# Hire Me or Not? Examining Language Model’s Behavior with Occupation
Attributes

Damin Zhang, Yi Zhang, Geetanjali Bihani, Julia Rayz
Department of Computer and Information Technology
Purdue University
West Lafayette, IN 47907, USA
{zhan4060, zhan3050, gbihani<EMAIL_ADDRESS>

###### Abstract

With the impressive performance in various downstream tasks, large language
models (LLMs) have been widely integrated into production pipelines, like
recruitment and recommendation systems. A known issue of models trained on
natural language data is the presence of human biases, which can impact the
fairness of the system. This paper investigates LLMs’ behavior with respect to
gender stereotypes, in the context of occupation decision making. Our
framework is designed to investigate and quantify the presence of gender
stereotypes in LLMs’ behavior via multi-round question answering. Inspired by
prior works, we construct a dataset by leveraging a standard occupation
classification knowledge base released by authoritative agencies. We tested
three LLMs (RoBERTa-large, GPT-3.5-turbo, and Llama2-70b-chat) and found that
all models exhibit gender stereotypes analogous to human biases, but with
different preferences. The distinct preferences of GPT-3.5-turbo and
Llama2-70b-chat may imply the current alignment methods are insufficient for
debiasing and could introduce new biases contradicting the traditional gender
stereotypes.

##  1 Introduction

Large language models (LLMs) have become well-known to public users due to
their impressive performance across multiple tasks (Tan et al., 2023; Wang et
al., 2023; Lee et al., 2023) that are scalable with model size (Kaplan et al.,
2020). Along with different prompting techniques to improve the responses and
the simple interaction analogous to human communication, companies have
started integrating LLMs into downstream pipelines to assist users in
completing generation tasks via natural language (Microsoft, 2023).

However, a known issue of language models (LMs) is the human biases traced
back to the large training corpus (Bender et al., 2021; Blodgett et al., 2020;
Nozza et al., 2022; Smith et al., 2022; Solaiman et al., 2019; Talat et al.,
2022), which can impact the fairness of the downstream tasks (Rudinger et al.,
2018; Zhao et al., 2018a; Stanovsky et al., 2019; Dev et al., 2020; Liang et
al., 2021; He et al., 2021). Various methods have been proposed to mitigate
human biases, for example, data augmentation using counterfactuals (Zhao et
al., 2019; Maudslay et al., 2019; Zmigrod et al., 2019), adjusting model
parameters (Lauscher et al., 2021; Garimella et al., 2021; Kaneko & Bollegala,
2021; Guo et al., 2022), and modify decoding step to decrease the harmful
generations (Schick et al., 2021). Unlike open-source LLMs, applying these
methods to closed-source LLMs is challenging due to inaccessibility of model
weights. Additionally, even if one can access the model weights, fine-tuning
LLMs to mitigate a certain human bias may introduce new biases (Van Der Wal et
al., 2022), as demonstrated by prior works in embedding debiasing methods
(Bordia & Bowman, 2019; Gonen & Goldberg, 2019; Nissim et al., 2020).
Alternatively, in-context methods have been proposed to mitigate biases
through stereotypical and anti-stereotypical contexts, such as interventions
(Zhao et al., 2021) and preambles (Oba et al., 2024). Recent work has found
that LLMs still exhibit gender biases even after removing explicit signals,
such as co-occurrences of “female” and “nurse”, suggesting the measured bias
is not necessarily relevant to explicit gender-associated words (Belém et al.,
2023).

Figure 1: An example of multi-step gender stereotypes verification dataset.

In this work, we focus on gender stereotypes related to occupation.
Particularly, we investigate LLMs’ behavior with the appearance of implicit
neutral occupation-relevant attribtues. For this purpose, we propose a
framework for multi-step gender stereotype verification to examine how often
LLMs’ behavior conforms to stereotypes across different contexts and answer
spaces, as shown in Figure 1. As human biases change along with time and
environment (Kozlowski et al., 2020), we leverage the latest standard
occupation classification taxonomy released by O*NET (Gregory et al., 2019) as
the source of implicit neutral occupation-relevant attributes.

Our experimental results show that most tested LLMs demonstrate different
gender stereotypes by violating their previous neutral selections. Our
findings of RoBERTa-large align with prior works that the model demonstrates
gender stereotypes (Li et al., 2020; Zhao et al., 2021), but additionally show
such stereotypes are relevant to the consistency of the model. The results of
GPT-3.5-turbo and Llama2-70b-chat show some gender stereotypes are analogous
to humans and some contradict traditional stereotypes. There are also distinct
preferences between these two LLMs, which may imply that current alignment
methods require additional research to explore advanced techniques capable of
enhancing bias mitigation performance even further.

##  2 Related Work

Repetitive co-occurrences between genders and certain occupations could
perpetuate and be transmitted through natural language then forming gender
biases, for example, male doctors and female nurses. Such relationships are
then passed on to LMs that are trained on large textual corpora explicitly or
implicitly containing such gender biases. Extensive literature has shown that
gender biases exist in the input representations to pre-trained language
models (PLMs) (Bolukbasi et al., 2016; Caliskan et al., 2017; Zhao et al.,
2017; Garg et al., 2018; Zhao et al., 2018b; May et al., 2019; Swinger et al.,
2019; Zhao et al., 2019; Chaloner & Maldonado, 2019; Bordia & Bowman, 2019;
Tan & Celis, 2019; Zhao et al., 2020), and downstream tasks, for example,
coreference resolution (Rudinger et al., 2018; Zhao et al., 2018a; Kurita et
al., 2019), machine translation (Vanmassenhove et al., 2018; Stanovsky et al.,
2019; Cho et al., 2019), textual entailment (Sap et al., 2020; Dev et al.,
2020), and so on (Tatman, 2017; Kiritchenko & Mohammad, 2018; Park et al.,
2018; Sheng et al., 2019; Lu et al., 2020).

Some recent works have focused on probing models’ behavior via alternating the
input (Wallace et al., 2019; Gardner et al., 2020; Sheng et al., 2020; Emelin
et al., 2021; Ye & Ren, 2021; Schick & Schütze, 2021; Oba et al., 2024), as
well as via underspecified questions (Li et al., 2020; Zhao et al., 2021).

A range of recent works investigates human biases in LLMs. Acerbi &
Stubbersfield (2023) use transmission chain-like methodology to reveal that
ChatGPT-3 shows biases analogous to humans for stereotypical content over
other content. Gupta et al. (2024) find that LLMs are deeply biased and
suggest they manifest implicit stereotypical and often erroneous presumptions
when taking on a persona. Wan et al. (2023) show that LLMs have distinct
language styles and lexical content in generating recommendation letters for
males and females. Belém et al. (2023) demonstrate that measured gender bias
is not necessarily due to the explicit signals, suggesting the implicit
factors that contribute to the biased behavior of LLMs. Kotek et al. (2023)
reveal that gender bias about occupations in LLMs is due to imbalanced
training datasets, and LLMs tend to reflect the imbalances even with
Reinforcement Learning with Human Feedback (RLHF). Chain-of-Thought technique
has also been used to evaluate gender bias in LLMs by counting the number of
feminine or masculine words (Kaneko et al., 2024).

Consistency of a model is a desirable property in NLP tasks that is equally
important to model accuracy (Elazar et al., 2021). There are many prior works
exploring consistency of PLMs for question answering (Rajpurkar et al., 2016;
Ribeiro et al., 2019; Alberti et al., 2019; Asai & Hajishirzi, 2020; Kassner
et al., 2021), robust evaluation(Li et al., 2019), natural language inference
(Camburu et al., 2018; 2020), and more (Du et al., 2019).

##  3 Multi-step Gender Stereotypes Verification

In this paper, we introduce Multi-step Gender Stereotype Verification that
involves three consecutive steps providing different contexts of occupation-
relevant attributes, stereotypical occupation titles, underspecified
questions, and different answer spaces, as shown in Figure 2. All tested LLMs
were investigated by comparing responses of three steps with respect to gender
stereotypes and consistency of the model. Rather than presuming ground truth
stereotypical associations, such as executive is stereotypical toward male, we
analyzed how LLMs’ behavior changed under different conditions and compared
them with stereotypical associations to gain insights.

In order to provide background information conducive to multi-step question
answering, we integrated structured human knowledge about occupations from
authoritative labor statistics. The integration was facilitated through the
utilization of the O*NET-SOC taxonomy (Gregory et al., 2019), constructed upon
data gathered from the Bureau of Labor Statistics and the Census Bureau. In a
job recruitment setting, a neutral evaluation process should assess candidates
based on their relevant skills, knowledge, and abilities matched to the role’s
requirements. We therefore used these occupation attributes from the taxonomy
to provide grounded background information and probe the LLMs’ decision-making
processes.

###  3.1 Dataset Construction

Our dataset focuses on two subject categories: male and female, aiming to
investigate potential gender biases regarding various occupation titles. We
leveraged existing collections of unique gender-specific names and occupation
title lists from prior work (Li et al., 2020; Zhao et al., 2021). To ensure
alignment with real-world occupation titles, we cross-referenced the
occupation titles against the O*NET-SOC taxonomy and retained 62 matching
occupations.

From the taxonomy, we selected the skills, knowledge, and abilities categories
as broadly applicable and gender-neutral minimum requirements for occupations.
For each occupation, we extracted the top 5 most frequently occurring
attributes within each category to account for skewed distributions. These
attributes consisted of a name $A_{name}$, type $A_{type}$
(skill/knowledge/ability), and short description $A_{desp}$.

Our final dataset contains instances with a male-female subject pair, the
occupation title, and 15 associated attribute triplets (5 per
skill/knowledge/ability) representing relevant background information about
that occupation’s requirements. To ensure consistent and controlled response
generation, we used the Outlines library (Outlines, 2024) to restrict LLM
outputs to a predetermined answer space.

Figure 2: Multi-rounds of questions

###  3.2 Extracting Background Attribute Associations

To begin, we probed the relevant attributes of each occupation (skills,
knowledge, abilities) associated with each subject in a gender pair using
LLMs. Following the format of the UnQover dataset (Li et al., 2020), each
instance contained a base context paragraph $p$ and an underspecified question
$q$:

> Paragraph: [$s_{1}$] and [$s_{2}$] are neighbors.
> Question: Who was a [$occupation$]?

We used $p$ as a template to generate the base context $C_{base}$ by iterating
over subject pairs such as Andrew and Shirley. The underspecified question $q$
was then adjusted with specific attribute descriptions from the O*NET
taxonomy:

> Cbase: [$s_{1}$] and [$s_{2}$] are neighbors.
> Question: [$a_{desp}$]. Does [$s_{1}$/$s_{2}$] have [$a_{name}$]
> [$a_{type}$]?

An example of the attribute question is listed below:

> Given [$s_{1}$]=Andrew, [$s_{2}$]=Shirley, [$a_{name}$, $a_{type}$,
> $a_{desp}$]=[Mathematics, Knowledge, Mathematics is defined as …]
> Cbase: Andrew and Shirley are neighbors.
> Attribute Question: Mathematics is defined as … Does Shirley have
> Mathematics knowledge?

In Step 1, as illustrated in Figure 2, each LLM outputs Yes/No to indicate
whether the subject possesses a particular attribute. We collected all
attributes receiving “Yes” as $f_{A}(s_{x})$, representing the background
information the LLM associates with subject $s_{x}$.

###  3.3 Assessing Individual Qualifications

Next, for Step 2 in Figure 2, we evaluated whether the LLM deems each subject
individually qualified for the occupation based solely on their inferred
background information $f_{A}(s_{x})$:

> Q1: [$C_{base}$]. [$f_{A}(s_{x})$]. Is [$s_{x}$] qualified for
> [$occupation$]?

Each LLM outputted a binary Yes/No response $R_{binary}(s_{x})$, indicating
its assessment of the subject’s qualifications given their associated
attributes.

###  3.4 Comparing Subject Selections

Finally, in Step 3 as illustrated in Figure 2, we probed which subject the LLM
favors when considering background information $f_{A}(s_{male})$ and
$f_{A}(s_{female})$ for both subjects. We used two meaning-preserved variants:

> Q2: [$C_{base}$]. [$f_{A}(s_{female})$]. [$f_{A}(s_{male})$]. Who is more
> qualified to be a/an [$occupation$]?
> Q3: [$C_{base}$]. [$f_{A}(s_{female})$]. [$f_{A}(s_{male})$]. Who will be
> a/an [$occupation$]?

$Q_{2}$ restricts selection to [$s_{male}$, $s_{female}$], while $Q_{3}$
allows [$s_{male}$, $s_{female}$, $both$, $neither$]. If the LLM keeps
selecting the same subject across $Q_{2}$ and $Q_{3}$ despite the expanded
neutral options in $Q_{3}$, it suggests a gender stereotype.

The LLM outputs are denoted as $R_{single}(s_{x})$ for $Q_{2}$ and
$R_{multi}(s_{x})$ for $Q_{3}$. The right part of Figure 2 shows the process.

###  3.5 Metric

A key aspect of our multi-step verification framework is the ability to
systematically analyze both potential gender stereotypes and consistency in
LLMs’ behaviors. To achieve this objective, we established two metrics,
confirmation, and consistency, that compared the responses from three
questions under different conditions.

