# Modelling the Socialization of Creative Agents in a Master-Apprentice
Setting: The Case of Movie Title Puns

Mika Hämäläinen
Department of Digital Humanities
Faculty of Arts
University of Helsinki
<EMAIL_ADDRESS>Khalid Alnajjar
Department of Computer Science
Faculty of Science
University of Helsinki
<EMAIL_ADDRESS>
###### Abstract

> This paper presents work on modelling the social psychological aspect of
> socialization in the case of a computationally creative master-apprentice
> system. In each master-apprentice pair, the master, a genetic algorithm, is
> seen as a parent for its apprentice, which is an NMT based sequence-to-
> sequence model. The effect of different parenting styles on the creative
> output of each pair is in the focus of this study. This approach brings a
> novel view point to computational social creativity, which has mainly
> focused in the past on computationally creative agents being on a socially
> equal level, whereas our approach studies the phenomenon in the context of a
> social hierarchy.

## Introduction

The master-apprentice approach, as introduced by (?), to computational
creativity has been shown to achieve creative autonomy and its creativity has
been thoroughly discussed and motivated. However, the question that has
remained without an answer has been the social nature of a master-apprentice
pair and its effect on the creative outcome.

The approach consists of two parts: a master, which is a genetic algorithm,
and an apprentice, which is an LSTM sequence-to-sequence model. While the
master is in charge of the internal appreciation of the overall system as
implemented in its fitness function, the apprentice plays a crucial role in
the creative autonomy as it can learn its standards partially from its master
and partially from its peers.

This paper focuses on the exploration of the master-apprentice approach from a
social psychological point of view. By modelling the socialization of the
apprentice into a creative society consisting of the master and peers, we seek
to gain a deeper understanding of the phenomenon in terms of the overall
creativity of the system. In addition, modelling the social aspects of a
computationally creative system can help in understanding creativity as a
social phenomenon in a broader sense (?).

We motivate the model of socialization based on research conducted on the
field of social psychology, namely developmental psychology. We select the
categorization of parenting styles presented by (?) as the theoretical
foundation of our work.

The creative task we are tackling in this paper is the creation of humorous
movie titles delivering a food-related pun. This consists of taking an
existing movie title such as Beauty and the Beast and making a pun out of it
such as Beauty and the Beets. As people have been writing funny movie titles
of this sort in a great abundance on the social media, we can gather parallel
data easily.

## Related Work

While pun generation has been vastly studied in the field of computational
creativity (?; ?; ?), we see that the most important contribution of our paper
lies in the realm of social creativity. Therefore, we dedicate this section in
describing some of the practical research conducted in the computational
social creativity.

Research on an agent community consisting of self-organizing maps (?),
although outside of the computational creativity paradigm, presents a way of
simulating the emergence of language. The agents are capable of meaning
negotiation and converging into a common language to communicate about
edibility of different food items in their shared world.

Multi-agent systems have been studied in the context of novelty seeking in
creative artifact generation (?). In their study, the agents exert self-
criticism and they can vote and veto on creative artifacts. Their findings
suggest that multiple creative agents can reach to a higher number of novelty
in their output than a single agent system.

A recent study (?) has been conducted in social creativity in agent societies
where the individuals are goal-aware. The individuals create artifacts of
their own and peer up to collaborate with another agent. The agents are
capable of learning a peer model that guides them in selecting a collaboration
partner.

The papers discussed in this section, as well as other similar previously
conducted work (?; ?; ?), study mostly the collaboration of agents that have
an equal social status, in contrast to our case where the social status is
hierarchical. Therefore we find that there’s need for conducting the study
presented in this paper to shed some light into asymmetrical social relations
in computational creativity.

## Social Development

The master-apprentice approach gives us an intriguing test bed for modelling
different social interactions between the master and the apprentice. With such
a complex phenomenon as human social behavior, we are bound to limit our focus
on a subarea of the phenomenon. In this section, we describe different
psychological approaches in understanding socialization.

Socialization, i.e. becoming a part of a social group, is an important part of
the psychological development of an individual. Even to such a degree that a
child who is never exposed to other people will not develop a language nor an
understanding of self. Socialization, thus seems to play a crucial role in
higher-level cognitive development of everything that we consider to separate
a man from an animal. Perhaps this great level of importance has been the
reason a great many researchers have dedicated effort in unraveling this
mystery.

The ecological systems theory of social development (?) highlights the
importance of bidirectionality of different social groups. An individual child
is in the middle of the model, but just as the immediate close family affects
on the child, the child is also an actor in the process of socialization. The
theory identifies multiple different systems from close family all the way to
the level of the society that play a role in the social development of a
child. This theory is quite complex to model computationally.

A take, simpler to model, on the social development is that of parenting
styles (?). We find these findings more suitable as a starting point for
modelling the socialization of the apprentice in our master-apprentice
approach. The parenting styles can be divided into four main categories:
authoritative, authoritarian, permissive and rejecting-neglecting. These
categories deviate from each other on the two-fold axis of demandingness and
responsiveness as seen in Figure 1.

Figure 1:  Parenting styles

The authoritative parents are high on both demandingness and responsiveness.
They set rules, but the rules are negotiable. The parenting is more supportive
than punitive in nature. The authoritarian parents, on the other hand, are low
on responsiveness and high on demandingness. They set non-negotiable rules and
expect obedience without explanation.

The permissive parents are low on demandingess and high on responsiveness.
They are very lenient and avoid confrontation. The rejecting-neglecting
parents, however, are low on both axis. They hardly engage in parenting, they
offer little support and do not set any rules.

## Creativity

The original research on the master-apprentice approach (?) used the creative
tripod (?) to define creativity in general and in the context of their work on
creating movie titles satirical towards Saudi-Arabia by following the notions
of the SPECS approach (?). We use the same creative tripod framework to adapt
their definition into our similar task of creating movie titles with food
puns. This definition provides us with a reasoned way of conducting evaluation
of the overall creativity of our systems.

The creative tripod requires three key notions to be present in a system in
order for it to achieve creativity. These are skill, imagination and
appreciation. All of these components must be present simultaneously in a
creative system, or the system will lack creativity.

For our systems to exhibit skill, they will need to take a movie title as
input and produce a new one with a food pun. As in the case of the earlier
master-apprentice approach, the new humorous title should still communicate
the original title, i.e. the original name of the movie should be
recognizable.

Requirements for a pun are that it reassembles the original word in
pronunciation and that it is humorous. According to (?), incongruity results
in humour if it is delivered in a playful fashion and accompanied by its
resolution. Another, maybe a bit more concrete way of looking at humour, is
seeing incongruity as a surprise and resolution as coherence c.f. (?).

Surprise in the context of humor means that the brain forms an expectation and
this expectation is then broken by the humorous element of the pun. Such is
the case in the pun Harry Potter and the Deathly Marshmallows where the
surprise is caused by the fact that the expected word Hallows is replaced by
Marshmallows. For the pun to be coherent, it should make sense in the context
of the original movie. In this case, a thought of deathly marshmallows
attacking the Hogwarts, although bizarre, can still be seen as coherent.

For achieving appreciation, the systems will need to be able to assess the
humorousness of the created pun in terms of surprise, coherence and sound
similarity. In addition to the humour, the system should be able to evaluate
the recognizability of the original title.

We define imagination by using the dichotomy of creativity introduced by (?).
This way of understanding creativity divides it in two different types:
P-creativity and H-creativity. P-creativity is the minimal requirement we set
for imagination of the systems, and it means that a creative entity should be
able to come up with something that is novel to itself. H-creativity, on the
other hand, refers to an innovation that is novel in a more global scale, i.e.
nobody else has come up with a similar creative artifact before. While
P-creativity is the minimum requirement, we consider H-creativity as a more
desired requirement for imagination.

## The Data

As our approach is to generate food related puns, we need a vocabulary
consisting of food related terms. For this purpose we use the Historical
Thesaurus of the Oxford English Dictionary111http://www.oed.com/thesaurus. We
use all the nouns recorded under the topic food and drink in the external
world taxonomy. This list contains 15,314 different nouns.

We extract real movie titles from the IMDB222Dumps from
https://datasets.imdbws.com/ (Internet movie database). As we want our movie
title corpus to consist only of well known movies, we want to filter out all
the less known indie movies. To achieve this, we filter out movies that have
received less than 100,000 votes, leaving us with 1,661 movie titles. For the
master and apprentices this is further limited to 1276 titles by filtering out
the titles that consisted only of one word.

For parallel humorous movie title data (later peer data), we crawl comments on
an Instagram post for an entertainment
account333https://www.instagram.com/p/BsWki-PFbMO/. People were encouraged to
come up with creative movie titles containing a pun related to food. The total
number of comments crawled is 16,088444Crawled on the second of February.
Then, we follow the same approach applied in (?) to map the crawled data to
movie titles. In summary, we preprocess the text to remove any hashtags and
mentions, and then we measure the character and word edit distances between
the comments and movie titles. Finally, a comment is considered to be a punny
variation of the matched movie title with the least edit distance, only if it
had at most three word differences while ensuring that there exist at least
one word matching the movie title. This process yields 9,294 human-authored
movie titles containing a pun.

## The Master-Apprentice Model

The master-apprentice model consists of a computationally creative genetic
algorithm that implements the criteria set for appreciation in its fitness
function and an apprentice that is an NMT (neural machine translation) model.
The master generates parallel data for the apprentice to learn from, while the
apprentice can also learn from its peers. In our setting, we have four
different apprentices; one for each parenting style.

### Master

Inspired by the work on slogan generation presented by ? (?), we employ a
similar generator to act as a master in our model. In our case, the generator,
which is a genetic algorithm, receives an original movie title as input and
outputs an entire population of movie titles carrying a pun, based on the
input movie title. The master makes use of the food related vocabulary
described earlier to replace words in the original title while optimizing
multiple parameters to increase the aptness of the substitution and the
punniness of the title. The following subsections elucidate the algorithm.

#### Evolutionary algorithm

The first step in the evolutionary algorithm is producing the initial
population, which will go through the process of evolution during a certain
number of generations. The evolutionary algorithm employed is a standard
($\mu$ \+ $\lambda$)555We set both to 100 empirically. where mutation and
crossover are applied to the current population to produce $\lambda$
offspring. Individuals in the current population and their offspring are then
evaluated by the algorithm to find the fittest $\mu$ number of individuals to
survive to the next generation. Once the specified number of generations (10,
in our case) is reached, the evolutionary process ends and returns the final
population.

#### Initial Population

The initial population consists of $\mu$ copies of the input movie title. For
each copy, a randomly selected noun, adjective or verb is replaced with a
random word from the vocabulary. We used Spacy (?) to parse titles. We inflect
the substituting words using Pattern (?) to match the morphology of the
original word when needed. The altered titles assemble the initial population.

#### Mutation and Crossover

In our evolutionary algorithm, we implement one kind of mutation and
crossover. The mutation process substitutes words in the individual in the
same fashion as done in the creation of the initial population. The crossover
employed is a standard single-point crossover, i.e. a random point in
individuals is selected and words to the right of the point are switched
between them.

#### Evaluation

In our evaluation metric, we propose four internal evaluation dimensions to
measure the fitness of an individual. These dimensions are (1) prosody, (2)
semantic similarity to “food”, (3) semantic similarity to the original word,
and (4) number of altered words. The first two dimensions are maximized,
whereas the last two are minimized.

