. Second, our method also
achieves the best SA/RA at current tasks across all time steps (corresponding
to the diagonal of Tab. 1). The above two observations indicate that our
method is effective not only in mitigating the forgetting of accuracy and
robustness but also in achieving high-accuracy prediction based on the new
knowledge. Third, we note that all the methods suffer from a performance drop
as more tasks are encountered. This is not surprising, since more tasks bring
in more classes and thus raise greater challenges in adversarial training and
the mitigation of the forgetting issue.

Fig. 1: RA of different methods v.s. coreset sizes.

Fig. 1 shows the model robustness (over the entire test set) vs. the coreset
size (in terms of data number per class) adopted at a given time step, where
the curve and shaded area represent the mean value and standard deviation over
three independent trials. We observe that the advantage of our method becomes
more significant as the coreset size reduces. For example, compared to our
default setting (100 images per class), the performance improvement over the
IS baseline increases from $2\%$ to $3\%$. This is encouraging when the data
storage capacity is limited in CIL.

##  6 Conclusion

In this paper, we develop DERPLL, the new data-efficient robustness-preserving
lifelong learning framework. We design an advanced coreset selection method to
determine the most effective data for ease of storing continual data while
preserving model’s adversarial robustness. Empirically, we show the
effectiveness of DERPLL in DNN-involved lifelong learning for class-
incremental image classification.

## References

  * [1] Alex Krizhevsky, Ilya Sutskever, et al.,  “Imagenet classification with deep convolutional neural networks,”  Communications of the ACM, 2017.
  * [2] Jacob Devlin, Ming-Wei Chang, et al.,  “Bert: Pre-training of deep bidirectional transformers for language understanding,”  arXiv preprint arXiv:1810.04805, 2018.
  * [3] Jie Zhou, Ganqu Cui, et al.,  “Graph neural networks: A review of methods and applications,”  AI Open, 2020.
  * [4] Sebastian Thrun,  “Is learning the n-th thing any easier than learning the first?,”  NeurIPS, 1995.
  * [5] Zalán Borsos, Mojmir Mutny, and Andreas Krause,  “Coresets via bilevel optimization for continual learning and streaming,”  NeurIPS, 2020.
  * [6] Tianlong Chen, Sijia Liu, et al.,  “Queried unlabeled data improves and robustifies class-incremental learning,”  TMLR, 2022.
  * [7] Michael McCloskey and Neal J Cohen,  “Catastrophic interference in connectionist networks: The sequential learning problem,”  in Psychology of learning and motivation. Elsevier, 1989.
  * [8] Ian J Goodfellow, Mehdi Mirza, et al.,  “An empirical investigation of catastrophic forgetting in gradient-based neural networks,”  arXiv preprint arXiv:1312.6211, 2013.
  * [9] David Lopez-Paz and Marc’Aurelio Ranzato,  “Gradient episodic memory for continual learning,”  NeurIPS, 2017.
  * [10] Chen He, Ruiping Wang, et al.,  “Exemplar-supported generative reproduction for class incremental learning.,”  in BMVC, 2018, p. 98.
  * [11] Aristotelis Chrysakis and Marie-Francine Moens,  “Online continual learning from imbalanced data,”  in ICML. PMLR, 2020, pp. 1952–1961.
  * [12] Junting Zhang, Jie Zhang, et al.,  “Class-incremental learning via deep model consolidation,”  in Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, 2020, pp. 1131–1140.
  * [13] James Kirkpatrick, Razvan Pascanu, et al.,  “Overcoming catastrophic forgetting in neural networks,”  Proceedings of the national academy of sciences, 2017.
  * [14] Zhizhong Li and Derek Hoiem,  “Learning without forgetting,”  IEEE transactions on pattern analysis and machine intelligence, 2017\.
  * [15] Khurram Javed and Faisal Shafait,  “Revisiting distillation and incremental classifier learning,”  in ACCV. Springer, 2018.
  * [16] Ian J Goodfellow, Jonathon Shlens, and Christian Szegedy,  “Explaining and harnessing adversarial examples,”  arXiv preprint arXiv:1412.6572, 2014.
  * [17] Francesco Croce and Matthias Hein,  “Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks,”  in ICML. PMLR, 2020.
  * [18] Aleksander Madry, Aleksandar Makelov, et al.,  “Towards deep learning models resistant to adversarial attacks,”  arXiv preprint arXiv:1706.06083, 2017.
  * [19] Yihua Zhang, Guanhua Zhang, et al.,  “Revisiting and advancing fast adversarial training through the lens of bi-level optimization,”  in ICML. 2022, PMLR.
  * [20] Marc Masana, Xialei Liu, et al.,  “Class-incremental learning: survey and performance evaluation on image classification,”  arXiv preprint arXiv:2010.15277, 2020.
  * [21] Pankaj K Agarwal, Sariel Har-Peled, et al.,  “Geometric approximation via coresets,”  Combinatorial and computational geometry, 2005.
  * [22] Dan Feldman, Melanie Schmidt, and Christian Sohler,  “Turning big data into tiny data: Constant-size coresets for k-means, pca, and projective clustering,”  SIAM Journal on Computing, 2020.
  * [23] Dan Feldman and Michael Langberg,  “A unified framework for approximating and clustering data,”  in Proceedings of the forty-third annual ACM symposium on Theory of computing, 2011.
  * [24] Jonathan Huggins, Trevor Campbell, and Tamara Broderick,  “Coresets for scalable bayesian logistic regression,”  NeurIPS, 2016.
  * [25] Mario Lucic, Matthew Faulkner, Andreas Krause, and Dan Feldman,  “Training gaussian mixture models at scale via coresets,”  JMLR, 2017.
  * [26] Ozan Sener and Silvio Savarese,  “Active learning for convolutional neural networks: A core-set approach,”  in ICLR, 2018.
  * [27] Trevor Campbell and Tamara Broderick,  “Automated scalable bayesian inference via hilbert coresets,”  JMLR, 2019.
  * [28] Ryan Curtin, Benjamin Moseley, et al.,  “Rk-means: Fast clustering for relational data,”  in AISTATS. PMLR, 2020.
  * [29] Hongyang Zhang, Yaodong Yu, Jiantao Jiao, Eric P Xing, Laurent El Ghaoui, and Michael I Jordan,  “Theoretically principled trade-off between robustness and accuracy,”  ICML, 2019.
  * [30] Dinghuai Zhang, Tianyuan Zhang, Yiping Lu, Zhanxing Zhu, and Bin Dong,  “You only propagate once: Accelerating adversarial training via maximal principle,”  arXiv preprint arXiv:1905.00877, 2019.
  * [31] Anish Athalye, Nicholas Carlini, and David Wagner,  “Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples,”  arXiv preprint arXiv:1802.00420, 2018.
  * [32] R Dennis Cook and Sanford Weisberg,  Residuals and influence in regression,  New York: Chapman and Hall, 1982.
  * [33] Pang Wei Koh and Percy Liang,  “Understanding black-box predictions via influence functions,”  in ICML. PMLR, 2017.
  * [34] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun,  “Deep residual learning for image recognition,”  in CVPR, 2016.
  * [35] Yoshua Bengio, Nicholas Léonard, and Aaron Courville,  “Estimating or propagating gradients through stochastic neurons for conditional computation,”  arXiv preprint arXiv:1308.3432, 2013.
  * [36] Risheng Liu, Jiaxin Gao, Jin Zhang, Deyu Meng, and Zhouchen Lin,  “Investigating bi-level optimization for learning and vision from a unified perspective: A survey and beyond,”  IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021\.
  * [37] Alex Krizhevsky et al.,  “Learning multiple layers of features from tiny images,”  2009\.