####  3.5.1 Confirmation

Question pairs $Q_{2}$ and $Q_{1}$ investigate the LLM on the same subject,
but differ in implicitly neutral contexts and answer spaces. $Q_{2}$ examines
the model favored subject choice whereas $Q_{1}$ concerns with the subject
qualification. Jointly, we are able to evaluate if the LLM shows biased
behavior by selecting subject $s_{x}$ in $Q_{2}$ and violating the individual
qualification assessment in $Q_{1}$ across the evaluation set:

$Confirmation(L,Q_{1},Q_{2},D_{eval})=\frac{1}{|D_{eval}|}\sum_{(s_{f},s_{m})\in
D_{eval}}\Phi(L(Q_{2})==s_{x},L(Q_{1},s_{x})==Yes)$ (1)

where $L$ represents the LM and $\Phi(*)$ returns 1 if both conditions are met
(LLM selected $s_{x}$ in $Q_{2}$ and also answered Yes that $s_{x}$ is
qualified in $Q_{1}$), and 0 otherwise.

####  3.5.2 Consistency

The meaning-preserving question pairs $Q_{2}$ and $Q_{3}$ investigate the LLM
on the same decision, but alter the answer space from [$s_{male}$,
$s_{female}$] to [$s_{male}$, $s_{female}$, $both$, $neither$]. This enables
us to evaluate if the LLM exhibits biased behavior by persistently favoring
the same subject across $Q_{2}$ and $Q_{3}$ despite the additional neutral
options in $Q_{3}$’s answer space:

$Consistency(L,Q_{1},Q_{2},D_{eval})=\frac{1}{|D_{eval}|}\sum_{(s_{f},s_{m})\in
D_{eval}}\Phi(L(Q_{2}),L(Q_{3}))$ (2)

where $\Phi(*)$ outputs 1 if the responses to $Q_{2}$ and $Q_{3}$ are
identical for the [$s_{male}$, $s_{female}$] subject pair, and 0 otherwise.

Taken together, a high score on Confirmation and a low score on Consistency
would suggest an LLM exhibiting low gender bias in its occupational decision-
making process.

##  4 Results

We evaluated the following three LLMs. To compare with prior works, we used
RoBERTa-large as a baseline model and two LLMs with different alignment
methods, where GPT-3.5-turbo uses Reinforcement Learning with Human Feedback
(RLHF) and Llama2-70b-chat uses RLHF and Supervised Fine-Tuning (SFT). We
retained the default settings loaded with the LLMs and made no changes.

  * •

RoBERTa-large (Liu et al., 2019) fine-tuned on SQuAD v2.0 (Rajpurkar et al.,
2018)

  * •

GPT-3.5-turbo (OpenAI, 2021)

  * •

Llama2-70b-chat (Touvron et al., 2023)

Figure 3: Measurement between $Q_{1}$ and $Q_{2}$

Figure 3 displays the Confirmation metric, measured as whether the LLM’s
$Q_{2}$ subject selection matches its $Q_{1}$ individual qualification
assessment for that subject. Compared to RoBERTa-large, GPT-3.5-turbo exhibits
higher variance in Confirmation, indicating greater fluctuation in its
decision as additional subject background information is provided.
Llama2-70b-chat demonstrates lower variance but with some outliers, indicating
generally stable but occasional deviations from its own qualification
judgments.

Figure 4: Measurement between $Q_{2}$ and $Q_{3}$

Figure 4 shows the Consistency metric which evaluates whether LLMs maintain
consistent outputs across the meaning-preserving question pairs with different
answer choices. Notably, both GPT-3.5-turbo and Llama2-70b-chat exhibit lower
overall scores than RoBERTa-large. The Consistency score of GPT-3.5-turbo
distribution is concentrated toward 0, indicating the model tends to modify
its behavior when providing more neutral options. The score of Llama2-70b-chat
is between RoBERTa-large and GPT-3.5-turbo.

Figure 5: Scatterplot of $Q_{1}Q_{2}$ vs. $Q_{2}Q_{3}$

As shown in Figure 5, analyzing Confirmation and Consistency jointly shows
interesting patterns across LLMs. RoBERTa-large demonstrates a relatively
linear relationship, where occupations with high Confirmation scores also have
high Consistency scores. Its gender stereotypes appear to be more systematic,
as additional background information does not significantly alter its
behavior.

In contrast, GPT-3.5-turbo exhibits a nearly vertical Confirmation-Consistency
pattern heavily concentrated at 0 Consistency. This suggests providing
additional neutral information successfully mitigates gender stereotypes in
many cases, but inconsistently compared to its initial qualification
decisions.

The scatter of Llama2-70b-chat is focused on the top-left quadrant, with
reasonably high Confirmation but low Consistency scores across occupations.

##  5 Discussion and Future Work

Finally, we would like to discuss these observations and where they could lead
to understanding of LLMs.

The distributions of three LLMs’ joint confirmation and consistency scores in
Figure 5 are quite intriguing. As expected, RoBERTa-large as a PLM has
relatively fewer parameters and thus unable to capture some implicit factors
that contribute to mitigating the biased behavior. GPT-3.5-turbo utilized RLHF
to align its responses with human preference. From the low consistency rate,
we could tell the contribution of RLHF to mitigate the biased behavior but the
high variance of confirmation rate raises the question: has RLHF introduced
new gender biases?

According to our metric definitions, a LLM’s behavior is biased if it has a
low Confirmation score or a high Consistency score. To further examine whether
RLHF introduces new gender biases, we calculated the difference of frequencies
that the model’s behavior is biased towards female and male subjects for each
occupation. A positive difference score indicates that the model’s behavior
favors female subjects and a negative score shows a preference for male
subjects.

Occupation | Diff
---|---
politician | 0.840
senator | 0.815
piano player | 0.590
violin player | 0.590
film director | 0.525
guitar player | 0.520
doctor | 0.320
poet | 0.285
lawyer | -0.235
plumber | -0.260
janitor | -0.290
butcher | -0.330
driver | -0.360
hunter | -0.396
athlete | -0.400
mechanic | -0.620
pilot | -0.660
Table 1: Occupation with high difference via GPT-3.5-turbo. Positive values
indicate the model favors female subjects and negative values indicate the
model favors male subjects.

Table 1 shows the occupations that have high difference scores larger than 0.2
from the results of GPT-3.5-turbo. The threshold value is determined by
observing all difference scores and 0.2 is an explicit boundary. We define a
fair value as 0 which indicates the model does not favor either gendered
subject. Among the occupations that the model favors female subjects, most are
art-related occupations with around 0.5 values. The two highest values come
from political occupations that are traditionally seen as male-stereotypical
occupations. We attribute such high value to the effort of RLHF which also
introduces a new gender bias against the male subjects. Similarly, doctor is a
traditionally male-stereotypical occupation of which GPT-3.5-turbo now favors
the female subjects. On the other hand, almost all occupations that the model
favors male subjects are stereotypes.

Occupation | Diff
---|---
piano player | 0.085
journal editor | 0.075
film director | 0.065
carpenter | 0.050
manager | 0.045
scientist | 0.040
detective | 0.035
writer | 0.035
architect | 0.030
assistant professor | 0.030
hunter | -0.045
violin player | -0.045
bodyguard | -0.050
model | -0.090
pilot | -0.090
athlete | -0.130
Table 2: Occupation with high difference via Llama2-70b-chat. Positive values
indicate the model favors female subjects and negative values indicate the
model favors male subjects.

Table 2 shows the occupations that have high difference scores larger than
0.02 from the results of Llama2-70b-chat. As we already stated, the threshold
value is determined by observing all difference scores and 0.02 is an explicit
boundary. Occupations that the model favors female subjects are mixed with
art-related occupations as well as science-related occupations, and the values
are all close to 0 which can be ignored as stereotypes. Similarly, occupations
that the model favors male subjects are from various domains and the values
are negligible.

Overall, our results suggest that gender biases still exist in the tested
LLMs, and RLHF might not be the ultimate solution for gender bias mitigation
in LLMs. It should either be replaced by or combined with other alignment
techniques.

###  5.1 Limitation

Our work is limited to investigating gender biases and stereotypes in English,
a morphologically limited language. Recent studies have found gender biases
existing in LLMs for different languages (Malik et al., 2022; Névéol et al.,
2022; Kaneko et al., 2022; Anantaprayoon et al., 2023; Levy et al., 2023). It
remains unclear whether our proposed methodology could effectively capture
biased behavior in other morphologically rich languages.

Moreover, we focused solely on gender biases in this work. However, prior
research has uncovered various other types of human biases in LMs, such as
ethnicity, nationality, and religion biases Li et al. (2020); Zhao et al.
(2021). While our proposed methodology could potentially be extended to these
other domains, it may require incorporating additional structured knowledge
from reliable sources to effectively extract relevant attributes.

Furthermore, we only considered gender-specific names in our work. The
efficacy of using gender-neutral names, which could be used by individuals of
any gender, in revealing LLMs’ biased behavior under our proposed methodology
remains unexplored. Additionally, our work only addresses binary gender
biases, whereas non-binary gender biases have also been explored in recent
literature (Cao & Daumé III, 2020; Dev et al., 2021).

##  6 Conclusion

In this work, we proposed a multi-step gender stereotypes verification
framework to investigate LLMs’ potentially biased behavior across different
implicitly neutral contexts and answer spaces. Our methodology does not
require access to LLMs’ weights, making it broadly applicable. Our carefully
crafted dataset leverages a reliable taxonomy to provide up-to-date structured
knowledge of occupation-relevant attributes. Additionally, we introduced two
novel metrics, Confirmation and Consistency, to systematically evaluate both
potential gender stereotypes and consistency in LLMs’ behavior.

Our experimental results show that LLMs still possess gender stereotypes
analogous to human biases. Our findings for RoBERTa-large align with prior
works. Additional experiments show that Reinforcement Learning with Human
Feedback (RLHF) alone is insufficient to mitigate human biases and could
potentially introduce new biases that contradict traditional stereotypes. The
differences between the distributions of GPT-3.5-turbo and Llama2-70b-chat
suggest that current alignment methods may require additional research to
further explore advanced techniques capable of enhancing bias mitigation
performance.

We urge caution in using LLMs in bias-sensitive domains without thorough
testing to understand the potential impact and corresponding solutions for
safe and equal treatment of all subjects. Our work provides a systematic
framework for investigating and quantifying gender stereotypes in LLMs,
contributing to future research in human bias mitigation and responsible AI
development.