The prosody dimension is a weighted sum of four prosody sub-features, which
are consonance, assonance, rhyme and alliteration. This dimension measures the
sound similarity between the original word and its substitution. To measure
the sound similarity, we use espeak-ng tool666https://github.com/espeak-
ng/espeak-ng to generate IPA (international phonetic alphabet) transcriptions
for assessing the prosody.

To measure the semantic similarity between two words, we employ a pre-trained
Glove model777https://nlp.stanford.edu/projects/glove/ with 6 billion tokens
and a dimension size of 300. The model is trained on Wikipedia and English
Gigaword Fifth Edition corpus. Using the semantic model, the next dimension
computes the maximum semantic similarity of words in the title to the word
“food”.

The third dimension measures the mean of the semantic similarity of new words
to their original corresponding word. We minimize this dimension to increase
surprise, with the idea that a lower semantic similarity between the original
word and its substitute would result in a bigger surprise.

The last dimension keeps track of the number of words modified in comparison
to the original title. Minimizing this dimension motivates that less
substitutions are made to the title, which makes it more recognizable.

These are the criteria based on which the fitness of individuals is evaluated
at the end of each generation to let only the best ones survive to the next
generation.

#### Selection and Filtering

To reduce having a dominating dimension and motivate generating titles with
diverse and balanced scores on all four dimensions, we opt for a non-dominant
sorting algorithm –NSGA-II– (?) as the selection algorithm.

During each iteration of the evolution, the current population and its
offspring go through a filtering phase which filters out any duplicate titles.

#### Final Verdict

On top of individual evaluation metrics, we introduce master’s final verdict,
which is a way of telling whether the master likes the generated title. The
final verdict of the master is a binary decision, i.e. an individual is either
good or not. In practice, the final verdict is defined as conditional
thresholds on each dimension. These thresholds are 1) a positive non-zero
value for prosody, 2) a positive non-zero semantic similarity to “food”, 3) a
semantic similarity less than 0.5 of the new word to its original and 4) not
more than 50 percent change of content words.

The master uses this functionality to express its liking to titles outside of
its own creations such as those created by the apprentice. Whenever we talk
about the master liking something in this paper, we mean that the final
verdict has a Boolean value of true.

### Apprentice

For the apprentices we use OpenNMT (?), which implements an RNN based sequence
to sequence model. The model has two RNN encoding layers and two RNN decoding
layers.

The attention mechanism is the general global attention formulated by (?). The
difference to the OpenNMT default parameters in our system is that we use the
copy attention mechanism which makes it possible for the model to copy words
from the source. This is useful since the task is to translate within the same
language.

All of the apprentice models described in this paper have been trained by
using the same random seed to make their intercomparison possible.

### Different Parenting Styles

We model computationally the four different parenting styles, authoritarian,
authoritative, permissive and rejecting-neglecting, in the way the master
interacts with the apprentice during the training process of the NMT model.

The training process is done iteratively. In each iteration, the apprentice is
trained for 1000 training steps. After each iteration, the apprentice produces
an output based on the 1276 popular IMDB movie titles. This output is then
evaluated by the master accordingly to the parenting style in question and
adjustments are made to the training data based on the master’s parenting. The
apprentices are trained for 20 iterations.

#### The Authoritarian Master

only lets the apprentice learn from its own output. The apprentice is not
exposed to any of the peer data and the apprentice’s own creations are not
taken into account.

#### The Authoritative Master

lets the apprentice learn from its own creations and those peers who it
considers good enough by the final verdict (this means 2446 titles). The
apprentice can show its creations to the master after each training iteration,
out of which the master picks the ones it likes and adds them to the training
material of the apprentice. The training of the NMT model continues with the
modified corpus.

#### The Permissive Master

lets the apprentice learn from its own creations and all of the peer data.
When the apprentice presents its own creations at the end of a training
iteration, the master praises them all and adds them to the training data.

#### The Rejecting-Neglecting Master

does not care about the apprentice. The apprentice has no choice but to learn
from its peers. The apprentice does not learn from its own creations because
it receives no support from the master.

### Training the Apprentice

The master is run once to create its own movie titles with food related puns.
This parallel data of 8306 titles is shared across the different parenting
styles. During the training process of the apprentice, the master does not
generate new titles of its own, but only interferes in the selection of the
parallel data used in the next training iteration as described in the sections
above.

After each iteration, we calculate BLEU score (?) and a uni-gram PINC score
(?) for the outputs of the apprentices. We compare the outputs both to the
training material coming from the master and the material from the peers. For
each title generated by the apprentice, we take the maximum BLEU and minimum
PINC score and take an average of them for each iteration.

BLEU score is traditionally used in machine translation to evaluate how good
the final translation is in terms of a gold standard. We, however, do not use
BLEU as a final evaluation metric, but rather use it to shed some light into
how closely the outputs of the apprentices resemble those of the master or the
peer written titles. BLEU measures the similarity, whereas PINC measures
divergence from the original data. In other words, the higher the BLEU, the
more closely the apprentice imitates and the higher the PINC the less it
imitates the master or the peers.

Figure 2:  BLEU when comparing to the master Figure 3:  PINC when comparing to
the master

As indicated by Figure 2, the authoritarian scenario, where the training data
consists only of the master’s output, starts quickly producing the output most
similar to the master. Where as the authoritative scenario leads to a bit less
similarity to the master. The effect of the peer data is very well visible in
the permissive and neglecting scenarios. The PINC scores in Figure 3 show the
other side of the coin where the authoritative and authoritarian scenarios are
the least divergent and the permissive and neglecting ones the most divergent.

Figure 4:  BLEU when comparing to peers Figure 5:  PINC when comparing to
peers

When we do the BLEU comparison to the peer data as seen in Figure 4, we can
see that only the neglecting scenario leads to high similarity with the peers,
where as the other scenarios are still quite low, the lowest being the
authoritarian scenario. The PINC scores tell a similar story in Figure 5,
where the neglecting scenario leads to the least amount of divergence, leaving
the authoritarian scenario the most divergent.

## Results and Evaluation

In this section, we show some of the results produced by the different
systems. In addition, we evaluate the different parenting style scenarios
after each iteration with the master’s appreciation function. Later, an
evaluation is conducted by humans.

### Results and Master’s Liking

Results from the approaches can be seen in Table 1. The master did not produce
any training data for the last two titles in the examples. Looking at these
results qualitatively, in broad lines, the permissive and neglecting scenarios
produced worse output than the ones exposed to the master’s training data. The
apprentice exposed to authoritarian parenting struggles in producing output
for titles not present in the training data. The authoritative scenario leads
to the most consistent results. The quantitative human evaluation in the next
section is used to verify these initial observations.

original | master | authoritarian | authoritative | permissive | neglecting
---|---|---|---|---|---
the butterfly effect | the brewery effect | the butterfly kimchi | the butterfly chicken | the butterfly effect |  | the lasagna
---
effect lazarus
| how to train
---
your dragon
| how to train
---
your pepperoni
| how to train
---
your avocado
| how to train
---
your pepperoni
| how to train
---
your bacon
| how to train
---
your bacon
| fantastic beasts and
---
where to find them
—- |  | fantastic ordinary
---
and where to find
| fantastic beets and
---
where to find them
| fantastic beefs and
---
where to find them
| fantastic beets and
---
where to find them
under the skin | —- | under the cereals | under the silver cake | under the 13th | fryday the 13

Table 1: Examples of the final output of the different models

Another way to look at the results is to use the appreciation metrics
implemented in the master. Figure 6 shows the percentage of how many titles
the master liked after each training iteration.

Figure 6:  Master’s liking of the output

As we can see, the appreciation the master has ranks the authoritarian and
authoritative scenarios higher than the permissive and neglecting ones. Even
in the authoritarian case, the master does not like all of the output produced
by the apprentice, which shows that the appreciation learned by the
apprentices is different from the one implemented in the master.

It is interesting to see to what extent the master’s liking correlates with
the evaluation results of the human judges. This can reveal more information
about the adequacy of the appreciation of the master in this creative task. Or
does the master’s appreciation only tell about obedience when applied to the
apprentices’ output?

### Evaluation Questions

In this section we provide some reasoning in our selection of the evaluation
questions that are presented to the human judges. Earlier, we defined the
creativity in the case of pun generation using the creative tripod as our
theoretical framework. This means that on a higher level, our evaluation
questions should evaluate skill, appreciation and imagination.

#### Skill

Our definition for skill stated that the system should be able to take an
existing movie title and produce a food related pun as an output. A further
requirement was that the original title should be recognizable from the
generated one.

  1. 1.

The title has a pun in it

  2. 2.

The title is related to food

  3. 3.

The original title is recognizable

The evaluation questions described above are designed to evaluate the
requirements set for skill. We evaluate whether a pun is perceived and whether
the new title relates to food separately, as it might be that the replacement
word delivers a pun, but is not food related or vice-versa.

#### Appreciation

We defined appreciation from the humor stand point. A good title with a pun is
also funny. For something to be funny, i.e. humorous, the pun has to exhibit
coherence and surprise.

  1. 4.

The title is humorous

  2. 5.

The pun is surprising

  3. 6.

The pun makes sense in the context of the original movie

We choose to evaluate the overall humor value of the title separately from the
components that constitute it. The last two questions are designed to evaluate
surprise and coherence respectively.

#### Imagination

We used Boden’s dichotomy to establish the definition of imagination. The
minimal requirement was set to P-creativity. However, P-creativity can easily
be verified by looking at the training data and the final output, if the
output is different from the training material, there is P-creativity.
Therefore, we use human judges to assess the H-creativity of the outputs.

  1. 7.

The pun in the title is obvious

  2. 8.

The pun in the title sounds familiar

If the pun is obvious, it probably is not too H-creative, as an obvious pun
could be said by just about anyone, also if the pun sounds familiar, it has
probably been said by someone before.

### Human Evaluation

We take a random sample of 20 original movie titles that were only present in
the training data provided by the master, 20 titles that were only present in
the peer data and 20 titles that were in both sources of parallel data. We
evaluate the creative output of each apprentice for these randomly sampled
titles. In addition, we evaluate the master’s output for the 40 titles of the
sample it had generated movie title puns for. As the master has generated
multiple creative titles per original title, we pick one randomly for each
original title. Altogether, we are evaluating 280 computer created titles.