#  Characterizations of a Lorentzian Manifold with a semi-symmetric metric
connection

Uday Chand De , Krishnendu De and SİNEM Güler ∗ Department of Pure
Mathematics, University of Calcutta, West Bengal, India. ORCID iD:
https://orcid.org/0000-0002-8990-4609<EMAIL_ADDRESS>Department of
Mathematics, Kabi Sukanta Mahavidyalaya, The University of Burdwan.
Bhadreswar, P.O.-Angus, Hooghly, Pin 712221, West Bengal, India.
ORCID iD: https://orcid.org/0000-0001-6520-4520<EMAIL_ADDRESS>Department of Industrial Engineering, Istanbul Sabahattin Zaim University,
Halkali̇, Istanbul, Turkey<EMAIL_ADDRESS>
###### Abstract.

In this article, we characterize a Lorentzian manifold $\mathcal{M}$ with a
semi-symmetric metric connection. At first, we consider a semi-symmetric
metric connection whose curvature tensor vanishes and establish that if the
associated vector field is a unit time-like torse-forming vector field, then
$\mathcal{M}$ becomes a perfect fluid spacetime. Moreover, we prove that if
$\mathcal{M}$ admits a semi-symmetric metric connection whose Ricci tensor is
symmetric and torsion tensor is recurrent, then $\mathcal{M}$ represents a
generalized Robertson-Walker spacetime. Also, we show that if the associated
vector field of a semi-symmetric metric connection whose curvature tensor
vanishes is a $f-$ Ric vector field, then the manifold is Einstein and if the
associated vector field is a torqued vector field, then the manifold becomes a
perfect fluid spacetime. Finally, we apply this connection to investigate
Ricci solitons.

###### Key words and phrases:

Lorentzian manifolds; semi-symmetric metric connection; perfect fluid
spacetimes.

###### 1991 Mathematics Subject Classification:

The Mathematics subject classification 2020: 83C05, 53C05, 53C20, 53C50.

00footnotetext: ∗ Corresponding author

##  1\. Introduction

In this study, we investigate several vector fields and solitons on a
Lorentzian manifold $\mathcal{M}$ obeying a semi-symmetric metric connection
(briefly, $\mathrm{SSMC}$). The notion of a semi-symmetric linear connection
was first introduced by Friedman and Schouten on a differentiable manifold
many years ago. The concept of metric connection with torsion tensor $T$ was
then presented by Hayden [16] in 1932 on a Riemannian manifold. Yano [23] made
a methodical investigation of $\mathrm{SSMC}$ in 1970.