## References

  * Acerbi & Stubbersfield (2023) Alberto Acerbi and Joseph M Stubbersfield.  Large language models show human-like content biases in transmission chain experiments.  _Proceedings of the National Academy of Sciences_ , 120(44):e2313790120, 2023.
  * Alberti et al. (2019) Chris Alberti, Daniel Andor, Emily Pitler, Jacob Devlin, and Michael Collins.  Synthetic QA corpora generation with roundtrip consistency.  In Anna Korhonen, David Traum, and Lluís Màrquez (eds.), _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pp. 6168–6173, Florence, Italy, July 2019. Association for Computational Linguistics.  doi: 10.18653/v1/P19-1620.  URL https://aclanthology.org/P19-1620.
  * Anantaprayoon et al. (2023) Panatchakorn Anantaprayoon, Masahiro Kaneko, and Naoaki Okazaki.  Evaluating gender bias of pre-trained language models in natural language inference by considering all labels.  _arXiv preprint arXiv:2309.09697_ , 2023.
  * Asai & Hajishirzi (2020) Akari Asai and Hannaneh Hajishirzi.  Logic-guided data augmentation and regularization for consistent question answering.  In Dan Jurafsky, Joyce Chai, Natalie Schluter, and Joel Tetreault (eds.), _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pp. 5642–5650, Online, July 2020. Association for Computational Linguistics.  doi: 10.18653/v1/2020.acl-main.499.  URL https://aclanthology.org/2020.acl-main.499.
  * Belém et al. (2023) Catarina G Belém, Preethi Seshadri, Yasaman Razeghi, and Sameer Singh.  Are models biased on text without gender-related language?  In _The Twelfth International Conference on Learning Representations_ , 2023.
  * Bender et al. (2021) Emily M Bender, Timnit Gebru, Angelina McMillan-Major, and Shmargaret Shmitchell.  On the dangers of stochastic parrots: Can language models be too big?  In _Proceedings of the 2021 ACM conference on fairness, accountability, and transparency_ , pp. 610–623, 2021.
  * Blodgett et al. (2020) Su Lin Blodgett, Solon Barocas, Hal Daumé III, and Hanna Wallach.  Language (technology) is power: A critical survey of “bias” in NLP.  In Dan Jurafsky, Joyce Chai, Natalie Schluter, and Joel Tetreault (eds.), _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pp. 5454–5476, Online, July 2020. Association for Computational Linguistics.  doi: 10.18653/v1/2020.acl-main.485.  URL https://aclanthology.org/2020.acl-main.485.
  * Bolukbasi et al. (2016) Tolga Bolukbasi, Kai-Wei Chang, James Y Zou, Venkatesh Saligrama, and Adam T Kalai.  Man is to computer programmer as woman is to homemaker? debiasing word embeddings.  _Advances in neural information processing systems_ , 29, 2016.
  * Bordia & Bowman (2019) Shikha Bordia and Samuel R. Bowman.  Identifying and reducing gender bias in word-level language models.  In Sudipta Kar, Farah Nadeem, Laura Burdick, Greg Durrett, and Na-Rae Han (eds.), _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Student Research Workshop_ , pp. 7–15, Minneapolis, Minnesota, June 2019. Association for Computational Linguistics.  doi: 10.18653/v1/N19-3002.  URL https://aclanthology.org/N19-3002.
  * Caliskan et al. (2017) Aylin Caliskan, Joanna J Bryson, and Arvind Narayanan.  Semantics derived automatically from language corpora contain human-like biases.  _Science_ , 356(6334):183–186, 2017.
  * Camburu et al. (2018) Oana-Maria Camburu, Tim Rocktäschel, Thomas Lukasiewicz, and Phil Blunsom.  e-snli: Natural language inference with natural language explanations.  _Advances in Neural Information Processing Systems_ , 31, 2018.
  * Camburu et al. (2020) Oana-Maria Camburu, Brendan Shillingford, Pasquale Minervini, Thomas Lukasiewicz, and Phil Blunsom.  Make up your mind! adversarial generation of inconsistent natural language explanations.  In Dan Jurafsky, Joyce Chai, Natalie Schluter, and Joel Tetreault (eds.), _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pp. 4157–4165, Online, July 2020. Association for Computational Linguistics.  doi: 10.18653/v1/2020.acl-main.382.  URL https://aclanthology.org/2020.acl-main.382.
  * Cao & Daumé III (2020) Yang Trista Cao and Hal Daumé III.  Toward gender-inclusive coreference resolution.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pp. 4568–4595, 2020.
  * Chaloner & Maldonado (2019) Kaytlin Chaloner and Alfredo Maldonado.  Measuring gender bias in word embeddings across domains and discovering new gender bias word categories.  In Marta R. Costa-jussà, Christian Hardmeier, Will Radford, and Kellie Webster (eds.), _Proceedings of the First Workshop on Gender Bias in Natural Language Processing_ , pp. 25–32, Florence, Italy, August 2019. Association for Computational Linguistics.  doi: 10.18653/v1/W19-3804.  URL https://aclanthology.org/W19-3804.
  * Cho et al. (2019) Won Ik Cho, Ji Won Kim, Seok Min Kim, and Nam Soo Kim.  On measuring gender bias in translation of gender-neutral pronouns.  In Marta R. Costa-jussà, Christian Hardmeier, Will Radford, and Kellie Webster (eds.), _Proceedings of the First Workshop on Gender Bias in Natural Language Processing_ , pp. 173–181, Florence, Italy, August 2019. Association for Computational Linguistics.  doi: 10.18653/v1/W19-3824.  URL https://aclanthology.org/W19-3824.
  * Dev et al. (2020) Sunipa Dev, Tao Li, Jeff M Phillips, and Vivek Srikumar.  On measuring and mitigating biased inferences of word embeddings.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 34, pp. 7659–7666, 2020.
  * Dev et al. (2021) Sunipa Dev, Masoud Monajatipoor, Anaelia Ovalle, Arjun Subramonian, Jeff Phillips, and Kai-Wei Chang.  Harms of gender exclusivity and challenges in non-binary representation in language technologies.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pp. 1968–1994, 2021.
  * Du et al. (2019) Xinya Du, Bhavana Dalvi, Niket Tandon, Antoine Bosselut, Wen-tau Yih, Peter Clark, and Claire Cardie.  Be consistent! improving procedural text comprehension using label consistency.  In Jill Burstein, Christy Doran, and Thamar Solorio (eds.), _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pp. 2347–2356, Minneapolis, Minnesota, June 2019. Association for Computational Linguistics.  doi: 10.18653/v1/N19-1244.  URL https://aclanthology.org/N19-1244.
  * Elazar et al. (2021) Yanai Elazar, Nora Kassner, Shauli Ravfogel, Abhilasha Ravichander, Eduard Hovy, Hinrich Schütze, and Yoav Goldberg.  Measuring and improving consistency in pretrained language models.  _Transactions of the Association for Computational Linguistics_ , 9:1012–1031, 2021.
  * Emelin et al. (2021) Denis Emelin, Ronan Le Bras, Jena D. Hwang, Maxwell Forbes, and Yejin Choi.  Moral stories: Situated reasoning about norms, intents, actions, and their consequences.  In Marie-Francine Moens, Xuanjing Huang, Lucia Specia, and Scott Wen-tau Yih (eds.), _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pp. 698–718, Online and Punta Cana, Dominican Republic, November 2021. Association for Computational Linguistics.  doi: 10.18653/v1/2021.emnlp-main.54.  URL https://aclanthology.org/2021.emnlp-main.54.
  * Gardner et al. (2020) Matt Gardner, Yoav Artzi, Victoria Basmov, Jonathan Berant, Ben Bogin, Sihao Chen, Pradeep Dasigi, Dheeru Dua, Yanai Elazar, Ananth Gottumukkala, Nitish Gupta, Hannaneh Hajishirzi, Gabriel Ilharco, Daniel Khashabi, Kevin Lin, Jiangming Liu, Nelson F. Liu, Phoebe Mulcaire, Qiang Ning, Sameer Singh, Noah A. Smith, Sanjay Subramanian, Reut Tsarfaty, Eric Wallace, Ally Zhang, and Ben Zhou.  Evaluating models’ local decision boundaries via contrast sets.  In Trevor Cohn, Yulan He, and Yang Liu (eds.), _Findings of the Association for Computational Linguistics: EMNLP 2020_ , pp. 1307–1323, Online, November 2020. Association for Computational Linguistics.  doi: 10.18653/v1/2020.findings-emnlp.117.  URL https://aclanthology.org/2020.findings-emnlp.117.
  * Garg et al. (2018) Nikhil Garg, Londa Schiebinger, Dan Jurafsky, and James Zou.  Word embeddings quantify 100 years of gender and ethnic stereotypes.  _Proceedings of the National Academy of Sciences_ , 115(16):E3635–E3644, 2018.
  * Garimella et al. (2021) Aparna Garimella, Akhash Amarnath, Kiran Kumar, Akash Pramod Yalla, N Anandhavelu, Niyati Chhaya, and Balaji Vasan Srinivasan.  He is very intelligent, she is very beautiful? on mitigating social biases in language modelling and generation.  In _Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021_ , pp. 4534–4545, 2021.
  * Gonen & Goldberg (2019) Hila Gonen and Yoav Goldberg.  Lipstick on a pig: Debiasing methods cover up systematic gender biases in word embeddings but do not remove them.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pp. 609–614, 2019.
  * Gregory et al. (2019) Christina Gregory, Phil Lewis, Pamela Frugoli, and Alexander Nallin.  Updating the o*net-soc taxonomy: Incorporating the 2018 soc structure, 2019.  URL https://www.onetcenter.org/reports/Taxonomy2019.html.
  * Guo et al. (2022) Yue Guo, Yi Yang, and Ahmed Abbasi.  Auto-debias: Debiasing masked language models with automated biased prompts.  In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pp. 1012–1023, 2022.
  * Gupta et al. (2024) Shashank Gupta, Vaishnavi Shrivastava, Ameet Deshpande, Ashwin Kalyan, Peter Clark, Ashish Sabharwal, and Tushar Khot.  Bias Runs Deep: Implicit reasoning biases in persona-assigned LLMs.  In _The Twelfth International Conference on Learning Representations_ , 2024.
  * He et al. (2021) Zexue He, Bodhisattwa Prasad Majumder, and Julian McAuley.  Detect and perturb: Neutral rewriting of biased and sensitive text via gradient-based decoding.  In _Findings of the Association for Computational Linguistics: EMNLP 2021_ , pp. 4173–4181, 2021.
  * Kaneko & Bollegala (2021) Masahiro Kaneko and Danushka Bollegala.  Debiasing pre-trained contextualised embeddings.  In _Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume_ , pp. 1256–1266, 2021.
  * Kaneko et al. (2022) Masahiro Kaneko, Aizhan Imankulova, Danushka Bollegala, and Naoaki Okazaki.  Gender bias in masked language models for multiple languages.  In _Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pp. 2740–2750, 2022.
  * Kaneko et al. (2024) Masahiro Kaneko, Danushka Bollegala, Naoaki Okazaki, and Timothy Baldwin.  Evaluating gender bias in large language models via chain-of-thought prompting.  _arXiv preprint arXiv:2401.15585_ , 2024.
  * Kaplan et al. (2020) Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei.  Scaling laws for neural language models.  _arXiv preprint arXiv:2001.08361_ , 2020.
  * Kassner et al. (2021) Nora Kassner, Oyvind Tafjord, Hinrich Schütze, and Peter Clark.  BeliefBank: Adding memory to a pre-trained language model for a systematic notion of belief.  In Marie-Francine Moens, Xuanjing Huang, Lucia Specia, and Scott Wen-tau Yih (eds.), _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pp. 8849–8861, Online and Punta Cana, Dominican Republic, November 2021. Association for Computational Linguistics.  doi: 10.18653/v1/2021.emnlp-main.697.  URL https://aclanthology.org/2021.emnlp-main.697.
  * Kiritchenko & Mohammad (2018) Svetlana Kiritchenko and Saif Mohammad.  Examining gender and race bias in two hundred sentiment analysis systems.  In Malvina Nissim, Jonathan Berant, and Alessandro Lenci (eds.), _Proceedings of the Seventh Joint Conference on Lexical and Computational Semantics_ , pp. 43–53, New Orleans, Louisiana, June 2018. Association for Computational Linguistics.  doi: 10.18653/v1/S18-2005.  URL https://aclanthology.org/S18-2005.
  * Kotek et al. (2023) Hadas Kotek, Rikker Dockum, and David Sun.  Gender bias and stereotypes in large language models.  In _Proceedings of The ACM Collective Intelligence Conference_ , pp. 12–24, 2023.
  * Kozlowski et al. (2020) Diego Kozlowski, Gabriela Lozano, Carla M Felcher, Fernando Gonzalez, and Edgar Altszyler.  Gender bias in magazines oriented to men and women: a computational approach.  _arXiv preprint arXiv:2011.12096_ , 2020.
  * Kurita et al. (2019) Keita Kurita, Nidhi Vyas, Ayush Pareek, Alan W Black, and Yulia Tsvetkov.  Measuring bias in contextualized word representations.  In Marta R. Costa-jussà, Christian Hardmeier, Will Radford, and Kellie Webster (eds.), _Proceedings of the First Workshop on Gender Bias in Natural Language Processing_ , pp. 166–172, Florence, Italy, August 2019. Association for Computational Linguistics.  doi: 10.18653/v1/W19-3823.  URL https://aclanthology.org/W19-3823.
  * Lauscher et al. (2021) Anne Lauscher, Tobias Lueken, and Goran Glavaš.  Sustainable modular debiasing of language models.  In _Findings of the Association for Computational Linguistics: EMNLP 2021_ , pp. 4782–4797, 2021.
  * Lee et al. (2023) Gibbeum Lee, Volker Hartmann, Jongho Park, Dimitris Papailiopoulos, and Kangwook Lee.  Prompted llms as chatbot modules for long open-domain conversation.  In _Findings of the Association for Computational Linguistics: ACL 2023_ , pp. 4536–4554, 2023.
  * Levy et al. (2023) Sharon Levy, Neha John, Ling Liu, Yogarshi Vyas, Jie Ma, Yoshinari Fujinuma, Miguel Ballesteros, Vittorio Castelli, and Dan Roth.  Comparing biases and the impact of multilingual training across multiple languages.  In _Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing_ , pp. 10260–10280, 2023.
  * Li et al. (2019) Tao Li, Vivek Gupta, Maitrey Mehta, and Vivek Srikumar.  A logic-driven framework for consistency of neural models.  In Kentaro Inui, Jing Jiang, Vincent Ng, and Xiaojun Wan (eds.), _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pp. 3924–3935, Hong Kong, China, November 2019. Association for Computational Linguistics.  doi: 10.18653/v1/D19-1405.  URL https://aclanthology.org/D19-1405.
  * Li et al. (2020) Tao Li, Daniel Khashabi, Tushar Khot, Ashish Sabharwal, and Vivek Srikumar.  UNQOVERing stereotyping biases via underspecified questions.  In _Findings of the Association for Computational Linguistics: EMNLP 2020_ , pp. 3475–3489, 2020.
  * Liang et al. (2021) Paul Pu Liang, Chiyu Wu, Louis-Philippe Morency, and Ruslan Salakhutdinov.  Towards understanding and mitigating social biases in language models.  In _International Conference on Machine Learning_ , pp. 6565–6576. PMLR, 2021.
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov.  Roberta: A robustly optimized bert pretraining approach.  _arXiv preprint arXiv:1907.11692_ , 2019.