Style | Training | Q1 | Q2 | Q3 | Q4 | Q5 | Q6 | Q7 | Q8
---|---|---|---|---|---|---|---|---|---
|  | $\mu_{x}$ | $SD$ | $\mu_{x}$ | $SD$ | $\mu_{x}$ | $SD$ | $\mu_{x}$ | $SD$ | $\mu_{x}$ | $SD$ | $\mu_{x}$ | $SD$ | $\mu_{x}$ | $SD$ | $\mu_{x}$ | $SD$
authoritarian | both | 3.35 | 1.08 | 2.65 | 1.26 | 3.33 | 1.06 | 3.02 | 1.10 | 3.08 | 0.99 | 3.03 | 1.01 | 3.16 | 0.99 | 3.10 | 1.03
authoritarian | peer only | 2.97 | 1.18 | 2.14 | 1.17 | 3.44 | 1.13 | 2.66 | 1.15 | 2.82 | 1.08 | 3.10 | 1.11 | 3.03 | 1.12 | 3.11 | 1.13
authoritarian | master only | 3.34 | 1.07 | 2.89 | 1.28 | 3.30 | 1.05 | 3.00 | 1.12 | 3.07 | 1.05 | 3.04 | 1.02 | 3.12 | 1.02 | 3.07 | 1.04
authoritative | both | 3.43 | 1.08 | 3.09 | 1.31 | 3.28 | 1.12 | 3.08 | 1.16 | 3.08 | 1.05 | 3.02 | 1.05 | 3.22 | 1.06 | 3.13 | 1.05
authoritative | peer only | 3.41 | 1.14 | 3.23 | 1.35 | 3.37 | 1.21 | 3.13 | 1.16 | 3.07 | 1.08 | 3.08 | 1.09 | 3.24 | 1.10 | 3.16 | 1.13
authoritative | master only | 3.47 | 1.03 | 3.15 | 1.33 | 3.41 | 1.08 | 3.17 | 1.11 | 3.16 | 1.02 | 3.16 | 1.04 | 3.28 | 1.01 | 3.24 | 1.06
master | both | 3.38 | 1.07 | 2.79 | 1.28 | 3.29 | 1.06 | 3.07 | 1.12 | 3.09 | 1.04 | 3.10 | 1.06 | 3.22 | 1.02 | 3.14 | 1.05
master | master only | 3.40 | 1.07 | 2.61 | 1.30 | 3.34 | 1.11 | 3.10 | 1.15 | 3.09 | 1.02 | 3.04 | 1.03 | 3.17 | 1.04 | 3.11 | 1.06
neglecting | both | 3.45 | 1.11 | 3.28 | 1.32 | 3.34 | 1.11 | 3.28 | 1.12 | 3.16 | 1.06 | 3.12 | 1.04 | 3.21 | 1.08 | 3.22 | 1.07
neglecting | peer only | 3.36 | 1.07 | 3.02 | 1.37 | 3.31 | 1.11 | 3.12 | 1.15 | 3.09 | 1.02 | 3.14 | 1.04 | 3.19 | 1.02 | 3.14 | 1.06
neglecting | master only | 3.28 | 1.13 | 2.87 | 1.35 | 3.34 | 1.12 | 3.09 | 1.14 | 3.05 | 1.05 | 3.07 | 1.06 | 3.15 | 1.06 | 3.18 | 1.08
permissive | both | 3.23 | 1.18 | 2.67 | 1.38 | 3.59 | 1.06 | 3.06 | 1.13 | 3.08 | 1.08 | 3.30 | 1.04 | 3.21 | 1.07 | 3.30 | 1.10
permissive | peer only | 3.05 | 1.19 | 2.87 | 1.39 | 3.25 | 1.18 | 2.88 | 1.13 | 2.88 | 1.09 | 3.00 | 1.08 | 2.99 | 1.11 | 3.04 | 1.14
permissive | master only | 3.09 | 1.23 | 2.32 | 1.24 | 3.64 | 1.11 | 2.88 | 1.15 | 2.91 | 1.12 | 3.07 | 1.15 | 2.98 | 1.13 | 3.04 | 1.12

Table 2: Mean and standard deviation.

Style | Q1 | Q2 | Q3 | Q4 | Q5 | Q6 | Q7 | Q8
---|---|---|---|---|---|---|---|---
authoritarian | 78.33% | 31.67% | 83.33% | 31.67% | 48.33% | 68.33% | 70.00% | 68.33%
authoritative | 93.33% | 60.00% | 83.33% | 56.67% | 63.33% | 66.67% | 90.00% | 70.00%
master | 92.50% | 37.50% | 87.50% | 62.50% | 60.00% | 65.00% | 80.00% | 67.50%
neglecting | 86.67% | 60.00% | 81.67% | 66.67% | 65.00% | 73.33% | 75.00% | 78.33%
permissive | 66.67% | 33.33% | 85.00% | 43.33% | 46.67% | 73.33% | 68.33% | 71.67%

Table 3: Percentage of movie titles having an average score by judges greater
than 3

The evaluation was conducted on a crowd-sourcing platform called Figure
Eight888https://www.figure-eight.com/. The platform assigned people to conduct
evaluation in such a way that each title was evaluated by 35 different users.
The users could choose how many titles they wanted to evaluate. The results of
the evaluation are show in Table 2. In the Training column, both, peer only
and master only indicate whether the original title was only present in the
master produced training data, peer produced training data or in both
respectively.

The authoritarian scenario didn’t get the best average score for any of the
test questions and neither did the master. They both score particularly low on
the Q2, which reflects the fact that some of the words in the HTOED food and
drink taxonomy were only loosely related to food such as steam and spit. It is
interesting to note that the authoritarian scenario gets the best results for
Q3, Q6 and Q7 for titles it did not encounter in the training data, in other
words it has developed an appreciation of its own that does not just mimic
what the master produces and fail otherwise. In light of these results, we can
deduce that the master produced worse titles with food puns than real people,
which left both the master and the authoritarian scenario without the first
place on any of the test questions.

The authoritative scenario, which was the highest ranking one according to
master’s liking as seen in Figure 6, got the best results for Q1 and Q5. This
means that it succeeds the best in the main task of generating puns and they
end up being the most surprising ones. It is also the only one that produces
consistently good results (above 3 on the average) for all training test sets
for Q1-Q6, unfortunately the results for Q7 and Q8 are also above 3 on the
average meaning that it does not rank high on H-creativity.

The same consistency can not be perceived in the the permissive case as scores
below 3 are common across the test questions. It however, manages to score the
best for Q3 and Q7-Q8, in other words, it can achieve the best H-creativity
and the original titles can be the most recognizable, although not
consistently so. This shows, that even though the appreciation the master has
might not be spot on, as it is not able to produce the best scoring titles,
having moderation on the peer data and critical assessment of the apprentice
generated results during the training by the master, has a positive effect on
the consistency of the results. In the permissive scenario, the apprentice was
exposed to everything without criticism and in the authoritative some
criticism was used to filer the training data, which made the authoritative
scenario more consistent, but less H-creative.

Finally, the neglecting scenario gets the best scores for the Q2, Q4 and Q5.
It is the best one at producing humorous, surprising and food related titles.
It is quite consistent with only the results for Q2 with previously unseen
titles giving a score that is inferior to 3. The good results of the
neglecting scenario serve as an additional proof to the fact that the output
of the master is worse than human written titles.

Table 3 shows the results form another stand point. The table shows overall
how many titles got the average rating above 3 for each test question. These
numbers are in line with what was previously discussed about the Table 2. The
authoritarian scenario leads to the worst performance, but this time master
gets the highest percentage point of titles above 3 for Q3. In the
authoritative scenario most of the titles have a clear pun and are related to
food with the highest percentage point. The permissive scenario holds the best
percentage points for Q6 and Q7. And the neglecting gets the best percentage
points in Q2, Q4, Q5 and Q6.

## Discussion and Future Work

The evaluation results were not completely in line with what we can observe by
looking at the titles output by the different methods by ourselves. This
raises the question whether our definition for creativity in movie title puns
is adequate and whether the evaluation questions we formulated based on the
definition really measure what they were designed to measure. Because we have
worked with a clear definition for creativity in this paper, it is possible to
take this under a critical study in the future. We also find evident that
qualitative research on the output titles with respect to the quantitative
results we got from the human judges is needed to evaluate the evaluation
itself.

Having a master with appreciation filter the parallel data of the apprentice
was beneficial for consistency (see authoritative vs permissive). Although the
evaluation results showed that the appreciation is not in par with that of a
real human, the implication remains that a good external appreciation can be
beneficial for the learning outcome of the apprentice model. As we used a
rather generic NMT model for the apprentice, our findings might be of a use in
more traditional context of sequence-to-sequence models such as machine
translation, text summarization or paraphrasing.

For now, the master and apprentice have been studied in a social vacuum, where
peer data is the only link to the surrounding world. However, in the future it
would be fruitful to see how the creative outcome changes when the master and
the apprentice are exposed to a more complex social system such as the one
described by Bronfenbrenner (?). In such a society, the master would also be
under a social pressure in changing its own standards of appreciation.

## Conclusions

This work has presented one of the first contributions to the field of
computational social creativity where the computationally creative agents are
in a hierarchical social relation. This asymmetry offers an intriguing setting
for studying socialization of computational agents from the creativity
perspective.

Despite building our definition of creativity upon an existing theory and
formulating the test questions based on the definition, the quantitative
evaluation left many questions unanswered. The results presented in this paper
call for qualitative evaluation to understand the phenomenon of evaluation in
this particular context.

Nevertheless, our findings suggest that having appreciation in parenting, or
training, an NMT model can be of a benefit. The applicability of these finding
into sequence-to-sequence deep learning models in a more generalized fashion
is an interesting research question on its own right.

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00footnotetext: $\mathbf{2000}$ Mathematics Subject Classification, 53C15,
53C25.
Key words and phrases : Sasakian manifold, locally $\phi$-symmetric,
$\phi$-semisymmetric, Ricci $\phi$-semisymmetric, projectively
$\phi$-semisymmetric, conformally $\phi$-semisymmetric, manifold of constant
curvature, $B$-tensor, $B$-$\phi$-semisymmetric.

#  ON LOCALLY $\phi$-SEMISYMMETRIC SASAKIAN MANIFOLDS

Absos Ali Shaikh and Helaluddin Ahmad
Department of Mathematics,
University of Burdwan, Golapbag,
Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Generalizing the notion of local $\phi$-symmetry of Takahashi [20], in the
present paper, we introduce the notion of local $\phi$-semisymmetry of a
Sasakian manifold along with its proper existence and characterization. We
also study the notion of local Ricci (resp., projective, conformal)
$\phi$-semisymmetry of a Sasakian manifold and obtain its characterization. It
is shown that the local $\phi$-semisymmetry, local projective
$\phi$-semisymmetry and local concircular $\phi$-semisymmetry are equivalent.
It is also shown that local conformal $\phi$-semisymmetry and local
conharmonical $\phi$-semisymmetry are equivalent.