On a Lorentzian manifold $\mathcal{M}$, a linear connection $\nabla$ is named
semi-symmetric if $T$ described by

$T(U_{1},V_{1})=\nabla_{U_{1}}V_{1}-\nabla_{V_{1}}U_{1}-[U_{1},V_{1}]$

obeys

(1.1) $T(U_{1},V_{1})=\pi(V_{1})U_{1}-\pi(U_{1})V_{1},$

where $\pi$ is defined by $\pi(U_{1})=g(U_{1},\xi)$, for a fixed vector field
$\xi$ (also, named associated vector field ). If we replace $U_{1}$ and
$V_{1}$, respectively, by $\phi U_{1}$ and $\phi V_{1}$ in the right side of
(1.1), then the $\mathrm{SSMC}$ $\nabla$ changes into a quarter-symmetric
connection [13], $\phi$ being a (1,1)- tensor field.

Again, if a $\mathrm{SSMC}$ $\nabla$ on $\mathcal{M}$ fulfills

(1.2) $(\nabla_{U_{1}}g)(V_{1},Z_{1})=0,$

then $\nabla$ is named metric [23], if not, it is non-metric [16].

We know [23] that if $\nabla$ is a $\mathrm{SSMC}$ and $\mathrm{D}$ be a Levi-
Civita connection, then

(1.3)
$\nabla_{U_{1}}V_{1}=\mathrm{D}_{U_{1}}V_{1}+\eta(V_{1})U_{1}-g(U_{1},V_{1})\xi.$

Further, it is also known [23] that if $\overline{R}$ and $R$ denote the
curvature tensors of $\nabla$ and $\mathrm{D}$, respectively, then

(1.4) $\displaystyle\overline{R}(U_{1},V_{1})Z_{1}$ $\displaystyle=$
$\displaystyle
R(U_{1},V_{1})Z_{1}-L_{1}(V_{1},Z_{1})U_{1}+L_{1}(U_{1},Z_{1})V_{1}$
$\displaystyle-g(V_{1},Z_{1})L_{2}U_{1}+g(U_{1},Z_{1})L_{2}V_{1},$

where $L_{1}$, a (0,2) tensor and $L_{2}$, a (1,1) tensor are defined by

(1.5)
$L_{1}(U_{1},V_{1})=(\mathrm{D}_{U_{1}}\pi)(V_{1})-\pi(U_{1})\pi(V_{1})+\frac{1}{2}\pi(\xi)g(U_{1},V_{1})$

and

(1.6) $g(L_{2}U_{1},V_{1})=L_{1}(U_{1},V_{1})$

for any vector fields $U_{1}$ and $V_{1}$.

In a local coordinate system, equations (1.4) and (1.5) can be written as
follows:

(1.7)
$\overline{R}_{hijk}=R_{hijk}-g_{hk}\pi_{ij}+g_{ik}\pi_{hj}-g_{ij}\pi_{hj}+g_{hk}\pi_{ik},$
(1.8)
$\pi_{ik}=\mathrm{D}_{i}v_{k}-v_{i}v_{k}+\frac{1}{2}g_{ik}v,\;\;\;\;\;v=v^{h}v_{h}$

in which $\overline{R}_{hijk}$ and $R_{hijk}$ denote the curvature tensor of
$\nabla$ and $\mathrm{D}$, respectively.

The idea of the $\mathrm{SSMC}$ was investigated by many researchers (see,
[3], [23], [25], [26], [27], [28]).

The current paper concerns with n-dimensional spacetimes (that is, a connected
time-oriented Lorentz manifold) ($\mathcal{M},g$). For $\psi>0$, a smooth
function (also called scale factor, or warping function), if
$\mathcal{M}=-I\times_{\psi},\mathrm{M}$, where $I$ is the open interval of
$\mathbb{R}$ , $\mathrm{M}^{n-1}$ denotes the Riemannian manifold, then
$\mathcal{M}$ is named a generalized Robertson-Walker $(\text{briefly},GRW)$
spacetime [1]. When the dimension of $\mathcal{M}$ is three and of constant
sectional curvature, this spacetime represents a Robertson-Walker (briefly,
$RW$) spacetime. In this context we may mention the work of Ünal and Shenawy
on warped product manifolds ([21], [22]).

Due to the absence of a stress tensor and heat conduction terms corresponding
to viscosity, the fluid is referred to as perfect and the energy momentum
tensor $\mathcal{T}_{hk}$ is written by

(1.9) $\mathcal{T}_{hk}=(\sigma+p)v_{h}v_{k}+pg_{hk},$

in which $p$ and $\sigma$ stand for the perfect fluid’s isotropic pressure and
energy density, respectively [20].

For a gravitational constant $\kappa$, the Einstein’s field equations without
a cosmological constant is given by

(1.10) $R_{hk}-\frac{R}{2}g_{hk}=\kappa\mathcal{T}_{hk},$

where $R_{hk}=R^{i}_{hki}$ and $R=g^{hk}R_{hk}$ are the Ricci tensor and the
scalar curvature, respectively.

A spacetime $\mathcal{M}$ is named a perfect fluid (briefly $PF$) spacetime if
the non-vanishing Ricci tensor $R_{hk}$ fulfills

(1.11) $R_{hk}=\alpha g_{hk}+\beta v_{h}v_{k}$

where $\alpha$ and $\beta$ indicate smooth functions on $\mathcal{M}$. In the
last equation, $g$ is the Lorentzian metric of $\mathcal{M}$ and $v_{h}$, the
velocity vector is defined by $g_{hk}v^{h}v^{k}=-1$ and $v_{h}=g_{hk}v^{k}$.