  * Lu et al. (2020) Kaiji Lu, Piotr Mardziel, Fangjing Wu, Preetam Amancharla, and Anupam Datta.  Gender bias in neural natural language processing.  _Logic, language, and security: essays dedicated to Andre Scedrov on the occasion of his 65th birthday_ , pp. 189–202, 2020.
  * Malik et al. (2022) Vijit Malik, Sunipa Dev, Akihiro Nishi, Nanyun Peng, and Kai-Wei Chang.  Socially aware bias measurements for hindi language representations.  In _Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pp. 1041–1052, 2022.
  * Maudslay et al. (2019) Rowan Hall Maudslay, Hila Gonen, Ryan Cotterell, and Simone Teufel.  It’s all in the name: Mitigating gender bias with name-based counterfactual data substitution.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pp. 5267–5275, 2019.
  * May et al. (2019) Chandler May, Alex Wang, Shikha Bordia, Samuel R. Bowman, and Rachel Rudinger.  On measuring social biases in sentence encoders.  In Jill Burstein, Christy Doran, and Thamar Solorio (eds.), _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pp. 622–628, Minneapolis, Minnesota, June 2019. Association for Computational Linguistics.  doi: 10.18653/v1/N19-1063.  URL https://aclanthology.org/N19-1063.
  * Microsoft (2023) Microsoft.  Announcing microsoft copilot, your everyday ai companion, 2023.  URL https://blogs.microsoft.com/blog/2023/09/21/announcing-microsoft-copilot-your-everyday-ai-companion/.
  * Névéol et al. (2022) Aurélie Névéol, Yoann Dupont, Julien Bezançon, and Karën Fort.  French crows-pairs: Extending a challenge dataset for measuring social bias in masked language models to a language other than english.  In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pp. 8521–8531, 2022.
  * Nissim et al. (2020) Malvina Nissim, Rik van Noord, and Rob Van Der Goot.  Fair is better than sensational: Man is to doctor as woman is to doctor.  _Computational Linguistics_ , 46(2):487–497, 2020.
  * Nozza et al. (2022) Debora Nozza, Federcio Bianchi, Dirk Hovy, et al.  Pipelines for social bias testing of large language models.  In _Proceedings of BigScience Episode# 5–Workshop on Challenges & Perspectives in Creating Large Language Models_. Association for Computational Linguistics, 2022.
  * Oba et al. (2024) Daisuke Oba, Masahiro Kaneko, and Danushka Bollegala.  In-contextual gender bias suppression for large language models.  In _Findings of the Association for Computational Linguistics: EACL 2024_ , pp. 1722–1742, 2024.
  * OpenAI (2021) OpenAI.  Openai gpt-3.5-turbo, 2021.  URL https://platform.openai.com/docs/models/gpt-3-5-turbo.
  * Outlines (2024) Outlines.  outlines-dev/outlines, March 2024.  URL https://github.com/outlines-dev/outlines.  original-date: 2023-03-17T16:01:18Z.
  * Park et al. (2018) Ji Ho Park, Jamin Shin, and Pascale Fung.  Reducing gender bias in abusive language detection.  In Ellen Riloff, David Chiang, Julia Hockenmaier, and Jun’ichi Tsujii (eds.), _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pp. 2799–2804, Brussels, Belgium, October-November 2018. Association for Computational Linguistics.  doi: 10.18653/v1/D18-1302.  URL https://aclanthology.org/D18-1302.
  * Rajpurkar et al. (2016) Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang.  SQuAD: 100,000+ questions for machine comprehension of text.  In Jian Su, Kevin Duh, and Xavier Carreras (eds.), _Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing_ , pp. 2383–2392, Austin, Texas, November 2016. Association for Computational Linguistics.  doi: 10.18653/v1/D16-1264.  URL https://aclanthology.org/D16-1264.
  * Rajpurkar et al. (2018) Pranav Rajpurkar, Robin Jia, and Percy Liang.  Know what you don’t know: Unanswerable questions for SQuAD.  In Iryna Gurevych and Yusuke Miyao (eds.), _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , pp. 784–789, Melbourne, Australia, July 2018. Association for Computational Linguistics.  doi: 10.18653/v1/P18-2124.  URL https://aclanthology.org/P18-2124.
  * Ribeiro et al. (2019) Marco Tulio Ribeiro, Carlos Guestrin, and Sameer Singh.  Are red roses red? evaluating consistency of question-answering models.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pp. 6174–6184, 2019.
  * Rudinger et al. (2018) Rachel Rudinger, Jason Naradowsky, Brian Leonard, and Benjamin Van Durme.  Gender bias in coreference resolution.  In Marilyn Walker, Heng Ji, and Amanda Stent (eds.), _Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Short Papers)_ , pp. 8–14, New Orleans, Louisiana, June 2018. Association for Computational Linguistics.  doi: 10.18653/v1/N18-2002.  URL https://aclanthology.org/N18-2002.
  * Sap et al. (2020) Maarten Sap, Saadia Gabriel, Lianhui Qin, Dan Jurafsky, Noah A. Smith, and Yejin Choi.  Social bias frames: Reasoning about social and power implications of language.  In Dan Jurafsky, Joyce Chai, Natalie Schluter, and Joel Tetreault (eds.), _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pp. 5477–5490, Online, July 2020. Association for Computational Linguistics.  doi: 10.18653/v1/2020.acl-main.486.  URL https://aclanthology.org/2020.acl-main.486.
  * Schick & Schütze (2021) Timo Schick and Hinrich Schütze.  Few-shot text generation with natural language instructions.  In Marie-Francine Moens, Xuanjing Huang, Lucia Specia, and Scott Wen-tau Yih (eds.), _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pp. 390–402, Online and Punta Cana, Dominican Republic, November 2021. Association for Computational Linguistics.  doi: 10.18653/v1/2021.emnlp-main.32.  URL https://aclanthology.org/2021.emnlp-main.32.
  * Schick et al. (2021) Timo Schick, Sahana Udupa, and Hinrich Schütze.  Self-diagnosis and self-debiasing: A proposal for reducing corpus-based bias in nlp.  _Transactions of the Association for Computational Linguistics_ , 9:1408–1424, 2021.
  * Sheng et al. (2019) Emily Sheng, Kai-Wei Chang, Premkumar Natarajan, and Nanyun Peng.  The woman worked as a babysitter: On biases in language generation.  In Kentaro Inui, Jing Jiang, Vincent Ng, and Xiaojun Wan (eds.), _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pp. 3407–3412, Hong Kong, China, November 2019. Association for Computational Linguistics.  doi: 10.18653/v1/D19-1339.  URL https://aclanthology.org/D19-1339.
  * Sheng et al. (2020) Emily Sheng, Kai-Wei Chang, Prem Natarajan, and Nanyun Peng.  Towards Controllable Biases in Language Generation.  In Trevor Cohn, Yulan He, and Yang Liu (eds.), _Findings of the Association for Computational Linguistics: EMNLP 2020_ , pp. 3239–3254, Online, November 2020. Association for Computational Linguistics.  doi: 10.18653/v1/2020.findings-emnlp.291.  URL https://aclanthology.org/2020.findings-emnlp.291.
  * Smith et al. (2022) Eric Michael Smith, Melissa Hall, Melanie Kambadur, Eleonora Presani, and Adina Williams.  “I’m sorry to hear that”: Finding new biases in language models with a holistic descriptor dataset.  In Yoav Goldberg, Zornitsa Kozareva, and Yue Zhang (eds.), _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pp. 9180–9211, Abu Dhabi, United Arab Emirates, December 2022. Association for Computational Linguistics.  doi: 10.18653/v1/2022.emnlp-main.625.  URL https://aclanthology.org/2022.emnlp-main.625.
  * Solaiman et al. (2019) Irene Solaiman, Miles Brundage, Jack Clark, Amanda Askell, Ariel Herbert-Voss, Jeff Wu, Alec Radford, Gretchen Krueger, Jong Wook Kim, Sarah Kreps, et al.  Release strategies and the social impacts of language models.  _arXiv preprint arXiv:1908.09203_ , 2019.
  * Stanovsky et al. (2019) Gabriel Stanovsky, Noah A. Smith, and Luke Zettlemoyer.  Evaluating gender bias in machine translation.  In Anna Korhonen, David Traum, and Lluís Màrquez (eds.), _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pp. 1679–1684, Florence, Italy, July 2019. Association for Computational Linguistics.  doi: 10.18653/v1/P19-1164.  URL https://aclanthology.org/P19-1164.
  * Swinger et al. (2019) Nathaniel Swinger, Maria De-Arteaga, Neil Thomas Heffernan IV, Mark DM Leiserson, and Adam Tauman Kalai.  What are the biases in my word embedding?  In _Proceedings of the 2019 AAAI/ACM Conference on AI, Ethics, and Society_ , pp. 305–311, 2019.
  * Talat et al. (2022) Zeerak Talat, Aurélie Névéol, Stella Biderman, Miruna Clinciu, Manan Dey, Shayne Longpre, Sasha Luccioni, Maraim Masoud, Margaret Mitchell, Dragomir Radev, et al.  You reap what you sow: On the challenges of bias evaluation under multilingual settings.  In _Proceedings of BigScience Episode# 5–Workshop on Challenges & Perspectives in Creating Large Language Models_, pp. 26–41, 2022.
  * Tan & Celis (2019) Yi Chern Tan and L Elisa Celis.  Assessing social and intersectional biases in contextualized word representations.  _Advances in neural information processing systems_ , 32, 2019.
  * Tan et al. (2023) Yiming Tan, Dehai Min, Yu Li, Wenbo Li, Nan Hu, Yongrui Chen, and Guilin Qi.  Can chatgpt replace traditional kbqa models? an in-depth analysis of the question answering performance of the gpt llm family.  In _International Semantic Web Conference_ , pp. 348–367. Springer, 2023.
  * Tatman (2017) Rachael Tatman.  Gender and dialect bias in youtube’s automatic captions.  In _Proceedings of the first ACL workshop on ethics in natural language processing_ , pp. 53–59, 2017.
  * Touvron et al. (2023) Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al.  Llama 2: Open foundation and fine-tuned chat models.  _arXiv preprint arXiv:2307.09288_ , 2023.
  * Van Der Wal et al. (2022) Oskar Van Der Wal, Jaap Jumelet, Katrin Schulz, and Willem Zuidema.  The birth of bias: A case study on the evolution of gender bias in an english language model.  In _Proceedings of the 4th Workshop on Gender Bias in Natural Language Processing (GeBNLP)_ , pp. 75–75, 2022.
  * Vanmassenhove et al. (2018) Eva Vanmassenhove, Christian Hardmeier, and Andy Way.  Getting gender right in neural machine translation.  In Ellen Riloff, David Chiang, Julia Hockenmaier, and Jun’ichi Tsujii (eds.), _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pp. 3003–3008, Brussels, Belgium, October-November 2018. Association for Computational Linguistics.  doi: 10.18653/v1/D18-1334.  URL https://aclanthology.org/D18-1334.
  * Wallace et al. (2019) Eric Wallace, Shi Feng, Nikhil Kandpal, Matt Gardner, and Sameer Singh.  Universal adversarial triggers for attacking and analyzing NLP.  In Kentaro Inui, Jing Jiang, Vincent Ng, and Xiaojun Wan (eds.), _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pp. 2153–2162, Hong Kong, China, November 2019. Association for Computational Linguistics.  doi: 10.18653/v1/D19-1221.  URL https://aclanthology.org/D19-1221.
  * Wan et al. (2023) Yixin Wan, George Pu, Jiao Sun, Aparna Garimella, Kai-Wei Chang, and Nanyun Peng.  “kelly is a warm person, joseph is a role model”: Gender biases in LLM-generated reference letters.  In _Findings of the Association for Computational Linguistics: EMNLP 2023_ , pp. 3730–3748, 2023.
  * Wang et al. (2023) Jiaan Wang, Yunlong Liang, Fandong Meng, Beiqi Zou, Zhixu Li, Jianfeng Qu, and Jie Zhou.  Zero-shot cross-lingual summarization via large language models.  In _Proceedings of the 4th New Frontiers in Summarization Workshop_ , pp. 12–23, 2023.
  * Ye & Ren (2021) Qinyuan Ye and Xiang Ren.  Learning to generate task-specific adapters from task description.  In Chengqing Zong, Fei Xia, Wenjie Li, and Roberto Navigli (eds.), _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 2: Short Papers)_ , pp. 646–653, Online, August 2021. Association for Computational Linguistics.  doi: 10.18653/v1/2021.acl-short.82.  URL https://aclanthology.org/2021.acl-short.82.
  * Zhao et al. (2017) Jieyu Zhao, Tianlu Wang, Mark Yatskar, Vicente Ordonez, and Kai-Wei Chang.  Men also like shopping: Reducing gender bias amplification using corpus-level constraints.  In Martha Palmer, Rebecca Hwa, and Sebastian Riedel (eds.), _Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing_ , pp. 2979–2989, Copenhagen, Denmark, September 2017. Association for Computational Linguistics.  doi: 10.18653/v1/D17-1323.  URL https://aclanthology.org/D17-1323.
  * Zhao et al. (2018a) Jieyu Zhao, Tianlu Wang, Mark Yatskar, Vicente Ordonez, and Kai-Wei Chang.  Gender bias in coreference resolution: Evaluation and debiasing methods.  In Marilyn Walker, Heng Ji, and Amanda Stent (eds.), _Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Short Papers)_ , pp. 15–20, New Orleans, Louisiana, June 2018a. Association for Computational Linguistics.  doi: 10.18653/v1/N18-2003.  URL https://aclanthology.org/N18-2003.
  * Zhao et al. (2018b) Jieyu Zhao, Yichao Zhou, Zeyu Li, Wei Wang, and Kai-Wei Chang.  Learning gender-neutral word embeddings.  In Ellen Riloff, David Chiang, Julia Hockenmaier, and Jun’ichi Tsujii (eds.), _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pp. 4847–4853, Brussels, Belgium, October-November 2018b. Association for Computational Linguistics.  doi: 10.18653/v1/D18-1521.  URL https://aclanthology.org/D18-1521.
  * Zhao et al. (2019) Jieyu Zhao, Tianlu Wang, Mark Yatskar, Ryan Cotterell, Vicente Ordonez, and Kai-Wei Chang.  Gender bias in contextualized word embeddings.  In Jill Burstein, Christy Doran, and Thamar Solorio (eds.), _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pp. 629–634, Minneapolis, Minnesota, June 2019. Association for Computational Linguistics.  doi: 10.18653/v1/N19-1064.  URL https://aclanthology.org/N19-1064.
  * Zhao et al. (2020) Jieyu Zhao, Subhabrata Mukherjee, Saghar Hosseini, Kai-Wei Chang, and Ahmed Hassan Awadallah.  Gender bias in multilingual embeddings and cross-lingual transfer.  In Dan Jurafsky, Joyce Chai, Natalie Schluter, and Joel Tetreault (eds.), _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pp. 2896–2907, Online, July 2020. Association for Computational Linguistics.  doi: 10.18653/v1/2020.acl-main.260.  URL https://aclanthology.org/2020.acl-main.260.
  * Zhao et al. (2021) Jieyu Zhao, Daniel Khashabi, Tushar Khot, Ashish Sabharwal, and Kai-Wei Chang.  Ethical-advice taker: Do language models understand natural language interventions?  In _Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021_ , pp. 4158–4164, 2021.
  * Zmigrod et al. (2019) Ran Zmigrod, Sabrina J Mielke, Hanna Wallach, and Ryan Cotterell.  Counterfactual data augmentation for mitigating gender stereotypes in languages with rich morphology.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pp. 1651–1661, 2019.