Typeset by AmS-LaTeX

##  1\. Introduction

Let $M$ be an $n$-dimensional, $n\geq 3$, connected smooth Riemannian manifold
endowed with the Riemannian metric $g$. Let $\nabla$, $R$, $S$ and $r$ be the
Levi-Civita connection, curvature tensor, Ricci tensor and the scalar
curvature of $M$ respectively. The manifold $M$ is called locally symmetric
due to Cartan ([2], [3]) if the local geodesic symmetry at $p\in M$ is an
isometry, which is equivalent to the fact that $\nabla R=0$. Generalizing the
concept of local symmetry, the notion of semisymmetry was introduced by Cartan
[4] and fully classified by Szabó ([17], [18], [19]). The manifold $M$ is said
to be semisymmetric if

$(R(U,V).R)(X,Y)Z=0$

for all vector fields $X$, $Y$, $Z$, $U$, $V$ on $M$, where $R(U,V)$ is
considered as the derivation of the tensor algebra at each point of $M$. Every
locally symmetric manifold is semisymmetric but the converse is not true, in
general. However, the converse is true only for $n=3$. As a weaker version of
local symmetry, in 1977 Takahashi [20] introduced the notion of local
$\phi$-symmetry on a Sasakian manifold. A Sasakian manifold is said to be
locally $\phi$-symmetric if

$\phi^{2}((\nabla_{W}R)(X,Y)Z)=0$

for all horizontal vector fields $X$, $Y$, $Z$, $W$ on $M$, where $\phi$ is
the structure tensor of the manifold $M$. The concept of local $\phi$-symmetry
on various structures and their generalizations or extensions are studied in
[6], [8], [9], [10], [11], [12], [13], [14], [15]. By extending the notion of
semisymmetry and generalizing the concept of local $\phi$-symmetry of
Takahashi [20], in the present paper, we introduce the notion of local
$\phi$-semisymmetry on a Sasakian manifold. A Sasakian manifold $M$, $n\geq
3$, is said to be locally $\phi$-semisymmetric if

$\phi^{2}((R(U,V).R)(X,Y)Z)=0$

for all horizontal vector fields $X$, $Y$, $Z$, $U$, $V$ on $M$. We note that
every locally $\phi$-symmetric as well as semisymmetric Sasakian manifold is
locally $\phi$-semisymmetric but not conversely. The object of the present
paper is to study the geometric properties of a locally $\phi$-semisymmetric
Sasakian manifold along with its proper existence and characterization. The
paper is organized as follows. Section 2 deals with the rudiments of Sasakian
manifolds. By extending the definition of local $\phi$-symmetry, in Section 3,
we derive the defining condition of a locally $\phi$-semisymmetric Sasakian
manifold and proved that a Sasakian manifold is locally $\phi$-semisymmetric
if and only if each Kählerian manifold, which is a base space of a local
fibering, is Hermitian locally semisymmetric. We cite an example of a locally
$\phi$-semisymmetric Sasakian manifold which is not locally $\phi$-symmetric.
We also obtain a characterization of locally $\phi$-semisymmetric Sasakian
manifold by considering the horizontal vector fields. Section 4 is devoted to
the characterization of locally $\phi$-semisymmetric Sasakian manifold for
arbitrary vector fields. As the generalization of Ricci (resp., projectively,
conformally) semisymmetric Sasakian manifold, in the last section, we
introduce the notion of locally Ricci (resp., projectively, conformally)
$\phi$-semisymmetric Sasakian manifold and obtain the characterization of such
notions. Recently Shaikh and Kundu [16] defined a generalized curvature
tensor, called $B$-tensor, by the linear combination of $R$, $S$ and $g$ which
includes various curvature tensors as particular cases. We study the
characterization of locally $B$-$\phi$-semisymmetric Sasakian manifolds. It is
shown that local $\phi$-semisymmetry, local projective $\phi$-semisymmetry and
local concircular $\phi$-semisymmetry are equivalent and hence they are of the
same characterization. Also it is proved that local conformal
$\phi$-semisymmetry and local conharmonical $\phi$-semisymmetry are
equivalent. Finally, we conclude that the study of local $\phi$-semisymmetry
and local conformal $\phi$-semisymmetry are meaningful as they are not
equivalent. However, the study of local $\phi$-semisymmetry with any other
generalized curvature tensor of type (1,3) (which are the linear combination
of $R$, $S$ and $g$) is either meaningless or redundant due to their
equivalency.

##  2\. Sasakian manifolds

An $n(=2m+1,m\geq 1)$-dimensional $C^{\infty}$ manifold $M$ is said to be a
contact manifold if it carries a global 1-form $\eta$ such that
$\eta\wedge(d\eta)^{m}\neq 0$ everywhere on $M$. Given a contact form $\eta$,
it is well-known that there exists a unique vector field $\xi$, called the
characterstic vector field of $\eta$, satisfying $\eta(\xi)=1$ and
$d\eta(X,\xi)=0$ for any vector field $X$ on $M$. A Riemannian metric $g$ is
said to be an associated metric if there exists a tensor field $\phi$ of type
(1,1) such that

(2.1) $\phi^{2}=-I+\eta\otimes\xi,\ \ \eta(\cdot)=g(\cdot,\xi),\ \ \
d\eta(\cdot,\cdot)=g(\cdot,\phi\cdot).$

Then the structure $(\phi,\xi,\eta,g)$ on $M$ is called a contact metric
stucture and the manifold $M$ equipped with such a stucture is called a
contact metric manifold [1].
From (2.1) it is easy to check that the following holds:

(2.2) $\displaystyle\phi\xi=0,\ \ \ \eta\circ\phi=0,\ \ \
g(\phi\cdot,\cdot)=-g(\cdot,\phi\cdot),$ (2.3) $\displaystyle
g(\phi\cdot,\phi\cdot)=g(\cdot,\cdot)-\eta\otimes\eta.$

Given a contact metric manifold $M$ there is an (1,1) tensor field $h$ given
by $h=\frac{1}{2}\pounds_{\xi}\phi$, where $\pounds$ denotes the operator of
Lie differentiation. Then $h$ is symmetric. The vector field $\xi$ is a
Killing vector field with respect to $g$ if and only if $h=0$. A contact
metric manifold $M$ for which $\xi$ is a Killing vector is said to be a
$K$-contact manifold. A contact structure on $M$ gives rise to an almost
complex structure $J$ on the product $M\times\mathbb{R}$ defined by

$J\Big{(}X,f\frac{d}{dt}\Big{)}=\Big{(}\phi
X-f\xi,\eta(X)\frac{d}{dt}\Big{)},$

where $f$ is a real valued function, is integrable, then the structure is said
to be normal and the manifold $M$ is a Sasakian manifold. Equivalently, a
contact metric manifold is Sasakian if and only if

(2.4) $R(X,Y)\xi=\eta(Y)X-\eta(X)Y$

holds for all $X$, $Y$ on $M$.
In an $n$-dimensional Sasakian manifold $M$ the following relations hold ([1],
[22]):

(2.5) $\displaystyle
R(\xi,X)Y=(\nabla_{X}\phi)(Y)=g(X,Y)\xi-\eta(Y)X=-R(X,\xi)Y,$ (2.6)
$\displaystyle\nabla_{X}\xi=-\phi X,\ \ \ (\nabla_{X}\eta)(Y)=g(X,\phi Y),$
(2.7) $\displaystyle\eta(R(X,Y)Z)=g(Y,Z)\eta(X)-g(X,Z)\eta(Y),$ (2.8)
$\displaystyle(\nabla_{W}R)(X,Y)\xi=g(W,\phi Y)X-g(W,\phi X)Y+R(X,Y)\phi W,$
(2.9) $\displaystyle(\nabla_{W}R)(X,\xi)Z=g(X,Z)\phi W-g(Z,\phi W)X+R(X,\phi
W)Z,$ (2.10) $\displaystyle S(X,\xi)=(n-1)\eta(X),\ \ S(\xi,\xi)=(n-1)$

for all vector fields $X$, $Y$, $Z$ and $W$ on $M$. In a Sasakian manifold,
for any $X$, $Y$, $Z$ on $M$, we also have [21]

(2.11) $R(X,Y)\phi W=g(W,\phi X)Y-g(W,Y)\phi X-g(W,\phi Y)X+g(W,X)\phi Y+\phi
R(X,Y)W.$

From (2.8) and (2.11), it follows that

(2.12) $(\nabla_{W}R)(X,Y)\xi=g(W,X)\phi Y-g(W,Y)\phi X+\phi R(X,Y)W.$

##  3\. Locally $\phi$-Semisymmetric Sasakian Manifolds

Let $M$ be an $n(=2m+1,m\geq 1)$-dimensional Sasakian manifold endowed with
the structure $(\phi,\xi,\eta,g)$. Let $\tilde{U}$ be an open neighbourhood of
$x\in M$ such that the induced Sasakian structure on $\tilde{U}$, denoted by
the same letters, is regular. Let $\pi:\tilde{U}\rightarrow N=\tilde{U}/\xi$
be a (local) fibering and let $(J,\bar{g})$ be the induced Kählerian structure
on $N$ [7]. Let $R$ and $\bar{R}$ be the curvature tensors constructed by $g$
and $\bar{g}$ respectively. For a vector field $\bar{X}$ on $N$, we denote its
horizontal lift (with respect to the connection form $\eta$) by $\bar{X}^{*}$.
Then we have, for any vector fields $\bar{X}$, $\bar{Y}$ and $\bar{Z}$ on $N$,

(3.1)
$(\bar{\nabla}_{\bar{X}}{\bar{Y}})^{*}=\nabla_{{\bar{X}}^{*}}{\bar{Y}}^{*}-\eta(\nabla_{{\bar{X}}^{*}}{\bar{Y}}^{*})\xi,$
(3.2)
$(\bar{R}(\bar{X},\bar{Y})\bar{Z})^{*}=R(\bar{X}^{*},\bar{Y}^{*})\bar{Z}^{*}+g(\phi{\bar{Y}^{*}},\bar{Z}^{*})\phi{\bar{X}^{*}}-g(\phi{\bar{X}^{*}},\bar{Z}^{*})\phi{\bar{Y}^{*}}-2g(\phi{\bar{X}^{*}},\bar{Y}^{*})\phi{\bar{Z}^{*}},$
(3.3)
$(({\bar{\nabla}}_{\bar{V}}\bar{R})(\bar{X},\bar{Y})\bar{Z})^{*}=-\phi^{2}[(\nabla_{\bar{V}^{*}}R)(\bar{X}^{*},\bar{Y}^{*})\bar{Z}^{*}]$

where $\bar{\nabla}$ is the Levi-Civita connection for $\bar{g}$. The
relations (3.1) and (3.2) are due to Ogiue [7] and the relation (3.3) is due
to Takahashi [20].
Making use of (2.1), (2.4)-(2.11) and (3.1)-(3.3), we get by straightforward
calculation

(3.4)
$((\bar{R}(\bar{U},\bar{V})\cdot\bar{R})(\bar{X},\bar{Y})\bar{Z})^{*}=-\phi^{2}[(R(\bar{U}^{*},\bar{V}^{*})\cdot
R)(\bar{X}^{*},\bar{Y}^{*})\bar{Z}^{*}]$

for any vector fields $\bar{X},\bar{Y},\bar{Z},\bar{U}$ and $\bar{V}$ on $N$,
where $R(U,V)$ is considered as the derivation of the tensor algebra at each
point of $N$. Hence from (3.4) it is natural to define the following:

######  Definition 3.1.

A Sasakian manifold is said to be a locally $\phi$-semisymmetric if

(3.5) $\phi^{2}[(R(U,V)\cdot R)(X,Y)Z]=0$

for any horizontal vector fields $X,Y,Z,U$ and $V$ on $M$, where a horizontal
vector is a vector which is horizontal with respect to the connection form
$\eta$ of the local fibering, that is, orthogonal to $\xi$.

Thus from (3.4) and (3.5), we can state the following:

######  Theorem 3.1.

A Sasakian manifold is locally $\phi$-semisymmetric if and only if each
Kählerian manifold, which is a base space of a local fibering, is a Hermitian
locally semisymmetric space.