Combining the equations (1.9), (1.10) and (1.11), we acquire

(1.12) $\beta=k^{2}(p+\sigma),\,\,\alpha=\frac{k^{2}(p-\sigma)}{2-n}.$

In [20], O’Neill established that a Robertson-Walker spacetime is a $PF$
spacetime. Every 4-dimensional generalized Robertson-Walker spacetime is a
$PF$ spacetime if and only if it is a Robertson-Walker spacetime [12]. For
additional insights about the $PF$ and $GRW$ spacetimes, we refer( [2], [4],
[5], [8], [9], [11], [19]) and its references.

######  Definition 1.1.

For a non vanishing 1-form $\omega_{k}$ and a scalar function $\phi$ if the
relation $\nabla_{k}u_{h}=\omega_{k}u_{h}+\phi g_{kh}$ holds, then the vector
field $u$ is called torse-forming.

This notion was introduced by Yano [24] on a Riemannian manifold. It is noted
that the foregoing torse-forming condition becomes
$\nabla_{k}u_{h}=\phi(u_{k}u_{h}+g_{kh})$, for a unit time-like vector.

In [19], Mantica and Molinari and in [4], Chen have proven the subsequent
theorems:

Let $\mathcal{M}$ be a Lorentzian manifold of dimension $n$ ($n\geq 3$).

######  Theorem 1.1.

[19] $\mathcal{M}$ is a $GRW$ spacetime if and only if it admits a unit time-
like torse-forming vector field: $\nabla_{k}v_{j}=\varphi(g_{ij}+v_{j}v_{i})$,
which is also an eigenvector of the Ricci tensor.

######  Theorem 1.2.

[4] $\mathcal{M}$ is a $GRW$ spacetime if and only if it admits a time-like
concircular vector field.

In this article, we consider a $\mathrm{SSMC}$ and prove the following:

######  Theorem 1.3.

If the associated vector field of $\mathcal{M}$ obeying a $\mathrm{SSMC}$
whose curvature tensor vanishes is a unit time-like torse-forming vector
field, then $\mathcal{M}$ becomes a $PF$ spacetime.

######  Definition 1.2.

In a semi-Riemannian manifold with a $\mathrm{SSMC}$ $\nabla$, the torsion
tensor $T$ is called recurrent if it obeys
$\nabla_{k}T^{h}_{ij}=A_{k}T^{h}_{ij}$, $A_{k}$ being a covariant vector.

Here, we choose a $\mathrm{SSMC}$ whose torsion tensor is recurrent and
establish the subsequent result:

######  Theorem 1.4.

Let $\mathcal{M}$ admit a $\mathrm{SSMC}$ whose Ricci tensor is symmetric and
torsion tensor is recurrent. Then the manifold $\mathcal{M}$ represents a
$\mathrm{GRW}$ spacetime.

######  Definition 1.3.

If $\nabla_{k}v_{h}=fR_{kh}$, then $v_{h}$ is named a $f$-Ric vector field,
where $f$ is a constant.

The foregoing idea of $f$-$Ric$ vector field was presented by Hinterleitner
and Kiosak [17] on a Riemannian manifold.

######  Definition 1.4.

[6]A vector field is called a torqued vector field if
$\nabla_{k}v_{h}=fg_{kh}+\omega_{k}v_{h}$ and $\omega_{k}v^{k}=0$, where $f$
is a real constant and 1-form $\omega$ is named the torqued form of $v$.

In this article, we consider a $\mathrm{SSMC}$ whose associated vector field
is $f-$ Ric and torqued vector field, respectively and prove the following
results:

######  Theorem 1.5.

If $\mathcal{M}$ admits a $\mathrm{SSMC}$ whose curvature tensor vanishes and
the associated vector field is $f-$ Ric, then $\mathcal{M}$ is Einstein.

######  Theorem 1.6.

Let $\mathcal{M}$ admit a $\mathrm{SSMC}$ whose curvature tensor vanishes. If
the associated vector field is torqued, then $\mathcal{M}$ becomes a $PF$
spacetime.

Hamilton [15] introduces the concept of Ricci flow as a solution to the
challenge of obtaining a canonical metric on a smooth manifold. Let
$\mathcal{M}$ be satisfied by an evolution equation $\frac{\partial}{\partial
t}g_{hk}(t)=-2R_{hk}$, then it is called Ricci flow [15]. If the metric of
$\mathcal{M}$ obeys the relation

(1.13) $\mathfrak{L}_{v}g_{hk}+2R_{hk}+2\overline{\lambda}g_{hk}=0,$

then it is called a Ricci soliton [14], where $\mathfrak{L}_{v}$ is the Lie
derivative operator and $\overline{\lambda}$ denotes a real constant. Here,
$v$ is called the potential vector field of the solitons. The Ricci soliton is
named shrinking, expanding or steady, if $\overline{\lambda}$ is negative,
positive, or zero, respectively.

Many mathematicians were fascinated in the geometry of Ricci solitons over the
past few years. Some interesting findings on this solitons have been
investigated in ([2], [7], [10], [18]) and also by others.