# Normal curves on a smooth immersed surface

Absos Ali Shaikh1, Mohamd Saleem Lone2 and Pinaki Ranjan Ghosh3 1Department of
Mathematics, University of Burdwan, Golapbag, Burdwan-713104, West Bengal,
India<EMAIL_ADDRESS><EMAIL_ADDRESS>2International Centre
for Theoretical Sciences, Tata Institute of Fundamental Research, 560089,
Bengaluru, India<EMAIL_ADDRESS><EMAIL_ADDRESS>3Department of Mathematics, University of Burdwan,Golapbag, Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to investigate the sufficient condition for the
invariance of a normal curve on a smooth immersed surface under isometry. We
also find the the deviations of the tangential and normal components of the
curve with respect to the given isometry.

###### Key words and phrases:

Isometry, normal curve, osculating curve, rectifying curve.

###### 2000 Mathematics Subject Classification:

53A04, 53A05, 53A15

##  1\. Introduction

When we talk of a manifold, one of the most elementary geometric object is its
position vector field, which reveals the position of an arbitrary point on
that manifold with respect to some origin. In case of curves, the position
vector field of a curve can be thought of as the motion of a particle with
respect to a parameter $s$ and intuitively the first and second derivatives of
the curve gives the velocity and the acceleration of the particle
respectively. So, we shall discuss a problem which altogether depends upon the
position vector field of a curve.

Let $\alpha:I\subset\mathbb{R}\rightarrow\mathbb{E}^{3}$ be a unit speed curve
having all the necessary properties such that $\\{t,n,b\\}$ acts as its
Serret-Frenet frame, where $t,n$, and $b$ are the tangent, the normal and the
binormal vectors(unitary), respectively. Then, the Serret-Frenet equations are
given by

(1.4) $\displaystyle\left\\{\begin{array}[]{ll}t^{\prime}=\kappa n\\\
n^{\prime}=-\kappa t+\tau b\\\ b^{\prime}=-\tau n,\end{array}\right.$

where $\kappa$ is the curvature and $\tau$ is the torsion of $\alpha$ with
$t=\alpha^{\prime}$, $n=\frac{t^{\prime}}{\kappa}$, $b=t\times n$, and
$\prime$ denotes the differentiation with respect to the parameter $s$. At
each point $\alpha(s)$ of $\alpha$, the planes spanned by $\\{t,n\\}$,
$\\{t,b\\}$ and $\\{n,b\\}$ are called as the osculating plane, the rectifying
plane and the normal plane, respectively. As it is evident from the names of
planes, a curve whose position vector field lies in the osculating plane is
called as an osculating curve. Similarly, a curve whose position vector filed
lies in the rectifying and the normal plane are called as rectifying and
normal curves, respectively.

It is well known that if at each point the position vector of $\alpha$ lies in
the osculating plane, then the curve lies in a plane. Similarly, if the the
position vector of $\alpha$ lies in the normal plane at each point, then the
curve lies on a sphere. So, in view of these facts, in 2003 Chen([3]) posed a
question: When does the position vector of a curve lies in the rectifying
curve? Therein, Chen obtained characterization results for rectifying curves.
In relation to these three types of curves, enormous study has been done. For
generic study, we refer the reader to see([4, 5, 6, 7]).

Before going to motivation and objective of this paper, we revisit a few
definitions ([2]) on surfaces.

######  Definition 1.1.

A diffeomorphism $J:{\bf M}\rightarrow\overline{{\bf M}}$ is an isometry if
for all $p\in{\bf M}$ and all pairs $x_{1},x_{2}\in T_{p}({\bf M})$, we have

$\langle x_{1},x_{2}\rangle_{p}=\langle
dJ_{p}(x_{1}),dJ_{p}(x_{2})\rangle_{J(p)}.$

The surfaces ${\bf M}$ and $\overline{{\bf M}}$ are then said to be isometric.

######  Definition 1.2.

A map $J:V\subset{\bf M}\rightarrow\overline{{\bf M}}$ of a neighborhood $V$
of $p\in{\bf M}$ is a local isometry at $p$ if there exists a neighborhood $U$
of $J(p)\in\overline{\bf M}$ such that $J:V\rightarrow U$ is an isometry. If
there exists local isometry at every point of ${\bf M}$, then the surfaces
${\bf M}$ and $\overline{{\bf M}}$ are said to be locally isometric. Clearly,
if $J$ is a diffeomorphism and a local isometry for every $p\in{\bf M}$, then
$J$ is global isometry.

It is straightforward to see that the first fundamental form coefficients are
preserved under isometry. So if $E,F,G$ and
$\overline{E},\overline{F},\overline{G}$ are the first fundamental form
coefficients of ${\bf M}$ and $\overline{{\bf M}}$, respectively and $J:{\bf
M}\rightarrow\overline{{\bf M}}$ is a local isometry then

$E=\overline{E},\quad F=\overline{F},\quad G=\overline{G}.$

In 2018 Shaikh and Ghosh ([8]) diverted the study of rectifying curves to a
new direction by questioning about the invariant properties of a rectifying
curve on a smooth surface under isometry. In addition to a sufficient
condition for a rectifying curve to remain invariant under isometry, they
showed that the component of the rectifying curve along the surface normal is
invariant under isometry. Again in ([9]) Shaikh and Ghosh studied osculating
curves and obtained their characterization along with invariancy under surface
isometry. Motivated by ([8], [9] and [10]), we shall investigate the similar
questions in case of normal curves, i.e.,

Question: What happens to a normal curve on a smooth surface under isometry?

In the section $2$, we give some of the basic notions about normal curves and
find the Frenet frame vectors of the normal curves with respect to the smooth
immersed surface. Section 3 is concerned with the main results and provided
the answer of above question.

##  2\. Preliminaries

Let $\alpha$ be a normal curve parameterized by arc with a Serret-Frenet frame
given in (1.4). The other way of interpreting a normal curve is: a curve is
said to be a normal curve if its position vector lies in the orthogonal
complement of tangent vector i.e., $\alpha\cdot t=0,$ or

(2.1) $\alpha(s)=\lambda(s)n(s)+\mu(s)b(s),$

where $\lambda,\mu$ are two smooth functions.

Suppose ${\bf M}$ is a regular surface(page no 52, [2]) with $\varphi(u,v)$
being its coordinate chart. Then, the curve $\alpha(s)=\alpha(u(s),v(s))$
defines a curve $\alpha(s)={\bf M}(u(s),v(s))$ on the surface ${\bf M}$. We
can easily find the derivatives of the curve $\alpha(s)$ as a curve on the
surface ${\bf M}$ using the chain rule:

$\displaystyle\alpha^{\prime}(s)$ $\displaystyle=$
$\displaystyle\varphi_{u}u^{\prime}+\varphi_{v}v^{\prime}$ or, (2.2)
$\displaystyle t(s)$ $\displaystyle=$
$\displaystyle\alpha^{\prime}(s)=\varphi_{u}u^{\prime}+\varphi_{v}v^{\prime}$
$\displaystyle{t^{\prime}}(s)$ $\displaystyle=$ $\displaystyle
u^{\prime\prime}\varphi_{u}+v^{\prime\prime}\varphi_{v}+{u^{\prime}}^{2}\varphi_{uu}+2u^{\prime}v^{\prime}\varphi_{uv}+{v^{\prime}}^{2}\varphi_{vv}.$

Now let ${\bf N}$ be the unit surface normal then we have

$\displaystyle n(s)$ $\displaystyle=$
$\displaystyle\frac{1}{k(s)}(u^{\prime\prime}\varphi_{u}+v^{\prime\prime}\varphi_{v}+u^{\prime
2}\varphi_{uu}+2u^{\prime}v^{\prime}\varphi_{uv}+v^{\prime 2}\varphi_{vv}).$
(2.3) $\displaystyle b(s)$ $\displaystyle=$ $\displaystyle t(s)\times
n(s)=t(s)\times\frac{t^{\prime}(s)}{k(s)}$ $\displaystyle=$
$\displaystyle\frac{1}{k(s)}\Big{[}(\varphi_{u}u^{\prime}+\varphi_{v}v^{\prime})\times(u^{\prime\prime}\varphi_{u}+v^{\prime\prime}\varphi_{v}+u^{\prime
2}\varphi_{uu}+2u^{\prime}v^{\prime}\varphi_{uv}+v^{\prime
2}\varphi_{vv})\Big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{k(s)}\Big{[}\\{u^{\prime}v^{\prime\prime}-u^{\prime\prime}v^{\prime}\\}{\bf
N}+u^{\prime 3}\varphi_{u}\times\varphi_{uu}+2u^{\prime
2}v^{\prime}\varphi_{u}\times\varphi_{uv}+u^{\prime}v^{\prime
2}\varphi_{u}\times\varphi_{vv}$ $\displaystyle+u^{\prime
2}v^{\prime}\varphi_{v}\times\varphi_{uu}+2u^{\prime}v^{\prime
2}\varphi_{v}\times\varphi_{uv}+v^{\prime
3}\varphi_{v}\times\varphi_{vv}\Big{]}.$

######  Definition 2.1.

[1] Let $\alpha$ be a unit speed curve on ${\bf M}$, then the unit tangent
vector $t=\alpha^{\prime}$ is orthogonal to the unit surface normal ${\bf N}$,
so $\alpha^{\prime}$, ${\bf N}$ and ${\bf N}\times\alpha^{\prime}$ are
mutually orthogonal vectors. Moreover, since $\alpha^{\prime\prime}$ is
orthogonal to $\alpha^{\prime}$, we can write $\alpha^{\prime\prime}$ as a
linear combination of ${\bf N}$ and ${\bf N}\times\alpha^{\prime}$, i.e.,

$\alpha^{\prime\prime}=\kappa_{n}{\bf N}+\kappa_{g}{\bf
N}\times\alpha^{\prime},$

where $\kappa_{n}$ and $\kappa_{g}$ are called as the normal curvature and the
geodesic curvature of $\alpha$, respectively and are given by

$\displaystyle\left\\{\begin{array}[]{ll}\kappa_{g}=\alpha^{\prime\prime}\cdot{\bf
N}\times\alpha^{\prime}\\\ \kappa_{n}=\alpha^{\prime\prime}\cdot{\bf
N}.\end{array}\right.$

Now since $\alpha^{\prime\prime}=\kappa(s)n(s)$, therefore we can write

$\kappa_{n}=\kappa(s)n(s)\cdot{\bf
N}=(u^{\prime\prime}\varphi_{u}+v^{\prime\prime}\varphi_{v}+u^{\prime
2}\varphi_{uu}+2u^{\prime}v^{\prime}\varphi_{uv}+v^{\prime
2}\varphi_{vv})\cdot{\bf N}$

or

(2.5) $\kappa_{n}={u^{\prime}}^{2}L+2u^{\prime}v^{\prime}M+{v^{\prime}}^{2}N,$

where $L,M,N$ are the second fundamental form coefficients of the surface. The
curve $\alpha$ on $\bf{M}$ is said to be asymptotic if and only if
$\kappa_{n}=0.$