Example 3.1. Suppose a Sasakian manifold is not of constant $\phi$-sectional
curvature. Then the Kählerian base manifold is not of constant sectional
curvature.
Now let $R(X,\phi X,Y,\phi Y)=f\in C^{\infty}(M)$. Then $(\nabla_{V}R)(X,\phi
X,Y,\phi Y)=(Vf)\neq 0$, i.e. the Kählerian manifold is not Hermitian locally
symmetric and therefore the Sasakian manifold is not locally $\phi$-symmetric.
Now $(\nabla_{U}\nabla_{V}R)(X,\phi X,Y,\phi Y)=U(Vf)\neq 0$, which implies
that $(R(U,V)\cdot R)(X,\phi X,Y,\phi Y)=0$, i.e. the Kählerian manifold is
Hermitian locally semisymmetric.
Hence the Sasakian manifold is locally $\phi$-semisymmetric but not locally
$\phi$-symmetric.
First we suppose that $M$ is a Sasakian manifold such that

(3.6) $\phi^{2}[(R(U,V)\cdot R)(X,Y)\xi]=0$

for any horizontal vector fields $X,Y,U$ and $V$ on $M$.
Differentiating (2.12) covariantly with respect to a horizontal vector field
$U$ we get

$\displaystyle(\nabla_{U}\nabla_{V}R)(X,Y)\xi$ $\displaystyle=$
$\displaystyle\\{g(Y,U)g(X,V)-g(X,U)g(Y,V)-R(X,Y,U,V)\\}\xi$ $\displaystyle+$
$\displaystyle\phi((\nabla_{U}R)(X,Y)V).$

Alternating $U$ and $V$ on (3) we get

$\displaystyle(\nabla_{V}\nabla_{U}R)(X,Y)\xi$ $\displaystyle=$
$\displaystyle\\{g(Y,V)g(X,U)-g(X,V)g(Y,U)-R(X,Y,V,U)\\}\xi$ $\displaystyle+$
$\displaystyle\phi((\nabla_{V}R)(X,Y)U).$

From (3) and (3) it follows that

$\displaystyle(R(U,V)\cdot R)(X,Y)\xi$ $\displaystyle=$ $\displaystyle
2\\{g(Y,U)g(X,V)-g(X,U)g(Y,V)-R(X,Y,U,V)\\}\xi$ $\displaystyle+$
$\displaystyle\phi\\{(\nabla_{U}R)(X,Y)V-(\nabla_{V}R)(X,Y)U\\}.$

Again from (3.6) we have

(3.10) $(R(U,V)\cdot R)(X,Y)\xi=0.$

From (3) and (3.10) we have

$\displaystyle 2\\{g(Y,U)g(X,V)$ $\displaystyle-$ $\displaystyle
g(X,U)g(Y,V)-R(X,Y,U,V)\\}\xi$ $\displaystyle+$
$\displaystyle\phi\\{(\nabla_{U}R)(X,Y)V-(\nabla_{V}R)(X,Y)U\\}=0.$

Applying $\phi$ on (3) and using (2.11), (2.12) and (2.2) we get

(3.12) $(\nabla_{U}R)(X,Y)V-(\nabla_{V}R)(X,Y)U=0.$

In view of (3.12), (3) yields

(3.13) $R(X,Y,U,V)=g(Y,U)g(X,V)-g(X,U)g(Y,V)$

for any horizontal vector fields $X,Y,U$ and $V$ on $M$. Hence $M$ is of
constant $\phi$-holomorphic sectional curvature 1 and hence of constant
curvature 1. This leads to the following:

######  Theorem 3.2.

If a Sasakian manifold $M$ satisfies the condition $\phi^{2}[(R(U,V)\cdot
R)(X,Y)\xi]=0$ for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$, then
it is a manifold of constant curvature 1.

Now we consider a locally $\phi$-semisymmetric Sasakian manifold. Then from
(3.5) we have

$(R(U,V)\cdot R)(X,Y)Z=g((R(U,V)\cdot R)(X,Y)Z,\xi)\xi,$

from which we get

(3.14) $(R(U,V)\cdot R)(X,Y)Z=-g((R(U,V)\cdot R)(X,Y)\xi,Z)\xi$

for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$.
In view of (3), it follows from (3.14) that

(3.15) $(R(U,V)\cdot R)(X,Y)Z=[(\nabla_{U}R)(X,Y,V,\phi
Z)-(\nabla_{V}R)(X,Y,U,\phi Z)]\xi.$

Now differentiating (2.11) covariantly with respect to a horizontal vector
field $V$, we obtain

$\displaystyle(\nabla_{V}R)(X,Y)\phi Z$ $\displaystyle=$
$\displaystyle[R(X,Y,Z,V)-\\{g(Y,Z)g(X,V)+g(X,Z)g(Y,V)]\xi$ $\displaystyle+$
$\displaystyle\phi((\nabla_{V}R)(X,Y)Z).$

Taking inner product of (3) with a horizontal vector field $U$, we obtain

(3.17) $g((\nabla_{V}R)(X,Y)\phi Z,U)=-g((\nabla_{V}R)(X,Y)Z,\phi U).$

Using (3.17) in (3.15) we get

(3.18) $(R(U,V)\cdot R)(X,Y)Z=[(\nabla_{U}R)(X,Y,Z,\phi
V)-(\nabla_{V}R)(X,Y,Z,\phi U)]\xi$

for any horizontal vector fields $X,Y,Z,U$ and $V$ on $M$. Hence we can state
the following:

######  Theorem 3.3.

A necessary and sufficient condition for a Sasakian manifold $M$ to be a
locally $\phi$-semisymmetric is that it satisfies the relation (3.18) for all
horizontal vector fields on $M$.

##  4\. Characterization of a Locally $\phi$-Semisymmetric Sasakian Manifold

In this section we investigate the condition of local $\phi$-semisymmetry of a
Sasakian manifold for arbitrary vector fields on $M$. To find the condition we
need the following lemmas.

######  Lemma 4.1.

[20] For any horizontal vector fields $X,Y$ and $Z$ on $M$, we get

(4.1) $(\nabla_{\xi}R)(X,Y)Z=0.$

Now Lemma 4.1, (2.9) and (2.12) together imply the following:

######  Lemma 4.2.

[20] For any vector fields $X,Y,Z,V$ on $M$, we get

$\displaystyle(\nabla_{\phi^{2}V}R)(\phi^{2}X,\phi^{2}Y)\phi^{2}Z$
$\displaystyle=$ $\displaystyle(\nabla_{V}R)(X,Y)Z$ $\displaystyle+$
$\displaystyle\eta(X)\\{g(Y,Z)\phi V-g(\phi V,Z)Y+R(Y,\phi V)Z\\}$
$\displaystyle-$ $\displaystyle\eta(Y)\\{g(X,Z)\phi V-g(\phi V,Z)X+R(X,\phi
V)Z\\}$ $\displaystyle-$ $\displaystyle\eta(Z)\\{g(X,V)\phi Y-g(Y,V)\phi
X+\phi R(X,Y)V\\}.$

Now let $X,Y,Z,U,V$ be arbitrary vector fields on $M$. We now compute
$(R(\phi^{2}U,\phi^{2}V)\cdot R)(\phi^{2}X,\phi^{2}Y)\phi^{2}Z$ in two
different ways. Firstly, from (3.18), (2.1) and (4.2) we get

(4.3) $\displaystyle(R(\phi^{2}U,\phi^{2}V)\cdot
R)(\phi^{2}X,\phi^{2}Y)\phi^{2}Z=[\\{(\nabla_{V}R)(X,Y,Z,\phi
U)-(\nabla_{U}R)(X,Y,Z,\phi V)\\}$ $\displaystyle+\eta(X)\\{g(Y,\phi V)g(\phi
U,Z)-g(Y,\phi U)g(\phi V,Z)-R(Y,Z,\phi U,\phi V)\\}$
$\displaystyle-\eta(Y)\\{g(X,\phi V)g(\phi U,Z)-g(X,\phi U)g(\phi
V,Z)-R(X,Z,\phi U,\phi V)\\}$
$\displaystyle-2\eta(Z)\\{g(X,V)g(U,Y)-g(X,U)g(V,Y)-R(X,Y,U,V)\\}]\xi.$

Again using (2.11) in (4.3), we obtain

(4.4) $\displaystyle(R(\phi^{2}U,\phi^{2}V)\cdot
R)(\phi^{2}X,\phi^{2}Y)\phi^{2}Z=[\\{(\nabla_{V}R)(X,Y,Z,\phi
U)-(\nabla_{U}R)(X,Y,Z,\phi V)\\}$
$\displaystyle-\eta(X)H(Y,Z,U,V)+\eta(Y)H(X,Z,U,V)+2\eta(Z)H(X,Y,U,V)]\xi$

where $H(X,Y,Z,U)=g(\mathcal{H}(X,Y)Z,U)$ and the tensor field $\mathcal{H}$
of type (1,3) is given by

(4.5) $\mathcal{H}(X,Y)Z=R(X,Y)Z-g(Y,Z)X+g(X,Z)Y$

for all vector fields $X,Y,Z$ on $M$. Secondly, we have

(4.6) $\displaystyle(R(\phi^{2}U,\phi^{2}V)\cdot
R)(\phi^{2}X,\phi^{2}Y)\phi^{2}Z=R(\phi^{2}U,\phi^{2}V)R(\phi^{2}X,\phi^{2}Y)\phi^{2}Z$
$\displaystyle-R(R(\phi^{2}U,\phi^{2}V)\phi^{2}X,\phi^{2}Y)\phi^{2}Z-R(\phi^{2}X,R(\phi^{2}U,\phi^{2}V)\phi^{2}Y)\phi^{2}Z$
$\displaystyle-R(\phi^{2}X,\phi^{2}Y)R(\phi^{2}U,\phi^{2}V)\phi^{2}Z.$

By straightforward calculation, from (4.6) we get

$\displaystyle(R(\phi^{2}U,\phi^{2}V)\cdot R)(\phi^{2}X,\phi^{2}Y)\phi^{2}Z$
$\displaystyle=$ $\displaystyle-(R(U,V)\cdot R)(X,Y)Z$ $\displaystyle+$
$\displaystyle\eta(U)\big{[}H(X,Y,Z,V)\xi+\eta(X)\mathcal{H}(V,Y)Z$
$\displaystyle+$
$\displaystyle\eta(Y)\mathcal{H}(X,V)Z+\eta(Z)\mathcal{H}(X,Y)V\big{]}$
$\displaystyle-$
$\displaystyle\eta(V)\big{[}H(X,Y,Z,U)\xi+\eta(X)\mathcal{H}(U,Y)Z$
$\displaystyle+$
$\displaystyle\eta(Y)\mathcal{H}(X,U)Z+\eta(Z)\mathcal{H}(X,Y)U\big{]}.$

From (4.4) and (4) it follows that

$\displaystyle(R(U,V)\cdot R)(X,Y)Z$ $\displaystyle=$
$\displaystyle\big{[}\\{(\nabla_{U}R)(X,Y,Z,\phi V)-(\nabla_{V}R)(X,Y,Z,\phi
U)\\}$ $\displaystyle+$
$\displaystyle\eta(X)H(Y,Z,U,V)-\eta(Y)H(X,Z,U,V)-2\eta(Z)H(X,Y,U,V)$
$\displaystyle+$ $\displaystyle\eta(U)H(X,Y,Z,V)-\eta(V)H(X,Y,Z,U)\big{]}\xi$
$\displaystyle+$
$\displaystyle\eta(U)\big{[}\eta(X)\mathcal{H}(V,Y)Z+\eta(Y)\mathcal{H}(X,V)Z+\eta(Z)\mathcal{H}(X,Y)V\big{]}$
$\displaystyle-$
$\displaystyle\eta(V)\big{[}\eta(X)\mathcal{H}(U,Y)Z+\eta(Y)\mathcal{H}(X,U)Z+\eta(Z)\mathcal{H}(X,Y)U\big{]}.$

Thus in a locally $\phi$-semisymmetric Sasakian manifold, the relation (4)
holds for any arbitrary vector fields $X,Y,Z,U$ and $V$ on $M$. Next, if the
relation (4) holds in a Sasakian manifold, then for any horizontal vector
fields $X,Y,Z,U$ and $V$ on $M$, we get the relation (3.18) and hence the
manifold is locally $\phi$-semisymmetric. Thus we can state the following:

######  Theorem 4.1.

A Sasakian manifold $M$ is locally $\phi$-semisymmetric if and only if the
relation (4) holds for any arbitrary vector fields $X,Y,Z,U$ and $V$ on $M$.