The above investigations motivate us to $\mathrm{SSMC}$ and establish the
subsequent:

######  Theorem 1.7.

Let us suppose that $\mathcal{M}$ admit a $\mathrm{SSMC}$ whose curvature
tensor vanishes and the associated vector field is a unit time-like torse-
forming vector field. If $\mathcal{M}$ admits a Ricci soliton
$(g,v,\overline{\lambda})$, then the soliton is shrinking if $\phi<1$,
expanding if $\phi>1$, or steady if $\phi=1$, respectively. Additionally, the
integral curves that $v$ generates are geodesics.

##  2\. Semi-symmetric metric connection

$\mathbf{Agreement}$: Throughout this article, we assume that the associated
vector $v_{i}$ is a unit time-like vector, that is, $v^{i}v_{i}=-1$.

Then, the equation (1.8) becomes

(2.1) $\pi_{ik}=\mathrm{D}_{i}v_{k}-v_{i}v_{k}-\frac{1}{2}g_{ik}.$

Multiplying (1.7) with $g^{hk}$, we acquire

(2.2) $\overline{R}_{ij}=R_{ij}-(n-2)\pi_{ij}-\pi g_{ij},$

where $\overline{R}_{ij}$ and $R_{ij}$ are the Ricci tensor of $\nabla$ and
$\mathrm{D}$, respectively and $\pi=g^{hk}\pi_{hk}$.

Also, we have

(2.3) $\nabla_{h}v_{k}=\mathrm{D}_{h}v_{k}-v_{h}v_{k}-g_{hk}.$

######  Proposition 2.1.

$\overline{R}_{ij}$ is symmetric if and only if $v_{i}$ is closed.

###### Proof.

From the equation (2.2), we obtain

(2.4) $\displaystyle\overline{R}_{ij}-\overline{R}_{ji}$ $\displaystyle=$
$\displaystyle-(n-2)\\{\pi_{ij}-\pi_{ji}\\}$
$\displaystyle=-(n-2)\\{\mathrm{D}_{i}v_{j}-v_{i}v_{j}-\frac{1}{2}g_{ij}-\mathrm{D}_{j}v_{i}+v_{j}v_{i}+\frac{1}{2}g_{ji}\\}$
$\displaystyle=-(n-2)\\{\mathrm{D}_{i}v_{j}-\mathrm{D}_{j}v_{i}\\}.$

If $v_{i}$ is closed, then the foregoing equation reduces to

(2.5) $\overline{R}_{ij}-\overline{R}_{ji}=0,$

that is, $\overline{R}_{ij}$ is symmetric and conversely. ∎

###  2.1. Proof of the Theorem 1.3

Since by hypothesis, $\overline{R}_{hijk}=0$, from (1.7), we get

(2.6) $R_{hijk}=g_{hk}\pi_{ij}-g_{ik}\pi_{hj}+g_{ij}\pi_{hj}-g_{hk}\pi_{ik}.$

Let

(2.7) $\nabla_{k}v_{j}=\phi(g_{kj}+v_{j}v_{k}).$

Then from (2.3) it follows that

(2.8) $\mathrm{D}_{k}v_{j}=(\phi+1)(g_{kj}+v_{j}v_{k}).$

Hence, using the foregoing equation in (2.1), we obtain

(2.9) $\pi_{ik}=(\phi-\frac{1}{2})g_{ik}+\phi v_{i}v_{k}).$

Now, making use of the previous equation in (2.6), we acquire

(2.10)
$R_{hijk}=(2\phi-1)\\{g_{hk}g_{ij}-g_{hj}g_{ik}\\}+\phi\\{g_{hk}v_{i}v_{j}+g_{ij}v_{h}v_{k}-g_{ik}v_{h}v_{j}-g_{hj}v_{i}v_{k}\\}.$

Multiplying with $g^{ij}$ the above equation yields

$R_{hk}=(2\phi-1)(n-1)g_{hk}+\phi\\{-g_{hk}+nv_{h}v_{k}-v_{h}v_{k}-v_{h}v_{k}\\},$

which implies

(2.11) $R_{hk}=\\{(n-1)(2\phi-1)-\phi\\}g_{hk}+(n-2)\phi v_{h}v_{k}.$

The foregoing equation entails that $\mathcal{M}$ is a $PF$ spacetime.

Hence, the proof is finished.

Now, using Theorem 1.1, we may state:

######  Corollary 2.1.

A $\mathrm{GRW}$-spacetime admitting a $\mathrm{SSMC}$ whose curvature tensor
vanishes is a $PF$ spacetime.

######  Remark 2.1.

In [19], Mantica and Molinari established that a $GRW$ spacetime satisfying
$divC=0$ represents a $PF$ spacetime, where $C$ indicates the conformal
curvature tensor. Here, curvature condition $divC=0$ is replaced by a
$\mathrm{SSMC}$ whose curvature tensor vanishes and acquires the same result.

For $n=4$, comparing the equations (1.11) and (2.11), we have

(2.12) $\alpha g_{hk}+\beta v_{h}v_{k}=(5\phi-3)g_{hk}+2\phi v_{h}v_{k}.$

Making use of (1.12), the foregoing equation yields

(2.13) $k^{2}(3p-2\sigma)=3,$

which implies

(2.14) $(3p-2\sigma)=c\,\,(say).$

######  Corollary 2.2.

A $PF$ spacetime admitting a $\mathrm{SSMC}$ satisfies the state equation
$\sigma=\frac{3}{2}p+$ constant.