##  3\. Normal curves

The equation of a normal curve is given by

(3.1) $\alpha(s)=\lambda(s)n(s)+\mu(s)b(s).$

Suppose this curve lies on a parametric surface $\varphi(u,v)$. Then (3.1) is
in the form:

$\displaystyle\alpha(s)$ $\displaystyle=$
$\displaystyle\frac{\lambda(s)}{\kappa(s)}\left[(u^{\prime\prime}\varphi_{u}+v^{\prime\prime}\varphi_{v})+({u^{\prime}}^{2}\varphi_{uu}+2u^{\prime}v^{\prime}\varphi_{uv}+{v^{\prime}}^{2}\varphi_{vv})\right]$
$\displaystyle+\frac{\mu(s)}{k(s)}\Big{[}\\{u^{\prime}v^{\prime\prime}-u^{\prime\prime}v^{\prime}\\}\vec{N}+u^{\prime
3}\varphi_{u}\times\varphi_{uu}+2u^{\prime
2}v^{\prime}\varphi_{u}\times\varphi_{uv}$ $\displaystyle+u^{\prime}v^{\prime
2}\varphi_{u}\times\varphi_{vv}+u^{\prime
2}v^{\prime}\varphi_{v}\times\varphi_{uu}+2u^{\prime}v^{\prime
2}\varphi_{v}\times\varphi_{uv}+v^{\prime
3}\varphi_{v}\times\varphi_{vv}\Big{]}.$

######  Theorem 3.1.

Let ${\bf M}$ and ${\overline{{\bf M}}}$ be two smooth surfaces and $J:{\bf
M}\rightarrow{\overline{\bf M}}$ be an isometry. Also, let $\alpha(s)$ be a
normal curve on ${\bf M}$. Then $\overline{\alpha}(s)=J\circ\alpha(s)$ is a
normal curve on $\overline{{\bf M}}$ if

$\displaystyle\overline{\alpha}(s)-J_{\ast}(\alpha(s))$ $\displaystyle=$
$\displaystyle\frac{\lambda(s)}{\kappa(s)}\Big{[}{u^{\prime}}^{2}\left(\frac{\partial
J_{\ast}}{\partial
u}\varphi_{u}\right)+2u^{\prime}v^{\prime}\left(\frac{\partial
J_{\ast}}{\partial u}\varphi_{v}\right)+{v^{\prime}}^{2}\left(\frac{\partial
J_{\ast}}{\partial v}\varphi_{v}\right)\Big{]}$
$\displaystyle+\frac{\mu(s)}{k(s)}\Big{[}u^{\prime
3}\Big{(}J_{*}\varphi_{u}\times\frac{\partial J_{*}}{\partial
u}\varphi_{u}\Big{)}+2u^{\prime
2}v^{\prime}\Big{(}J_{*}\varphi_{u}\times\frac{\partial J_{*}}{\partial
u}\varphi_{v}\Big{)}$ $\displaystyle+u^{\prime}v^{\prime
2}\Big{(}J_{*}\varphi_{u}\times\frac{\partial J_{*}}{\partial
v}\varphi_{v}\Big{)}+u^{\prime
2}v^{\prime}\Big{(}J_{*}\varphi_{v}\times\frac{\partial J_{*}}{\partial
u}\varphi_{u}\Big{)}$ $\displaystyle+2u^{\prime}v^{\prime
2}\Big{(}J_{*}\varphi_{v}\times\frac{\partial J_{*}}{\partial
u}\varphi_{v}\Big{)}+v^{\prime 3}\Big{(}J_{*}\varphi_{v}\times\frac{\partial
J_{*}}{\partial v}\varphi_{v}\Big{)}\Big{]}.$

###### Proof.

Suppose $\varphi$ and $\overline{\varphi}$ are the chart maps of ${\bf M}$ and
${\overline{\bf M}}$, respectively. Then, we have

$\overline{\varphi}=J\circ\varphi.$

$J:{\bf M}\rightarrow{\overline{\bf M}}$ is an isometry whose differential map
$dJ=J_{\ast}$ is a $3\times 3$ orthogonal matrix taking linearly independent
vectors of $T_{p}({\bf M})$ to linearly independent vectors of
$T_{J(p)}{\overline{\bf M}}$, i.e.,

$J_{\ast}:T_{p}({\bf M})\rightarrow T_{J(p)}{\overline{\bf M}}.$

Since $\\{\varphi_{u},\varphi_{v}\\}$ is a basis of tangent plane $T_{p}({\bf
M})$ at a point $p$ on ${\bf M}$, we have

(3.4)
$\displaystyle\bar{\varphi}_{u}(u,v)=J_{*}\varphi_{u}=J_{*}(\varphi(u,v))\varphi_{u},$
(3.5)
$\displaystyle\bar{\varphi}_{v}(u,v)=J_{*}\varphi_{v}=J_{*}(\varphi(u,v))\varphi_{v}.$

Differentiating (3.4) and (3.5) with respect to $u,v$, we get

$\displaystyle\bar{\varphi}_{uu}$ $\displaystyle=$
$\displaystyle\frac{\partial J_{*}}{\partial u}\varphi_{u}+J_{*}\varphi_{uu},$
(3.6) $\displaystyle\bar{\varphi}_{vv}$ $\displaystyle=$
$\displaystyle\frac{\partial J_{*}}{\partial v}\varphi_{v}+J_{*}\varphi_{vv},$
$\displaystyle\bar{\varphi}_{uv}$ $\displaystyle=$
$\displaystyle\frac{\partial J_{*}}{\partial
u}\varphi_{v}+J_{*}\varphi_{uv}=\frac{\partial J_{*}}{\partial
v}\varphi_{u}+J_{*}\varphi_{uv}.$

We can write

(3.7) $J_{*}\varphi_{u}\times\frac{\partial J_{*}}{\partial
u}\varphi_{u}=J_{*}\varphi_{u}\times\Big{(}\frac{\partial J_{*}}{\partial
u}\varphi_{u}+F*\varphi_{uu}\Big{)}-J_{*}(\varphi_{u}\times\varphi_{uu})=\bar{\varphi}_{u}\times\bar{\varphi}_{uu}-J_{*}(\varphi_{u}\times\varphi_{uu}).$

Similarly

$\displaystyle J_{*}\varphi_{u}\times\frac{\partial J_{*}}{\partial
u}\varphi_{v}$
$\displaystyle=\bar{\varphi}_{u}\times\bar{\varphi}_{uv}-J_{*}(\varphi_{u}\times\varphi_{uv}),$
$\displaystyle J_{*}\varphi_{u}\times\frac{\partial J_{*}}{\partial
v}\varphi_{v}$
$\displaystyle=\bar{\varphi}_{u}\times\bar{\varphi}_{vv}-J_{*}(\varphi_{u}\times\varphi_{vv}),$
(3.8) $\displaystyle J_{*}\varphi_{v}\times\frac{\partial J_{*}}{\partial
u}\varphi_{u}$
$\displaystyle=\bar{\varphi}_{v}\times\bar{\varphi}_{uu}-J_{*}(\varphi_{v}\times\varphi_{uu}),$
$\displaystyle J_{*}\varphi_{v}\times\frac{\partial J_{*}}{\partial
u}\varphi_{v}$
$\displaystyle=\bar{\varphi}_{v}\times\bar{\varphi}_{uv}-J_{*}(\varphi_{v}\times\varphi_{uv}),$
$\displaystyle J_{*}\varphi_{v}\times\frac{\partial J_{*}}{\partial
v}\varphi_{v}$
$\displaystyle=\bar{\varphi}_{v}\times\bar{\varphi}_{vv}-J_{*}(\varphi_{v}\times\varphi_{vv}).$

Therefore, with respect to $(3.2)$, (3.7) and (3), we get

$\displaystyle\overline{\alpha}(s)$ $\displaystyle=$
$\displaystyle\frac{\lambda(s)}{\kappa(s)}\Big{[}u^{\prime\prime}J_{\ast}\varphi_{u}+v^{\prime\prime}J_{\ast}\varphi_{v}+{u^{\prime}}^{2}\left(\frac{\partial
J_{\ast}}{\partial_{u}}\varphi_{u}+J_{\ast}\varphi_{uu}\right)+2u^{\prime}v^{\prime}\left(\frac{\partial
J_{\ast}}{\partial u}\varphi_{v}+J_{\ast}\varphi_{uv}\right)$
$\displaystyle+{v^{\prime}}^{2}\left(\frac{\partial J_{\ast}}{\partial
v}\varphi_{v}+J_{\ast}\varphi_{vv}\right)\Big{]}+\frac{\mu(s)}{k(s)}\Big{[}\\{u^{\prime}v^{\prime\prime}-u^{\prime\prime}v^{\prime}\\}J_{*}\vec{N}+u^{\prime
3}J_{*}(\varphi_{u}\times\varphi_{uu})$ $\displaystyle+2u^{\prime
2}v^{\prime}J_{*}(\varphi_{u}\times\varphi_{uv})+u^{\prime}v^{\prime
2}J_{*}(\varphi_{u}\times\varphi_{vv})+u^{\prime
2}v^{\prime}J_{*}(\varphi_{v}\times\varphi_{uu})+2u^{\prime}v^{\prime
2}J_{*}(\varphi_{v}\times\varphi_{uv})$ $\displaystyle+v^{\prime
3}J_{*}(\varphi_{v}\times\varphi_{vv})+u^{\prime
3}\Big{(}J_{*}\varphi_{u}\times\frac{\partial J_{*}}{\partial
u}\varphi_{u}\Big{)}+2u^{\prime
2}v^{\prime}\Big{(}J_{*}\varphi_{u}\times\frac{\partial J_{*}}{\partial
u}\varphi_{v}\Big{)}$ $\displaystyle+u^{\prime}v^{\prime
2}\Big{(}J_{*}\varphi_{u}\times\frac{\partial J_{*}}{\partial
v}\varphi_{v}\Big{)}+u^{\prime
2}v^{\prime}\Big{(}J_{*}\varphi_{v}\times\frac{\partial J_{*}}{\partial
u}\varphi_{u}\Big{)}+2u^{\prime}v^{\prime
2}\Big{(}J_{*}\varphi_{v}\times\frac{\partial J_{*}}{\partial
u}\varphi_{v}\Big{)}$ $\displaystyle+v^{\prime
3}\Big{(}J_{*}\varphi_{v}\times\frac{\partial J_{*}}{\partial
v}\varphi_{v}\Big{)}\Big{]},$

or

$\displaystyle\overline{\alpha}(s)$ $\displaystyle=$
$\displaystyle\frac{\lambda(s)}{\kappa(s)}\left[(u^{\prime\prime}\overline{\varphi}_{u}+v^{\prime\prime}\overline{\varphi}_{v})+({u^{\prime}}^{2}\overline{\varphi}_{uu}+2u^{\prime}v^{\prime}\overline{\varphi}_{uv}{v^{\prime}}^{2}\overline{\varphi}_{vv})\right]$
$\displaystyle+\frac{\mu(s)}{k(s)}\Big{[}\\{u^{\prime}v^{\prime\prime}-u^{\prime\prime}v^{\prime}\\}\vec{\overline{N}}+u^{\prime
3}\varphi_{u}\times\overline{\varphi}_{uu}+2u^{\prime
2}v^{\prime}\overline{\varphi}_{u}\times\overline{\varphi}_{uv}$
$\displaystyle+u^{\prime}v^{\prime
2}\overline{\varphi}_{u}\times\overline{\varphi}_{vv}+u^{\prime
2}v^{\prime}\overline{\varphi}_{v}\times\overline{\varphi}_{uu}+2u^{\prime}v^{\prime
2}\overline{\varphi}_{v}\times\overline{\varphi}_{uv}+v^{\prime
3}\overline{\varphi}_{v}\times\overline{\varphi}_{vv}\Big{]}.$

Therefore

(3.10)
$\overline{\alpha}(s)=\frac{\overline{\lambda}(s)}{\overline{\kappa}(s)}\overline{n}(s)+\frac{\overline{\mu}(s)}{\overline{\kappa}(s)}\overline{b}(s)$

for some functions $\overline{\lambda}(s)$ and $\overline{\mu}(s)$. Therefore,
$\overline{\alpha}(s)$ is a normal curve in ${\overline{\bf M}}$. ∎

######  Theorem 3.2.

Let $J:{\bf M}\rightarrow{\overline{\bf M}}$ be an isometry and $\alpha(s)$ be
a normal curve on ${\bf M}$. Then for the tangential components, we have

(3.11) $\displaystyle\bar{\alpha}\cdot{\bf\bar{T}}-\alpha\cdot{\bf
T}=\frac{\mu}{\kappa}(\bar{\kappa}_{n}-\kappa_{n})(av^{\prime}+bu^{\prime}),$

where ${\bf T}=a\varphi_{u}+b\varphi_{v}$ is any tangent vector to ${\bf M}$
for some $a,b\in\mathbb{R}$.