######  Corollary 4.1.

[21] A semisymmetric Sasakian manifold is a manifold of constant curvature 1.

##  5\. Locally Ricci (resp., Projectively, Conformally) $\phi$-Semisymmetric
Sasakian Manifolds

######  Definition 5.1.

A Sasakian manifold $M$ is said to be a locally Ricci $\phi$-semisymmetric if
the relation

(5.1) $\phi^{2}[(R(U,V)\cdot Q)(X)]=0$

holds for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$, $Q$ being the
Ricci operator of the manifold.

We know that

(5.2) $(R(U,V)\cdot Q)(X)=R(U,V)QX-QR(U,V)X.$

Applying $\phi^{2}$ on both sides of (5.2) we get

(5.3) $\phi^{2}[(R(U,V)\cdot Q)(X)]=-(R(U,V)\cdot Q)(X)$

for all horizontal vector fields $U,V$ and $X$ on $M$. This leads to the
following:

######  Theorem 5.1.

A Sasakian manifold $M$ is locally Ricci $\phi$-semisymmetric if and only if
$(R(U,V)\cdot Q)(X)=0$ for all horizontal vector fields $U$, $V$ and $X$ on
$M$.

Now let $M$ be a locally $\phi$-semisymmetric Sasakian manifold. Then the
relation (3.18) holds on $M$. Taking inner product of (3.18) with a horizontal
vector field $W$ and then contracting over $X$ and $W$, we get $(R(U,V)\cdot
S)(Y,Z)=0$ from which it follows that $(R(U,V)\cdot Q)(Y)=0$ for all
horizontal vector fields $U$, $V$ and $Y$ on $M$. Thus in view of the Theorem
5.1, we can state the following:

######  Theorem 5.2.

A locally $\phi$-semisymmetric Sasakian manifold $M$ is locally Ricci
$\phi$-semisymmetric.

Now let $U$, $V$ and $X$ are arbitrary vector fields on a Sasakian manifold
$M$. Then in view of (2.1), (2.4), (2.5) and (2.10), (5.2) yields

$\displaystyle(R(\phi^{2}U,\phi^{2}V)\cdot Q)(\phi^{2}X)$ $\displaystyle=$
$\displaystyle-(R(U,V)\cdot Q)(X)+\\{E(X,V)\eta(U)-E(X,U)\eta(V)\\}\xi$
$\displaystyle-$
$\displaystyle\eta(X)\\{\eta(V)\mathcal{E}U-\eta(U)\mathcal{E}V\\}$

where $g(\mathcal{E}X,Y)=E(X,Y)$ and $E$ is given by

(5.5) $E(X,Y)=S(X,Y)-(n-1)g(X,Y).$

Since $\phi^{2}U$, $\phi^{2}V$ and $\phi^{2}X$ are orthogonal to $\xi$, in a
locally Ricci $\phi$-semisymmetric Sasakian manifold $M$, from (5) we have

$\displaystyle(R(U,V)\cdot Q)(X)$ $\displaystyle=$
$\displaystyle\\{E(X,V)\eta(U)-E(X,U)\eta(V)\\}\xi$ $\displaystyle-$
$\displaystyle\eta(X)\\{\eta(V)\mathcal{E}U-\eta(U)\mathcal{E}V\\}.$

Thus in a locally Ricci $\phi$-semisymmetric Sasakian manifold $M$ the
relation (5) holds for any arbitrary vector fields $U$, $V$ and $X$ on $M$.
Next, if the relation (5) holds in a Sasakian manifold $M$, then for all
horizontal vector fields $U$, $V$ and $X$, we have $(R(U,V)\cdot Q)(X)=0$ and
hence $M$ is locally Ricci $\phi$-semisymmetric. Thus we can state the
following:

######  Theorem 5.3.

A Sasakian manifold $M$ is locally Ricci $\phi$-semisymmetric if and only if
the relation (5) holds for any arbitrary vector fields $U$, $V$ and $X$ on
$M$.

######  Corollary 5.1.

[21] A Ricci semisymmetric Sasakian manifold is an Einstein manifold.

######  Definition 5.2.

A Sasakian manifold $M$ is said to be a locally projectively (resp.
conformally) $\phi$-semisymmetric if the relation

(5.7) $\phi^{2}[(R(U,V)\cdot P)(X,Y)Z]\ \big{(}\textnormal{resp.}\
\phi^{2}[(R(U,V)\cdot C)(X,Y)Z]\big{)}=0$

holds for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$, $P$ (resp.
$C$) being the projective (resp. conformal) curvature tensor of the manifold.

The projective transformation is such that geodesics transformed into
geodesics [23] and as the invariant of such transformation the Weyl projective
curvature tensor $P$ of type (1,3) is given by [23]

(5.8) $P(X,Y)Z=R(X,Y)Z-\frac{1}{n-1}\big{[}S(Y,Z)X-S(X,Z)Y\big{]}.$

The conformal transformation is an angle preserving mapping and as the
invariant of such transformation the Weyl conformal curvature tensor $C$ of
type (1,3) on a Riemannian manifold $M$, $n>3$, is given by [23]

$\displaystyle C(X,Y)Z$ $\displaystyle=$ $\displaystyle
R(X,Y)Z-\frac{1}{n-2}\\{S(Y,Z)X-S(X,Z)Y+g(Y,Z)QX$ $\displaystyle-$
$\displaystyle g(X,Z)QY\\}+\frac{r}{(n-1)(n-2)}\\{g(Y,Z)X-g(X,Z)Y\\}.$

From (5.8) we get

$\displaystyle\ \ \ \ \ \ (R(U,V)\cdot P)(X,Y)Z$ $\displaystyle=(R(U,V)\cdot
R)(X,Y)Z-\frac{1}{n-1}\left[(R(U,V)\cdot S)(Y,Z)X\right.$
$\displaystyle\left.-(R(U,V)\cdot S)(X,Z)Y\right].$

Applying $\phi^{2}$ on both sides of (5) and using (3) we obtain

$\displaystyle\ \ \ \ \ \ \ \phi^{2}[(R(U,V)\cdot P)(X,Y)Z]$
$\displaystyle=-(R(U,V)\cdot R)(X,Y)Z$
$\displaystyle+\left[(\nabla_{U}R)(X,Y,Z,\phi V)-(\nabla_{V}R)(X,Y,Z,\phi
U)\right]\xi$ $\displaystyle+\frac{1}{n-1}\left[(R(U,V)\cdot
S)(Y,Z)X-(R(U,V)\cdot S)(X,Z)Y\right]$

for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$.
Now we suppose that $M$ is a locally projectively $\phi$-semisymmetric
Sasakian manifold. Then from (5) we obtain

$\displaystyle(R(U,V)\cdot R)(X,Y)Z$
$\displaystyle=\left[(\nabla_{U}R)(X,Y,Z,\phi V)-(\nabla_{V}R)(X,Y,Z,\phi
U)\right]\xi$ $\displaystyle+\frac{1}{n-1}\left[(R(U,V)\cdot
S)(Y,Z)X-(R(U,V)\cdot S)(X,Z)Y\right]$

for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$. Taking inner
product of (5) with a horizontal vector field $W$ and then contracting over
$X$ and $Z$, we get

(5.13) $(R(U,V)\cdot S)(Y,W)=0$

for all horizontal vetor fields $U,V,Y$ and $W$ on $M$ and hence by Theorem
5.1, it follows that the manifold $M$ is locally Ricci $\phi$-semisymmetric.
Using (5.13) in (5), it follows that the manifold $M$ is locally
$\phi$-semisymmetric.
Next, we suppose that $M$ is a locally $\phi$-semisymmetric Sasakian manifold.
Then the relation (3.18) holds on $M$. Taking inner product of (3.18) with a
horizontal vector field $W$ and then contracting over $X$ and $W$, we get
$(R(U,V)\cdot S)(Y,Z)=0$ for all horizontal vetor fields $U,V,Y$ and $Z$ on
$M$ and hence from (5) it follows that the manifold $M$ is locally
projectively $\phi$-semisymmetric. This leads to the following:

######  Theorem 5.4.

A locally projectively $\phi$-semisymmetric Sasakian manifold $M$ is locally
$\phi$-semi-symmetric and vice versa.

Now from (5) we get

$\displaystyle(R(U,V)\cdot C)(X,Y)Z$ $\displaystyle=$
$\displaystyle(R(U,V)\cdot R)(X,Y)Z$ $\displaystyle-$
$\displaystyle\frac{1}{n-2}[(R(U,V)\cdot S)(Y,Z)X-(R(U,V)\cdot S)(X,Z)Y$
$\displaystyle+$ $\displaystyle g(Y,Z)(R(U,V)\cdot Q)(X)-g(X,Z)(R(U,V)\cdot
Q)(Y)].$

Applying $\phi^{2}$ on both sides of (5) and using (3) and (5.3) we obtain

$\displaystyle\ \ \ \ \phi^{2}[(R(U,V)\cdot C)(X,Y)Z]$ $\displaystyle=$
$\displaystyle-(R(U,V)\cdot R)(X,Y)Z$ $\displaystyle+$
$\displaystyle[(\nabla_{U}R)(X,Y,Z,\phi V)-(\nabla_{V}R)(X,Y,Z,\phi U)]\xi$
$\displaystyle+$ $\displaystyle\frac{1}{n-2}\left[(R(U,V)\cdot
S)(Y,Z)X-(R(U,V)\cdot S)(X,Z)Y\right.$ $\displaystyle+$
$\displaystyle\left.g(Y,Z)(R(U,V)\cdot Q)(X)-g(X,Z)(R(U,V)\cdot Q)(Y)\right]$

for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$. This leads to the
following:

######  Theorem 5.5.

A Sasakian manifold $M$ is locally conformally $\phi$-semisymmetric if and
only if the relation

$\displaystyle(R(U,V)\cdot R)(X,Y)Z$ $\displaystyle=$
$\displaystyle\left[(\nabla_{U}R)(X,Y,Z,\phi V)-(\nabla_{V}R)(X,Y,Z,\phi
U)\right]\xi$ $\displaystyle+$ $\displaystyle\frac{1}{n-2}\left[(R(U,V)\cdot
S)(Y,Z)X-(R(U,V)\cdot S)(X,Z)Y\right.$ $\displaystyle+$
$\displaystyle\left.g(Y,Z)(R(U,V)\cdot Q)(X)-g(X,Z)(R(U,V)\cdot Q)(Y)\right]$

holds for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$.