###  2.2. Proof of the Theorem 1.4

It is known [3] that if a Riemannian manifold admits a $\mathrm{SSMC}$
$\nabla$ whose torsion tensor $T$ is recurrent, then the curvature tensor
$\overline{R}^{h}_{ijk}$ of $\nabla$ is given by

(2.15) $\displaystyle\overline{R}^{h}_{ijk}$ $\displaystyle=$ $\displaystyle
R^{h}_{ijk}+A_{k}[v_{i}\delta^{h}_{j}-g_{ij}v^{h}]$ $\displaystyle-
A_{j}[v_{k}\delta^{h}_{i}-g_{ik}v^{h}]+v^{l}v_{l}[g_{jk}\delta^{h}_{i}-g_{ik}\delta^{h}_{j}].$

For a Lorentzian manifold the above relation becomes

(2.16) $\displaystyle\overline{R}^{h}_{ijk}$ $\displaystyle=$ $\displaystyle
R^{h}_{ijk}+A_{k}[v_{i}\delta^{h}_{j}-g_{ij}v^{h}]$ $\displaystyle-
A_{j}[v_{k}\delta^{h}_{i}-g_{ik}v^{h}]-[g_{jk}\delta^{h}_{i}-g_{ik}\delta^{h}_{j}],$

since $v^{l}v_{l}=-1$.

First we prove the following lemma:

######  Lemma 2.1.

If the torsion tensor $T$ is given by

(2.17) $\nabla_{k}T^{h}_{ij}=A_{k}T^{h}_{ij},$

then $A_{k}$ is gradient.

###### Proof.

Let

(2.18) $f^{2}=T_{hij}T^{hij},$

where $T_{hij}=g_{hk}T^{k}_{ij}$.

From (2.18), we get

(2.19) $\displaystyle 2f\nabla_{l}f$ $\displaystyle=$
$\displaystyle\nabla_{l}T_{hij}T^{hij}+T_{hij}\nabla_{l}T^{hij}$
$\displaystyle=2A_{l}f^{2},$

which implies that

(2.20) $\nabla_{l}f=fA_{l}.$

Therefore, from the above equation, we obtain

(2.21)
$\nabla_{l}\nabla_{m}f-\nabla_{m}\nabla_{l}f=\nabla_{m}fA_{l}-\nabla_{l}fA_{m}+f(\nabla{m}A_{l}-\nabla_{l}A_{m}).$

Since $f$ is a scalar, using the equation (2.20) in (2.21), we acquire

(2.22) $f(\nabla{m}A_{l}-\nabla_{l}A_{m})=0.$

Since $f\neq 0$, $A_{l}$ is gradient. ∎

From $T^{h}_{ij}=\delta^{h}_{i}v_{j}-\delta^{h}_{j}v_{i}$, we get

(2.23) $T^{h}_{ij}=(n-1)v_{j},$

which implies

(2.24) $\nabla_{k}T^{h}_{ij}=(n-1)v_{k}v_{j}.$

Again from $\nabla_{k}T^{h}_{ij}=A_{k}T^{h}_{ij}$, we obtain

(2.25) $\nabla_{k}T^{h}_{ij}=A_{k}T^{h}_{ij}=(n-1)A_{k}v_{j}.$

Contracting $h$ and $k$ in (2.16) yields

(2.26) $\overline{R}_{ij}=R_{ij}+[A_{j}v_{i}-g_{ij}A_{h}v^{h}].$

By hypothesis $\overline{R}_{ij}$ is symmetric. Hence, we have from (2.26)

(2.27) $A_{i}v_{j}=A_{j}v_{i}.$

Multiplying the foregoing equation by $v^{i}$ gives

(2.28) $A_{j}=av_{j},$

where $a=-A_{i}v^{i}$.

From (2.24) and (2.25) it follows that

(2.29) $\nabla_{k}v_{j}=A_{k}v_{j}.$

Using (2.3) in the previous equation yields

(2.30) $\mathrm{D}_{h}v_{k}=g_{hk}+\mu_{h}v_{k},$

where $\mu_{h}=A_{h}+v_{h}$.

From the above equation we conclude that $v_{k}$ is a torseforming vector
field with $\alpha=\beta=1$.

Here, $v_{h}$ is closed, since by hypothesis the Ricci tensor of the
$\mathrm{SSMC}$ is symmetric. Also by Lemma 2.1, $A_{h}$ is closed and hence
$\mu_{h}$ is closed. Therefore, $v_{h}$ is concircular. Hence, using Theorem
1.2 we state that $\mathcal{M}$ represents a $GRW$ spacetime.

This completes the proof.

###  2.3. Proof of the Theorem 1.5

Since the associated vector field is $f-$ Ric, then we have

(2.31) $\nabla_{k}v_{h}=fR_{hk},$

$f$ is a constant.