###### Proof.

From (3), we see that

(3.12) $\displaystyle\alpha\cdot\varphi_{u}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{\kappa}\Big{[}u^{\prime\prime}E+v^{\prime\prime}F+{u^{\prime}}^{2}\varphi_{uu}\cdot\varphi_{u}+2u^{\prime}v^{\prime}\varphi_{uv}\cdot\varphi_{u}+{v^{\prime}}^{2}\varphi_{vv}\cdot\varphi_{u}\Big{]}$
$\displaystyle+\frac{\mu}{\kappa}\Big{[}{u^{\prime}}^{2}v^{\prime}L+2{v^{\prime}}^{2}u^{\prime}M+{v^{\prime}}^{3}N\Big{]}.$

Now for the isometric images of $\alpha$ and $\varphi_{u}$, we have

(3.13) $\displaystyle\overline{\alpha}\cdot\overline{\varphi}_{u}$
$\displaystyle=$
$\displaystyle\frac{\lambda}{\kappa}\Big{[}u^{\prime\prime}\overline{E}+v^{\prime\prime}\overline{F}+{u^{\prime}}^{2}\overline{\varphi}_{uu}\cdot\overline{\varphi}_{u}+2u^{\prime}v^{\prime}\overline{\varphi}_{uv}\cdot\overline{\varphi}_{u}+{v^{\prime}}^{2}\overline{\varphi}_{vv}\cdot\overline{\varphi}_{u}\Big{]}$
$\displaystyle+\frac{\mu}{\kappa}\Big{[}{u^{\prime}}^{2}v^{\prime}\overline{L}+2{v^{\prime}}^{2}u^{\prime}\overline{M}+{v^{\prime}}^{3}\overline{N}\Big{]}.$

Since we know that $\overline{E}=E,\overline{F}=F,\overline{G}=G$. In
particular

$E=\overline{E}=J_{\ast}\varphi_{u}\cdot
J_{\ast}\varphi_{u}=\varphi_{u}\cdot\varphi_{u}$

Differentiating the above equation with respect to $u$, we get

$\left(\frac{\partial J_{\ast}}{\partial
u}\varphi_{u}+J_{\ast}\varphi_{uu}\right)\cdot(J_{\ast}\varphi_{u})=\varphi_{uu}\cdot\varphi_{u}$

or

$\overline{\varphi}_{uu}\cdot\overline{\varphi}_{u}=\varphi_{uu}\cdot\varphi_{u}.$

Similarly, it is easy to show that

$\overline{\varphi}_{uv}\cdot\overline{\varphi}_{u}=\varphi_{uv}\cdot\varphi_{u},\quad\overline{\varphi}_{vv}\cdot\overline{\varphi}_{u}=\varphi_{vv}\cdot\varphi_{u}.$

Thus from (3.13), we get

$\displaystyle\overline{\alpha}\cdot\overline{\varphi}_{u}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{\kappa}\Big{[}u^{\prime\prime}E+v^{\prime\prime}F+{u^{\prime}}^{2}\varphi_{uu}\cdot\varphi_{u}+2u^{\prime}v^{\prime}\varphi_{uv}\cdot\varphi_{u}+{v^{\prime}}^{2}\varphi_{vv}\cdot\varphi_{u}\Big{]}$
$\displaystyle+\frac{\mu}{\kappa}\Big{[}{u^{\prime}}^{2}v^{\prime}\overline{L}+2{v^{\prime}}^{2}u^{\prime}\overline{M}+{v^{\prime}}^{3}\overline{N}\Big{]},$

or

(3.14)
$\overline{\alpha}\cdot\overline{\varphi}_{u}=\frac{\lambda}{\kappa}\Big{[}u^{\prime\prime}E+v^{\prime\prime}F+{u^{\prime}}^{2}\varphi_{uu}\cdot\varphi_{u}+2u^{\prime}v^{\prime}\varphi_{uv}\cdot\varphi_{u}+{v^{\prime}}^{2}\varphi_{vv}\cdot\varphi_{u}\Big{]}+v^{\prime}\frac{\mu}{\kappa}\overline{\kappa}_{n}.$

Taking the difference of (3.14) and (3.12), we get

(3.15)
$\overline{\alpha}\cdot\overline{\varphi}_{u}-\alpha\cdot\varphi_{u}=v^{\prime}\frac{\mu}{\kappa}(\overline{\kappa}_{n}-\kappa_{n}).$

Similarly the following relation hold

(3.16)
$\overline{\alpha}\cdot\overline{\varphi}_{v}-\alpha\cdot\varphi_{v}=u^{\prime}\frac{\mu}{\kappa}(\overline{\kappa}_{n}-\kappa_{n}).$

Now with the help of $(\ref{3.2.a})$ and $(\ref{3.2.b})$ we get

$\displaystyle\bar{\alpha}\cdot{\bf\bar{T}}-\alpha\cdot{\bf T}$
$\displaystyle=$
$\displaystyle\bar{\alpha}\cdot(a\bar{\varphi}_{u}+b\bar{\varphi}_{v})-\alpha\cdot(a\varphi_{u}+b\varphi_{v})$
$\displaystyle=$ $\displaystyle
a(\overline{\alpha}\cdot\overline{\varphi}_{u}-\alpha\cdot\varphi_{u})+b(\overline{\alpha}\cdot\overline{\varphi}_{v}-\alpha\cdot\varphi_{v})$
$\displaystyle=$ $\displaystyle
av^{\prime}\frac{\mu}{\kappa}(\overline{\kappa}_{n}-\kappa_{n})+bu^{\prime}\frac{\mu}{\kappa}(\overline{\kappa}_{n}-\kappa_{n})$
$\displaystyle=$
$\displaystyle\frac{\mu}{\kappa}(\bar{\kappa}_{n}-\kappa_{n})(av^{\prime}+bu^{\prime}).$

This proves our claim. ∎

######  Corollary 3.3.

Let $J:{\bf M}\rightarrow{\overline{\bf M}}$ be an isometry and $\alpha(s)$ be
a normal curve on ${\bf M}$. Then the component of the normal curve
$\alpha(s)$ along any tangent vector ${\bf T}$ to the surface ${\bf M}$ is
invariant if and only if any one of the following holds:

  * (i)

the position vector of $\alpha(s)$ is in the normal direction of $\alpha$.

  * (ii)

The normal curvature is invariant.

###### Proof.

From $(\ref{3.2.A})$, $\bar{\alpha}\cdot{\bf\bar{T}}=\alpha\cdot{\bf T}$ if
and only if

$\displaystyle\frac{\mu}{\kappa}(\bar{\kappa}_{n}-\kappa_{n})(av^{\prime}+bu^{\prime})=0$
$\displaystyle\text{i.e., if and only if }\mu=0\text{ or
}\bar{\kappa}_{n}-\kappa_{n}=0.$

If $\mu=0$ then from $(\ref{1})$, we see that $\alpha(s)=\lambda(s)n(s)$,
i.e., the position vector of the normal curve $\alpha(s)$ is in the normal
direction of itself. ∎

######  Corollary 3.4.

Let $J:{\bf M}\rightarrow{\overline{\bf M}}$ be an isometry and $\alpha(s)$ be
a normal curve on ${\bf M}$. The component of the normal curve $\alpha(s)$
along any tangent vector ${\bf T}$ to the surface ${\bf M}$ is invariant and
the position vector of $\alpha(s)$ is not in the normal direction of $\alpha$,
then $\alpha(s)$ is asymptotic if and only if $\bar{\alpha}(s)$ is asymptotic.

###### Proof.

From Corollary $3.3.$, $\bar{\alpha}\cdot{\bf\bar{T}}=\alpha\cdot{\bf T}$ and
the position vector of $\alpha(s)$ is not in the normal direction of $\alpha$
if and only if $\kappa_{n}=\bar{\kappa}_{n}$.
Therefore $\alpha(s)$ is asymptotic if and only if $\kappa_{n}=0$ if and only
if $\bar{\kappa}_{n}=0$ if and only if $\bar{\alpha}(s)$ is asymptotic. ∎

######  Theorem 3.5.

Let $J:{\bf M}\rightarrow{\overline{\bf M}}$ be an isometry and $\alpha(s)$ be
a normal curve on ${\bf M}$. Then for the component of $\alpha(s)$ along the
surface normal $N$, we have

(3.17) $\overline{\alpha}\cdot\overline{\bf N}-\alpha\cdot{\bf
N}=\frac{\lambda}{\kappa}(\overline{\kappa}_{n}-\kappa_{n}).$

###### Proof.

From (3), we have

$\displaystyle\alpha\cdot{\bf N}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{\kappa}\Big{[}{u^{\prime}}^{2}\varphi_{uu}\cdot(\varphi_{u}\times\varphi_{v})+{v^{\prime}}^{2}\varphi_{vv}\cdot(\varphi_{u}\times\varphi_{v})+2u^{\prime}v^{\prime}\varphi_{uv}\cdot(\varphi_{u}\times\varphi_{v})\Big{]}$
$\displaystyle+\frac{\mu}{\kappa}\Big{[}(u^{\prime}v^{\prime\prime}-v^{\prime}u^{\prime\prime})(EG-F^{2})+{u^{\prime}}^{3}(\varphi_{u}\times\varphi_{uu})\cdot(\varphi_{u}\times\varphi_{v})$
$\displaystyle+2{u^{\prime}}^{2}v^{\prime}(\varphi_{u}\times\varphi_{uv})\cdot(\varphi_{u}\times\varphi_{v})+{v^{\prime}}^{2}u^{\prime}(\varphi_{u}\times\varphi_{vv})\cdot(\varphi_{u}\times\varphi_{v})$
$\displaystyle+{u^{\prime}}^{2}v^{\prime}(\varphi_{v}\times\varphi_{uu})\cdot(\varphi_{u}\times\varphi_{v})+2{u^{\prime}}{v^{\prime}}^{2}(\varphi_{v}\times\varphi_{uv})\cdot(\varphi_{u}\times\varphi_{v})$
$\displaystyle+{v^{\prime}}^{3}(\varphi_{v}\times\varphi_{vv})\cdot(\varphi_{u}\times\varphi_{v})\Big{]}$

or

$\displaystyle\alpha\cdot{\bf N}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{\kappa}\Big{[}{u^{\prime}}^{2}L+{v^{\prime}}^{2}N+2u^{\prime}v^{\prime}M\Big{]}+\frac{\mu}{\kappa}\Big{[}(u^{\prime}v^{\prime\prime}-v^{\prime}u^{\prime\prime})(EG-F^{2})+{u^{\prime}}^{3}\\{E(\varphi_{uu}\cdot\varphi_{v})$
$\displaystyle-F(\varphi_{uu}\cdot\varphi_{u})\\}+2{u^{\prime}}^{2}v^{\prime}\\{E(\varphi_{uv}\cdot\varphi_{v})-F(\varphi_{uv}\cdot\varphi_{u})\\}+{u^{\prime}}{v^{\prime}}^{2}\\{E(\varphi_{vv}\cdot\varphi_{v})$
$\displaystyle-F(\varphi_{vv}\cdot\varphi_{u})\\}+{u^{\prime}}^{2}v^{\prime}\\{F(\varphi_{uu}\cdot\varphi_{v})-G(\varphi_{uu}\cdot\varphi_{u})\\}+2{u^{\prime}}{v^{\prime}}^{2}\\{F(\varphi_{uv}\cdot\varphi_{v})$
$\displaystyle-G(\varphi_{uv}\cdot\varphi_{u})\\}+{v^{\prime}}^{3}\\{F(\varphi_{vv}\cdot\varphi_{v})-G(\varphi_{vv}\cdot\varphi_{u})\\}\Big{]}.$

Since we know that with respect to isometry:
$E=\overline{E},F=\overline{F},G=\overline{G}$. Then, it easy to verify:

(3.20)
$\displaystyle\left\\{\begin{array}[]{ll}E_{u}=\overline{E}_{u},F_{u}=\overline{F}_{u},G_{u}=\overline{G}_{u}\\\
E_{v}=\overline{E}_{v},F_{v}=\overline{F}_{v},G_{v}=\overline{G}_{v}.\end{array}\right.$

Then we have
$E_{u}=(\varphi_{u}\cdot\varphi_{u})_{u}=\frac{1}{2}\varphi_{uu}\cdot\varphi_{u}$
or

(3.21) $\varphi_{uu}\cdot\varphi_{u}=\frac{E_{u}}{2}.$

On the similar lines, we can find

(3.24)
$\displaystyle\left\\{\begin{array}[]{ll}\varphi_{uu}\cdot\varphi_{v}=F_{u}-\frac{E_{v}}{2},\varphi_{vv}\cdot\varphi_{v}=\frac{G_{v}}{2},\varphi_{vv}\cdot\varphi_{u}=F_{v}=\frac{G_{u}}{2}\\\
\varphi_{uv}\cdot\varphi_{v}=\frac{G_{u}}{2},\varphi_{uv}\cdot\varphi_{u}=\frac{E_{v}}{2}\end{array}\right.$