Let $M$ be a locally $\phi$-semisymmetric Sasakian manifold. Then $M$ is
locally Ricci $\phi$-semi-symmetric and thus in view of (3.18), it follows
from (5) that $\phi^{2}[(R(U,V)\cdot C)(X,Y)Z]=0$ for all horizontal vector
fields $X,Y,Z,U$ and $V$ on $M$. Hence the manifold $M$ is locally conformally
$\phi$-semisymmetric.
Again, we consider $M$ as the locally conformally $\phi$-semisymmetric
Sasakian manifold. If $M$ is locally $\phi$-semisymmetric Sasakian manifold,
then from (5.5) it follows that $(R(U,V)\cdot S)(Y,Z)=0$ which implies that
$(R(U,V)\cdot Q)(Y)=0$ for all horizontal vector fields $U$, $V$ and $Y$ on
$M$ and hence by Theorem 5.1, the manifold $M$ is locally Ricci
$\phi$-semisymmetric. Again, if $M$ is locally Ricci $\phi$-semisymmetric,
then $(R(U,V)\cdot Q)(Y)=0$ for all horizontal vector fields $U$, $V$ and $Y$
on $M$ and hence by Theorem 3.3, it follows from (5.5) that the manifold $M$
is locally $\phi$-semisymmetric. This leads to the following:

######  Theorem 5.6.

A locally $\phi$-semisymmetric Sasakian manifold $M$ is locally conformally
$\phi$-semisymmetric. The converse is true if and only if the manifold $M$ is
locally Ricci $\phi$-semi-symmetric.

Now let $X,Y,Z,U$ and $V$ be any arbitrary vector fields on a Sasakian
manifold $M$. Then using (2.1), (2.10), (4.2) and (5.5) we obtain

$\displaystyle(R(\phi^{2}U,\phi^{2}V)\cdot R)(\phi^{2}X,\phi^{2}Y)\phi^{2}Z$
$\displaystyle=$ $\displaystyle[\\{(\nabla_{V}R)(X,Y,Z,\phi
U)-(\nabla_{U}R)(X,Y,Z,\phi V)\\}$ $\displaystyle-$
$\displaystyle\eta(X)H(Y,Z,U,V)+\eta(Y)H(X,Z,U,V)$ $\displaystyle+$
$\displaystyle 2\eta(Z)H(X,Y,U,V)]\xi$ $\displaystyle-$
$\displaystyle\frac{1}{n-2}\big{[}\\{(R(U,V)\cdot S)(Y,Z)X-(R(U,V)\cdot
S)(X,Z)Y\\}$ $\displaystyle-$ $\displaystyle\big{\\{}(R(U,V)\cdot
S)(Y,Z)\eta(X)-(R(U,V)\cdot S)(X,Z)\eta(Y)\big{\\}}\xi$ $\displaystyle-$
$\displaystyle\big{\\{}E(V,Z)\eta(U)-E(U,Z)\eta(V)\big{\\}}\\{\eta(Y)X-\eta(X)Y\\}$
$\displaystyle+$
$\displaystyle\big{\\{}E(Y,U)X-E(X,U)Y\big{\\}}\eta(Z)\eta(V)$
$\displaystyle-$
$\displaystyle\big{\\{}E(Y,V)X-E(X,V)Y\big{\\}}\eta(Z)\eta(U)\big{]}-\frac{1}{n-2}$
$\displaystyle\big{[}\\{g(Y,Z)(R(U,V).Q)(X)-g(X,Z)(R(U,V).Q)(Y)\\}$
$\displaystyle-$
$\displaystyle\\{\eta(Y)(R(U,V).Q)(X)-\eta(X)(R(U,V).Q)(Y)\\}\eta(Z)$
$\displaystyle+$
$\displaystyle\\{g(Y,Z)\eta(X)-g(X,Z)\eta(Y)\\}\\{\eta(V)\mathcal{E}U-\eta(U)\mathcal{E}V\\}$
$\displaystyle-$
$\displaystyle\big{\\{}E(X,V)\eta(U)-E(X,U)\eta(V)\big{\\}}g(Y,Z)\xi$
$\displaystyle+$
$\displaystyle\big{\\{}E(Y,V)\eta(U)-E(Y,U)\eta(V)\big{\\}}g(X,Z)\xi\big{]}$

where $g(\mathcal{E}U,V)=E(U,V)$ and $E$ is given by (5.5).
From (5) and (4) it follows that

$\displaystyle\ \ \ \ \ \ \ (R(U,V)\cdot R)(X,Y)Z$ $\displaystyle=$
$\displaystyle\big{[}\\{(\nabla_{U}R)(X,Y,Z,\phi V)-(\nabla_{V}R)(X,Y,Z,\phi
U)\\}$ $\displaystyle+$ $\displaystyle\eta(X)H(Y,Z,U,V)-\eta(Y)H(X,Z,U,V)$
$\displaystyle-$ $\displaystyle 2\eta(Z)H(X,Y,U,V)+\eta(U)H(X,Y,Z,V)$
$\displaystyle-$
$\displaystyle\eta(V)H(X,Y,Z,U)\big{]}\xi+\eta(U)\big{[}\eta(X)\mathcal{H}(V,Y)Z$
$\displaystyle+$
$\displaystyle\eta(Y)\mathcal{H}(X,V)Z+\eta(Z)\mathcal{H}(X,Y)V\big{]}$
$\displaystyle-$
$\displaystyle\eta(V)\big{[}\eta(X)\mathcal{H}(U,Y)Z+\eta(Y)\mathcal{H}(X,U)Z+\eta(Z)\mathcal{H}(X,Y)U\big{]}$
$\displaystyle+$ $\displaystyle\frac{1}{n-2}\big{[}\\{(R(U,V)\cdot
S)(Y,Z)X-(R(U,V)\cdot S)(X,Z)Y\\}$ $\displaystyle-$
$\displaystyle\big{\\{}(R(U,V)\cdot S)(Y,Z)\eta(X)-(R(U,V)\cdot
S)(X,Z)\eta(Y)\big{\\}}\xi$ $\displaystyle-$
$\displaystyle\big{\\{}E(V,Z)\eta(U)-E(U,Z)\eta(V)\big{\\}}\\{\eta(Y)X-\eta(X)Y\\}$
$\displaystyle+$
$\displaystyle\big{\\{}E(Y,U)X-E(X,U)Y\big{\\}}\eta(Z)\eta(V)$
$\displaystyle-$
$\displaystyle\big{\\{}E(Y,V)X-E(X,V)Y\big{\\}}\eta(Z)\eta(U)\big{]}$
$\displaystyle+$
$\displaystyle\frac{1}{n-2}\big{[}\\{g(Y,Z)(R(U,V).Q)(X)-g(X,Z)(R(U,V).Q)(Y)\\}$
$\displaystyle-$
$\displaystyle\\{\eta(Y)(R(U,V).Q)(X)-\eta(X)(R(U,V).Q)(Y)\\}\eta(Z)$
$\displaystyle+$
$\displaystyle\\{g(Y,Z)\eta(X)-g(X,Z)\eta(Y)\\}\\{\eta(V)\mathcal{E}U-\eta(U)\mathcal{E}V\\}$
$\displaystyle-$
$\displaystyle\big{\\{}E(X,V)\eta(U)-E(X,U)\eta(V)\big{\\}}g(Y,Z)\xi$
$\displaystyle+$
$\displaystyle\big{\\{}E(Y,V)\eta(U)-E(Y,U)\eta(V)\big{\\}}g(X,Z)\xi\big{]}$

where $H(X,Y,Z,U)=g(\mathcal{H}(X,Y)Z,U)$ and $g(\mathcal{E}U,V)=E(U,V)$,
$\mathcal{H}$ and $E$ are given by (4.5) and (5.5) respectively. Thus in a
locally conformally $\phi$-semisymmetric Sasakian manifold $M$ the relation
(5) holds for any arbitrary vector fields $X,Y,Z,U$ and $V$ on $M$. Next, if
the relation (5) holds in a Sasakian manifold $M$, then for all horizontal
vector fields $X,Y,Z,U$ and $V$ on $M$, we have (5.5), that is, the manifold
is locally conformally $\phi$-semisymmetric. This leads to the following:

######  Theorem 5.7.

A Sasakian manifold $M$ is locally conformally $\phi$-semisymmetric if and
only if the relation (5) holds for any arbitrary vector fields $X,Y,Z,U$ and
$V$ on $M$.

######  Corollary 5.2.

[5] A conformally semisymmetric Sasakian manifold is a manifold of constant
curvature 1.

######  Remark 5.1.

Since the skew-symmetric operator $R(X,Y)$ and the structure tensor $\phi$ of
the Sasakian manifold both are commutes with the contraction, it follows from
Theorem 6.6(ii) of Shaikh and Kundu [16] that the same conclusion of the
Theorem 5.5, Theorem 5.6 and Theorem 5.7 holds for locally conharmonically
$\phi$-semisymmetric Sasakian manifold.

Again, by linear combination of $R$, $S$ and $g$, Shaikh and Kundu [16]
defined a generalized curvature tensor $B$ (see, equation (2.1) of [16]) of
type (1,3), called $B$-tensor which includes various curvature tensors as
particular cases. Then Shaikh and Kundu (see, equation (5.5) of [16]) shows
that this $B$-tensor turns into the following form:

$\displaystyle B(X,Y)Z$ $\displaystyle=$ $\displaystyle
b_{0}R(X,Y)Z+b_{1}\\{S(Y,Z)X-S(X,Z)Y$ $\displaystyle+$ $\displaystyle
g(Y,Z)QX-g(X,Z)QY\\}+b_{2}r\\{g(Y,Z)X-g(X,Z)Y\\}$

where $b_{0}$, $b_{1}$ and $b_{2}$ are scalars.
We note that if
(a) $b_{0}=1$, $b_{1}=0$ and $b_{2}=-\frac{1}{n(n-1)}$;
(b) $b_{0}=1$, $b_{1}=-\frac{1}{(n-2)}$ and $b_{2}=\frac{1}{(n-1)(n-2)}$;
(c) $b_{0}=1$, $b_{1}=-\frac{1}{(n-2)}$ and $b_{2}=0$;
and (d) $b_{2}=-\frac{1}{n}\big{(}\frac{b_{0}}{n-1}+2b_{1}\big{)}$,
then from (5) it follows that the $B$-tensor turns into the (a) concircular,
(b) conformal, (c) conharmonic and (d) quasi-conformal curvature tensor
respectively. For details about the $B$-tensor we refer the reader to see
Shaikh and Kundu [16] and also references therein.

######  Definition 5.3.

A Sasakian manifold $M$ is said to be a locally $B$-$\phi$-semisymmetric if
the relation

(5.20) $\phi^{2}[(R(U,V)\cdot B)(X,Y)Z]=0$

holds for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$, $B$ being the
generalized curvature tensor of the manifold.