Then from (2.3), we acquire

(2.32) $\mathrm{D}_{h}v_{k}=fR_{hk}+g_{hk}+v_{h}v_{k}.$

Hence, using the previous equation in (2.1), we infer

(2.33) $\pi_{ik}=fR_{ik}+\frac{1}{2}g_{ik}.$

Since by assumption $\overline{R}_{hijk}=0$. From (1.7), we obtain

(2.34) $R_{hijk}=g_{hk}\pi_{ij}-g_{ik}\pi_{hj}+g_{ij}\pi_{hj}-g_{hk}\pi_{ik}.$

Now, using equation (2.33) in (2.34), we get

(2.35)
$R_{hijk}=f\\{g_{hk}R_{ij}-g_{ik}R_{hj}+g_{ij}R_{hk}-g_{hj}R_{ik}\\}+\\{g_{hk}g_{ij}-g_{hj}g_{ik}\\}.$

Multiplying with $g^{ij}$ the foregoing equation yields

$R_{hk}=f\\{nR_{hk}-R_{hk}+Rg_{hk}-R_{hk}\\}+(n-1)g_{hk},$

which entails

(2.36) $R_{hk}=\frac{fR+(n-1)}{1-(n-2)f}g_{hk}.$

The above equation states that $\mathcal{M}$ is Einstein.

This ends the proof.

######  Remark 2.2.

From the equation (2.35), we also get the spacetime is of constant curvature
$\frac{2f^{2}R+nf+1}{1-(n-2)f}$.

If $R=-\frac{nf+1}{f^{2}}$, then…

###  2.4. Proof of the Theorem 1.6

By hypothesis, $v$ is a torqued vector field. Therefore, we get

(2.37) $\nabla_{k}v_{h}=fg_{hk}+\omega_{k}v_{j}$

and $\omega_{k}v^{k}=0$. Then from (2.3), we obtain

(2.38) $\mathrm{D}_{h}v_{k}=(f+1)g_{hk}+(\omega_{h}+v_{h})v_{k}.$

Hence, using the above equation in (2.1), we acquire

(2.39) $\pi_{ik}=(f+\frac{3}{2})g_{ik}+\omega_{i}v_{k}.$

Since $\overline{R}_{hijk}=0$, from (1.7), we get

(2.40) $R_{hijk}=g_{hk}\pi_{ij}-g_{ik}\pi_{hj}+g_{ij}\pi_{hj}-g_{hk}\pi_{ik}.$

Now, using equation (2.39) in (2.40), we have

(2.41)
$R_{hijk}=2(f+\frac{3}{2})\\{g_{hk}g_{ij}-g_{hj}g_{ik}\\}+\\{g_{hk}\omega_{i}-g_{ik}\omega_{h}\\}v_{j}+\\{g_{ij}\omega_{h}-g_{hj}\omega_{i}\\}v_{k}.$

Multiplying with $g^{ij}$, we acquire

$R_{hk}=2n(f+\frac{3}{2})g_{hk}-2(f+\frac{3}{2})g_{hk}-(n-1)\omega_{h}\omega_{k}+\\{g_{hk}\omega_{h}v_{k}-\omega_{h}v_{k}\\}.$

(2.42) $R_{hk}=(2n-2)(f+\frac{3}{2})g_{hk}+(n-2)\omega_{h}v_{k}.$

Again, multiplying with $v^{h}$ yields

(2.43) $R_{hk}v^{h}=(2n-2)(f+\frac{3}{2})v_{k},$

since $\omega_{i}v^{i}=0$.

Also, from (2.42), we get $\omega_{h}v_{k}=\omega_{k}v_{h}$. Multiplying the
foregoing relation by $v^{k}$, we have $\omega_{h}=av_{h}$, where
$a=\omega_{k}v^{k}$. Hence, the equation (2.42) becomes

(2.44) $R_{hk}=(2n-2)(f+\frac{3}{2})g_{hk}+(n-2)av_{h}v_{k}.$

Foregoing equation tells that $\mathcal{M}$ is a $PF$ spacetime.

This accomplishes the proof.

##  3\. Applications to Ricci solitons

###  3.1. Proof of the Theorem 1.7

Now, suppose that $\mathcal{M}$ with $\mathrm{SSMC}$ admits a Ricci soliton.
Then from equation (1.13), we get

(3.1) $\nabla_{h}v_{k}+\nabla_{k}v_{h}+2R_{hk}+2\overline{\lambda}g_{hk}=0.$

Multiplying by $v^{h}v^{k}$, we obtain

(3.2) $R_{hk}v^{h}v^{k}-\overline{\lambda}=0,$

since $g_{hk}v^{h}v^{k}=-1$ and $v^{h}v^{k}\nabla_{h}v_{k}=0$. Again,
multiplying the equation (2.11) by $v^{h}v^{k}$, we infer

(3.3) $R_{hk}v^{h}v^{k}=-\\{(n-1)(2\phi-1)-\phi\\}+(n-2)\phi.$

Comparing the last two equations, we acquire
$\overline{\lambda}=(1-n)(\phi-1)$. This shows that the Ricci soliton is
shrinking if $\phi<1$, expanding if $\phi>1$, or steady if $\phi=1$,
respectively.

Again, multiplying (3.1) by $v^{h}$, we find

(3.4) $v^{h}\nabla_{k}v_{h}+2R_{hk}v^{h}+2\overline{\lambda}v_{k}=0,$

since $v^{h}\nabla_{h}v_{k}=0$.