From (5) we get

$\displaystyle\ \ \ \ \ \ (R(U,V)\cdot B)(X,Y)Z$
$\displaystyle=b_{0}(R(U,V)\cdot R)(X,Y)Z+b_{1}\left[(R(U,V)\cdot
S)(Y,Z)X\right.$ $\displaystyle\left.-(R(U,V)\cdot S)(X,Z)Y+g(Y,Z)(R(U,V)\cdot
Q)(X)\right.$ $\displaystyle\left.-g(X,Z)(R(U,V)\cdot Q)(Y)\right].$

Applying $\phi^{2}$ on both sides of (5) and using (3) and (5.3) we obtain

$\displaystyle\ \ \ \ \ \ \phi^{2}[(R(U,V)\cdot B)(X,Y)Z]$ $\displaystyle=$
$\displaystyle-b_{0}[(R(U,V)\cdot R)(X,Y)Z$ $\displaystyle-$
$\displaystyle\left\\{(\nabla_{U}R)(X,Y,Z,\phi V)-(\nabla_{V}R)(X,Y,Z,\phi
U)\right\\}\xi]$ $\displaystyle-$ $\displaystyle b_{1}\left[(R(U,V)\cdot
S)(Y,Z)X-(R(U,V)\cdot S)(X,Z)Y\right.$ $\displaystyle+$
$\displaystyle\left.g(Y,Z)(R(U,V)\cdot Q)(X)-g(X,Z)(R(U,V)\cdot Q)(Y)\right]$

for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$. This leads to the
following:

######  Theorem 5.8.

A Sasakian manifold $M$ is locally $B$-$\phi$-semisymmetric if and only if

$\displaystyle(R(U,V)\cdot R)(X,Y)Z$ $\displaystyle=$
$\displaystyle\left[(\nabla_{U}R)(X,Y,Z,\phi V)-(\nabla_{V}R)(X,Y,Z,\phi
U)\right]\xi$ $\displaystyle-$
$\displaystyle\frac{b_{1}}{b_{0}}\left[(R(U,V)\cdot S)(Y,Z)X-(R(U,V)\cdot
S)(X,Z)Y\right.$ $\displaystyle\left.+g(Y,Z)(R(U,V)\cdot
Q)(X)-g(X,Z)(R(U,V)\cdot Q)(Y)\right]$

for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$, provided $b_{0}\neq
0$.

Now taking inner product of (5.8) with a horizontal vector field $W$ and then
contracting over $X$ and $W$, we get

(5.24) $\\{b_{0}+(n-2)b_{1}\\}(R(U,V)\cdot S)(Y,Z)=0$

for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$.
From (5.24) following two cases arise:
Case-I: If $b_{0}+(n-2)b_{1}\neq 0$, then from (5.24) we have

(5.25) $(R(U,V)\cdot S)(Y,Z)=0$

for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$, from which it
follows that $(R(U,V)\cdot Q)(Y)=0$ for all horizontal vector fields $U$, $V$
and $Y$ on $M$. This leads to the following:

######  Theorem 5.9.

A locally $B$-$\phi$-semisymmetric Sasakian manifold $M$ is locally Ricci
$\phi$-semi-symmetric provided that $b_{0}+(n-2)b_{1}\neq 0$.

######  Corollary 5.3.

A locally concircularly $\phi$-semisymmetric Sasakian manifold $M$ is locally
Ricci $\phi$-semisymmetric.

######  Corollary 5.4.

A locally quasi-conformally $\phi$-semisymmetric Sasakian manifold $M$ is
locally Ricci $\phi$-semisymmetric provided that $b_{0}+(n-2)b_{1}\neq 0$.

Now if $b_{0}+(n-2)b_{1}\neq 0$, then in view of (5.25), (5.8) takes the form
(3.18) for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$ and hence the
manifold $M$ is locally $\phi$-semisymmetric. Again, if we consider the
manifold $M$ as locally $\phi$-semisymmetric, then the relation (3.18) holds
on $M$. Taking inner product of (3.18) with a horizontal vector field $W$ and
then contracting over $X$ and $W$, we get $(R(U,V)\cdot S)(Y,Z)=0$ for all
horizontal vetor fields $U,V,Y$ and $Z$ on $M$ and hence from (5) it follows
that the manifold $M$ is locally $B$-$\phi$-semisymmetric. Thus we can state
the following:

######  Theorem 5.10.

In a Sasakian manifold $M$, local $B$-$\phi$-semisymmetry and local
$\phi$-semisymmetry are equivalent provided that $b_{0}+(n-2)b_{1}\neq 0$.

######  Corollary 5.5.

In a Sasakian manifold $M$, local concircular $\phi$-semisymmetry and local
$\phi$-semisymmetry are equivalent.

######  Corollary 5.6.

In a Sasakian manifold $M$, local quasi-conformal $\phi$-semisymmetry and
local $\phi$-semisymmetry are equivalent provided that $b_{0}+(n-2)b_{1}\neq
0$.

######  Remark 5.2.

Since the skew-symmetric operator $R(X,Y)$ and the structure tensor $\phi$ of
the Sasakian manifold both are commutes with the contraction, it follows from
Theorem 6.6(i) of Shaikh and Kundu [16] that the same conclusion of the
Corollary 5.3 and Corollary 5.5 holds for locally projectively
$\phi$-semisymmetric Sasakian manifold as the contraction on projective
curvature tensor gives rise the Ricci operator although projective curvature
tensor is not a generalized curvature tensor.

Case-II: If $b_{0}+(n-2)b_{1}=0$, then from (5) we have

$\displaystyle\ \ \ \ \ \ \phi^{2}[(R(U,V)\cdot B)(X,Y)Z]$ $\displaystyle=$
$\displaystyle-b_{0}[(R(U,V)\cdot R)(X,Y)Z$ $\displaystyle-$
$\displaystyle\left\\{(\nabla_{U}R)(X,Y,Z,\phi V)-(\nabla_{V}R)(X,Y,Z,\phi
U)\right\\}\xi]$ $\displaystyle+$
$\displaystyle\frac{b_{0}}{n-2}\left[(R(U,V)\cdot S)(Y,Z)X-(R(U,V)\cdot
S)(X,Z)Y\right.$ $\displaystyle+$ $\displaystyle\left.g(Y,Z)(R(U,V)\cdot
Q)(X)-g(X,Z)(R(U,V)\cdot Q)(Y)\right]$

for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$. This leads to the
following:

######  Theorem 5.11.

A Sasakian manifold $M$ is locally $B$-$\phi$-semisymmetric if and only if

$\displaystyle(R(U,V)\cdot R)(X,Y)Z$ $\displaystyle=$
$\displaystyle\left[(\nabla_{U}R)(X,Y,Z,\phi V)-(\nabla_{V}R)(X,Y,Z,\phi
U)\right]\xi$ $\displaystyle+$ $\displaystyle\frac{1}{n-2}\left[(R(U,V)\cdot
S)(Y,Z)X-(R(U,V)\cdot S)(X,Z)Y\right.$ $\displaystyle\left.+g(Y,Z)(R(U,V)\cdot
Q)(X)-g(X,Z)(R(U,V)\cdot Q)(Y)\right]$

for all horizontal vector fields $X,Y,Z,U$ and $V$ on $M$ provided that
$b_{0}+(n-2)b_{1}=0$.

######  Corollary 5.7.

A Sasakian manifold $M$ is locally conformally (resp. conharmonically)
$\phi$-semisymmetric if and only if the relation (5.11) holds.

######  Corollary 5.8.

A Sasakian manifold $M$ is locally quasi-conformally $\phi$-semisymmetric if
and only if the relation (5.11) holds provided that $b_{0}+(n-2)b_{1}=0$.

Let $M$ be a locally $\phi$-semisymmetric Sasakian manifold. Then $M$ is
locally Ricci $\phi$-semi-symmetric and thus in view of (3.18), it follows
from (5) that $\phi^{2}[(R(U,V)\cdot B)(X,Y)Z]=0$ for all horizontal vector
fields $X,Y,Z,U$ and $V$ on $M$. Hence the manifold $M$ is locally
$B$-$\phi$-semisymmetric.
Again, we consider $M$ as the locally $B$-$\phi$-semisymmetric Sasakian
manifold. If $b_{0}+(n-2)b_{1}\neq 0$, then $M$ is locally
$\phi$-semisymmetric. So we suppose that $b_{0}+(n-2)b_{1}=0$. If $M$ is
locally $\phi$-semisymmetric, then from (5.11) it follows that $(R(U,V)\cdot
S)(Y,Z)=0$, which implies that $(R(U,V)\cdot Q)(Y)=0$ for all horizontal
vector fields $U$, $V$ and $Y$ on $M$. Thus in view of Theorem 5.1, the
manifold $M$ is locally Ricci $\phi$-semisymmetric. Again, if $M$ is locally
Ricci $\phi$-semisymmetric, then $(R(U,V)\cdot Q)(Y)=0$ for all horizontal
vector fields $U$, $V$ and $Y$ on $M$. Thus in view of Theorem 3.3, it follows
from (5.11) that the manifold $M$ is locally $\phi$-semisymmetric. This leads
to the following:

######  Theorem 5.12.

A locally $\phi$-semisymmetric Sasakian manifold $M$ is locally
$B$-$\phi$-semisymmetric. The converse is true for $b_{0}+(n-2)b_{1}=0$ if and
only if the manifold $M$ is locally Ricci $\phi$-semisymmetric.

If $X,Y,Z,U$ and $V$ are arbitrary vector fields on $M$, then proceeding
similarly as in the case of conformal curvature tensor, it is easy to check
that (5) holds for $b_{0}+(n-2)b_{1}=0$. Hence we can state the following:

######  Theorem 5.13.

A Sasakian manifold $M$ is locally $B$-$\phi$-semisymmetric if and only if the
relation (5) holds for any arbitrary vector fields $X,Y,Z,U$ and $V$ on $M$
provided that $b_{0}+(n-2)b_{1}=0$.

######  Corollary 5.9.

A Sasakian manifold $M$ is locally conformally (resp. conharmonically)
$\phi$-semisymmetric if and only if the relation (5) holds for any arbitrary
vector fields $X,Y,Z,U$ and $V$ on $M$.

######  Corollary 5.10.

A Sasakian manifold $M$ is locally quasi-conformally $\phi$-semisymmetric if
and only if the relation (5) holds for any arbitrary vector fields $X,Y,Z,U$
and $V$ on $M$ provided that $b_{0}+(n-2)b_{1}=0$.

Conclusion. From the above discussion and results, we conclude that the study
of local $\phi$-semisymmetry is meaningful as a generalized notion of local
$\phi$-symmetry and semisymmetry. From Theorem 6.6 and Corollary 6.2 of Shaikh
and Kundu [16] we also conclude that the same characterization of local
$\phi$-semisymmetry of a Sasakian manifold holds for the locally projectively
$\phi$-semisymmetric and locally concircularly $\phi$-semisymmetric Sasakian
manifolds as the contraction on projective or concircular curvature tensor
gives rise the Ricci operator. And also from Theorem 6.6 and Corollary 6.2 of
Shaikh and Kundu [16] we again conclude that the local conformal
$\phi$-semisymmetry and local conharmonical $\phi$-semisymmetry on a Sasakian
manifold are equivalent. However, the study of local $\phi$-semisymmetry and
local conformal $\phi$-semisymmetry are meaningful as they are not equivalent.
Finally, we conclude that the study of local $\phi$-semisymmetry on a Sasakian
manifold by considering any other generalized curvature tensor of type
(1,3)(which are the linear combination of $R$, $S$ and $g$ ) is either
meaningless or redundant due to their equivalency.

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