Again, multiplying (2.11) by $v^{h}$ yields

(3.5) $R_{hk}v^{h}=\\{(n-1)(2\phi-1)-\phi\\}-(n-2)\phi.$

Using the above equation in (3.4), we have $v^{h}\nabla_{k}v_{h}=0$, since
$\overline{\lambda}=(1-n)(\phi-1)$.

This ends the proof.

## References

  * [1] Alias, L., Romero, A. and Sánchez M., Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes, Gen. Relativ. Gravit. 27 (1995), 71-84.
  * [2] Blaga, A.M., Solitons and geometrical structures in a perfect fluid spacetime, Rocky Mountain J. Math. 50 (2020), 41-53.
  * [3] Chaki, M. C. and Konar, A., On a type of semi-symmetric metric on a Riemannian manifolds, J. Pure Math. 1 (1981), 77-80.
  * [4] B. Y. Chen, Pseudo-Riemannian Geometry, $\delta$-invariants and Applications, World Scientific, 2011.
  * [5] by same author, A simple characterization of generalized Robertson-Walker spacetimes, Gen. Relativ. Gravit. 46 (2014), 1833 (5 pages).
  * [6] by same author, Rectifying submanifolds of Riemannian manifolds and torqued vector fields, Kragujevac J. Math. 41(1) (2017), 93-103.
  * [7] Chen, B.Y. and Deshmukh, S., Ricci solitons and concurrent vector fields, Balkan J. Geom. Appl., 20 (2015), 14-25.
  * [8] De, U.C., Chaubey, S.K. and Shenawy, S., Perfect fluid spacetimes and Yamabe solitons, J Math Phys. 62, 032501 (2021); https://doi.org/10.1063/5.0033967
  * [9] De, K., De, U.C., Syied, A. A., Turki N. B. and Alsaeed, S., Perfect fluid spacetimes and gradient solitons, Journal of Nonlinear Mathematical Physics https://doi.org/10.1007/s44198-022-00066-5.
  * [10] Deshmukh, S., Alodan, H. and Al-Sodais, H., A Note on Ricci Soliton, Balkan J. Geom. Appl. 16, 1 (2011), 48-55.
  * [11] Duggal, K. L. and Sharma, R.,Symmetries of spacetimes and Riemannian manifolds, Mathematics and its Applications 487 (Kluwer Academic Press, Boston, London 1999).
  * [12] Guti$\acute{e}$rrez, M. and Olea, B., Global decomposition of a Lorentzian manifold as a generalized Robertson-Walker space, Differ. Geom. Appl. 27 (2009), 146-156.
  * [13] Golab, S., On semi-symmetric and quarter-symmetric linear connections, Tensor (N. S.) 29 (1975), 249-254.
  * [14] Hamilton, R. S., Three-manifolds with positive Ricci curvature, J. Differ. Geom. 17 (1982), 255-306.
  * [15] Hamilton, R. S., The Ricci flow on surfaces, Contemp. Math. 71 (1988), 237-261.
  * [16] Hayden, H.A., Subspace of space with torsion, Proc. Lon. Math. Soc. 34 (1932), 27-50.
  * [17] Hinterleitner, I. and Volodymyr, A. K., $\phi(Ric)$-vector fields in Riemannian spaces, Archivum Math. (Brno) 44 (2008), 385-390. 00
  * [18] Karaca, F. and Ozgur, C., Gradient Ricci solitons on multiply warped product manifolds, Filomat, 32 (2018), 4221-4228.
  * [19] Mantica, C.A. and Molinari, L.G., Generalized Robertson-Walker spacetimes-A survey, Int. J. Geom. Methods Mod. Phys. 14 (2017), 1730001 (27 pages).
  * [20] O’Neill, B., Semi-Riemannian Geometry with Applications to the Relativity, Academic Press, New York-London, 1983.
  * [21] Ünal, B. and Shenawy, S., 2-Killing vector fields on warped product manifolds, International Journal of Mathematics. 26 (2015). 1550065. 10.1142/S0129167X15500652.
  * [22] Ünal, B. and Shenawy, S., The $W_{2}$ curvature tensor on warped product manifolds and applications, Int. J. Geom. Methods Mod. Phys. 13 (2016). 10.1142/S0219887816500997.
  * [23] Yano, K., On semi-symmetric metric connection, Revue Roumaine de Math. Pures et Appliques. 15 (1970), 1579-1586.
  * [24] K. Yano, On torse forming direction in a Riemannian space, Proc. Imp. Acad. Tokyo 20 (1944), 340–345.
  * [25] Zengin, F.O., Demirbag, S. A., Uysal, S. A. and Yilmaz, H. B., Some vector fields on a Riemannian manifold with semi-symmetric metric connection, Bull. Iranian Math. Soc., 38(2012), 479-490.
  * [26] Zengin F.O., Uysal S. A. and Demirbag S. A., On sectional curvature of a Riemannian manifold with semi-symmetric metric connection, Ann. Polon. Math., 101(2011), 131-138.
  * [27] Zhao, P.B., Fengyun, F. and Yang, X., Some Properties of S-Semi-Symmetric Metric Connections, Int. J. Contemp. Math. Sciences, 6 (2011), 1817-1827.
  * [28] Zhao, P.B., Han, Y. and Fengyun, F., On semi-symmetric metric connection in sub-Riemannian manifolds, Tamkang Journal of Mathematics, 47 (2016), 373-384.