Therefore in view of (3.21) and (3.24), we get

$\displaystyle\alpha\cdot{\bf N}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{\kappa}\Big{[}{u^{\prime}}^{2}L+{v^{\prime}}^{2}N+2u^{\prime}v^{\prime}M\Big{]}$
$\displaystyle+\frac{\mu}{\kappa}\Big{[}(u^{\prime}v^{\prime\prime}-v^{\prime}u^{\prime\prime})(EG-F^{2})+{u^{\prime}}^{3}\left\\{E\left(F_{u}-\frac{E_{v}}{2}\right)-\frac{FE_{u}}{2}\right\\}$
$\displaystyle+2{u^{\prime}}^{2}v^{\prime}\left\\{\frac{EG_{u}}{2}-\frac{FE_{v}}{2}\right\\}+{u^{\prime}}{v^{\prime}}^{2}\left\\{\frac{EG_{v}}{2}-F\left(F_{v}-\frac{G_{u}}{2}\right)\right\\}$
$\displaystyle+{u^{\prime}}^{2}v^{\prime}\left\\{\frac{FG_{u}}{2}-\frac{GE_{u}}{2}\right\\}+2{u^{\prime}}{v^{\prime}}^{2}\left\\{FG_{u}-\frac{GE_{v}}{2}\right\\}$
$\displaystyle+{v^{\prime}}^{3}\left\\{\frac{FG_{v}}{2}-G\left(F_{v}-\frac{G_{u}}{2}\right)\right\\}\Big{]}.$

Now applying $J$ and with the help of (3.20), we get

$\displaystyle\overline{\alpha}\cdot\overline{{\bf N}}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{\kappa}\Big{[}{u^{\prime}}^{2}\overline{L}+{v^{\prime}}^{2}\overline{N}+2u^{\prime}v^{\prime}\overline{M}\Big{]}$
$\displaystyle+\frac{\mu}{\kappa}\Big{[}(u^{\prime}v^{\prime\prime}-v^{\prime}u^{\prime\prime})(EG-F^{2})+{u^{\prime}}^{3}\left\\{E\left(F_{u}-\frac{E_{v}}{2}\right)-\frac{FE_{u}}{2}\right\\}$
$\displaystyle+2{u^{\prime}}^{2}v^{\prime}\left\\{\frac{EG_{u}}{2}-\frac{FE_{v}}{2}\right\\}+{u^{\prime}}{v^{\prime}}^{2}\left\\{\frac{EG_{v}}{2}-F\left(F_{v}-\frac{G_{u}}{2}\right)\right\\}$
$\displaystyle+{u^{\prime}}^{2}v^{\prime}\left\\{\frac{FG_{u}}{2}-\frac{GE_{u}}{2}\right\\}+2{u^{\prime}}{v^{\prime}}^{2}\left\\{FG_{u}-\frac{GE_{v}}{2}\right\\}$
$\displaystyle+{v^{\prime}}^{3}\left\\{\frac{FG_{v}}{2}-G\left(F_{v}-\frac{G_{u}}{2}\right)\right\\}\Big{]}.$

On taking the difference of $\alpha\cdot{\bf N}$ and its isometric image and
with the help of $(3.2)$, we get

$\overline{\alpha}\cdot\overline{\bf N}-\alpha\cdot{\bf
N}=\frac{\lambda}{\kappa}(\overline{\kappa}_{n}-\kappa_{n}).$

This proves our claim. ∎

######  Corollary 3.6.

Let $J:{\bf M}\rightarrow{\overline{\bf M}}$ be an isometry and $\alpha(s)$ be
a normal curve on ${\bf M}$. Then the component of the normal curve
$\alpha(s)$ along the surface normal is invariant if and only if any one of
the following holds:

  * (i)

The position vector of $\alpha(s)$ is in the binormal direction of $\alpha$.

  * (ii)

The normal curvature is invariant.

###### Proof.

From $(\ref{3.4.B})$, $\bar{\alpha}\cdot{\bf\bar{N}}=\alpha\cdot{\bf N}$ if
and only if

$\displaystyle\frac{\lambda}{\kappa}(\overline{\kappa}_{n}-\kappa_{n})=0$
$\displaystyle\text{i.e., if and only if }\lambda=0\text{ or
}\overline{\kappa}_{n}-\kappa_{n}=0.$

If $\lambda=0$ then from $(\ref{1})$, we see that $\alpha(s)=\mu(s)b(s)$,
i.e., the position vector of the normal curve $\alpha(s)$ is in the binormal
direction of itself. ∎

######  Corollary 3.7.

Let $J:{\bf M}\rightarrow{\overline{\bf M}}$ be an isometry and $\alpha(s)$ be
a normal curve on ${\bf M}$. The component of the normal curve $\alpha(s)$
along surface normal ${\bf N}$ to the surface ${\bf M}$ is invariant and the
position vector of $\alpha(s)$ is not in the normal direction of $\alpha$,
then $\alpha(s)$ is asymptotic if and only if $\bar{\alpha}(s)$ is asymptotic.

###### Proof.

From Corollary $3.6.$, $\bar{\alpha}\cdot{\bf\bar{N}}=\alpha\cdot{\bf N}$ and
the position vector of $\alpha(s)$ is not in the normal direction of $\alpha$
if and only if $\kappa_{n}=\bar{\kappa}_{n}$.
Therefore $\alpha(s)$ is asymptotic if and only if $\kappa_{n}=0$ if and only
if $\bar{\kappa}_{n}=0$ if and only if $\bar{\alpha}(s)$ is asymptotic. ∎

Since $\bf T$ and $\bf N$ are perpendicular vector at $\alpha(s)$, hence
$\\{\bf T,\bf N,\bf T\times\bf N\\}$ form an orthogonal system at every point
of the normal curve $\alpha(s)$.

######  Theorem 3.8.

Let $J:{\bf M}\rightarrow{\overline{\bf M}}$ be an isometry and $\alpha(s)$ be
a normal curve on ${\bf M}$. Then for the component of $\alpha(s)$ along $\bf
T\times\bf N$, we have

(3.25) $\bar{\alpha}\cdot({\bf\bar{T}}\times{\bf\bar{N}})-\alpha\cdot({\bf
T}\times{\bf
N})=\frac{\mu}{\kappa}(\bar{\kappa}_{n}-\kappa_{n})\\{a(Fv^{\prime}-Eu^{\prime})+b(Gv^{\prime}-Fu^{\prime})\\}$

###### Proof.

From $(\ref{2})$, we have

$\displaystyle\alpha\cdot(\bf T\times\bf N)$ $\displaystyle=$
$\displaystyle\alpha\cdot\\{(a\varphi_{u}+b\varphi_{v})\times\bf N\\},$
$\displaystyle=$
$\displaystyle\alpha\cdot\\{a(F\varphi_{u}-E\varphi_{v})+b(G\varphi_{u}-F\varphi_{v})\\},$
$\displaystyle=$
$\displaystyle(aF+bG)\alpha\cdot\varphi_{u}-(aE+bF)\alpha\cdot\varphi_{v}.$

Therefore using $(\ref{3.2.a})$ and $(\ref{3.2.b})$ we get

$\displaystyle\bar{\alpha}\cdot(\bar{\bf T}\times\bar{\bf N})-\alpha\cdot(\bf
T\times\bf N)$ $\displaystyle=$
$\displaystyle(aF+bG)(\overline{\alpha}\cdot\overline{\varphi}_{u}-\alpha\cdot\varphi_{u})-(aE+bF)$
$\displaystyle(\overline{\alpha}\cdot\overline{\varphi}_{v}-\alpha\cdot\varphi_{v}),$
$\displaystyle=$
$\displaystyle\frac{\mu}{\kappa}(\bar{\kappa}_{n}-\kappa_{n})\\{a(Fv^{\prime}-Eu^{\prime})+b(Gv^{\prime}-Fu^{\prime})\\}.$

This proves our claim. ∎

######  Corollary 3.9.

Let $J:{\bf M}\rightarrow{\overline{\bf M}}$ be an isometry and $\alpha(s)$ be
a normal curve on ${\bf M}$. Then the component of the normal curve
$\alpha(s)$ along $\bf T\times\bf N$ is invariant if and only if any one of
the following holds:

  * (i)

The position vector of $\alpha(s)$ is in the normal direction of $\alpha$.

  * (ii)

The normal curvature is invariant.

###### Proof.

From $(\ref{3.6.C})$, $\bar{\alpha}\cdot(\bar{\bf T}\times\bar{\bf
N})=\alpha\cdot(\bf T\times\bf N)$ if and only if

$\displaystyle\frac{\mu}{\kappa}(\bar{\kappa}_{n}-\kappa_{n})\\{a(Fv^{\prime}-Eu^{\prime})+b(Gv^{\prime}-Fu^{\prime})\\}=0$
$\displaystyle\text{i.e., if and only if }\mu=0\text{ or
}\bar{\kappa}_{n}-\kappa_{n}=0.$

If $\mu=0$ then from $(\ref{1})$, we see that $\alpha(s)=\lambda(s)n(s)$,
i.e., the position vector of the normal curve $\alpha(s)$ is in the normal
direction of itself. ∎

######  Corollary 3.10.

Let $J:{\bf M}\rightarrow{\overline{\bf M}}$ be an isometry and $\alpha(s)$ be
a normal curve on ${\bf M}$. The component of the normal curve $\alpha(s)$
along $\bf T\times\bf N$ is invariant and the position vector of $\alpha(s)$
is not in the normal direction of $\alpha$, then $\alpha(s)$ is asymptotic if
and only if $\bar{\alpha}(s)$ is asymptotic.

###### Proof.

From Corollary $3.9.$, $\bar{\alpha}\cdot(\bar{\bf T}\times\bar{\bf
N})=\alpha\cdot(\bf T\times\bf N)$ and the position vector of $\alpha(s)$ is
not in the normal direction of $\alpha$ if and only if
$\kappa_{n}=\bar{\kappa}_{n}$.
Therefore $\alpha(s)$ is asymptotic if and only if $\kappa_{n}=0$ if and only
if $\bar{\kappa}_{n}=0$ if and only if $\bar{\alpha}(s)$ is asymptotic. ∎

######  Proposition 1.

The geodesic curvature of a smooth curve and in particular of a normal curve
remains invariant under isometry.

###### Proof.

Let $\alpha$ be a curve on a parametric surface ${\bf M}$, then the geodesic
curvature is given by Beltrami formula as:

(3.26)
$\kappa_{g}=\Big{[}\Gamma_{11}^{2}{u^{\prime}}^{3}+(2\Gamma_{12}^{2}-\Gamma_{11}^{1}){u^{\prime}}^{2}v^{\prime}+(\Gamma_{22}^{2}-2\Gamma_{12}^{1})u^{\prime}{v^{\prime}}^{2}-\Gamma_{22}^{1}{v^{\prime}}^{3}+u^{\prime}v^{\prime\prime}-u^{\prime\prime}v^{\prime}\Big{]}\sqrt{EG-F^{2}},$

where $\Gamma_{ij}^{k}$ are the Christoffel symbols of the second kind given
by

(3.27)
$\left\\{\begin{array}[]{ll}\Gamma_{11}^{1}=\frac{1}{2W^{2}}\left\\{GE_{u}+F[E_{v}-2F_{u}]\right\\},\quad\Gamma_{22}^{2}=\frac{1}{2W^{2}}\left\\{EG_{v}+F[G_{v}-2F_{v}]\right\\}\\\
\Gamma_{11}^{2}=\frac{1}{2W^{2}}\left\\{E[2F_{u}-E_{v}]-FE_{v}\right\\},\quad\Gamma_{22}^{1}=\frac{1}{2W^{2}}\left\\{G[2F_{v}-G_{u}]-FG_{v}\right\\}\\\
\Gamma_{12}^{2}=\frac{1}{2W^{2}}\left\\{EG_{u}-FE_{v}\right\\}=\Gamma_{21}^{2},\quad\Gamma_{21}^{1}=\frac{1}{2W^{2}}\left\\{GE_{v}-FG_{u}\right\\}=\Gamma_{12}^{1}\end{array}\right.$

and $W=\sqrt{EG-F^{2}}$. Thus, in view of (3.20), (3.26) and (3.27), we see
that $\overline{\kappa}_{g}=\kappa_{g}$. In particular the same holds for a
normal curve. ∎

##  4\. acknowledgment

The third author greatly acknowledges to The University Grants Commission,
Government of India for the award of Junior Research Fellow.

## References

  * [1] A. Pressley, Elementary differential geometry, Springer-Verlag, 2001.
  * [2] M. P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Inc, New Jersey, 1976.
  * [3] B.-Y. Chen, What does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly, 110 (2003), 147-152.
  * [4] B.-Y. Chen and F. Dillen, Rectfying curve as centrode and extremal curve., Bull. Inst. Math. Acad. Sinica, 33, no. 2, (2005), 77-90.
  * [5] S. Deshmukh, B.-Y. Chen and S. H. Alshammari, On a rectifying curves in euclidean 3-space, Turk. J. Math., 42 (2018), 609-620.
  * [6] M. Grbović and Emilija Nešović, Some relations between rectifying and normal curves in Minkowski 3-space, Math. Commun., 17 (2012), 655-664.
  * [7] K. Ilarslan, M. Sakaki and A. Uçum, On osculating, normal and rectifying bi-null curves in $\mathbb{R}_{2}^{5}$, Novi Sad J. Math. , 48 (2018), 9-20.
  * [8] A. A Shaikh and P. R. Ghosh Rectifying curves on a smooth surface immersed in the Euclidean space, to appear in Indian J. Pure Appl. Math., (2018).
  * [9] A. A Shaikh and P. R. Ghosh Rectifying and osculating curves on a smooth surface, to appear in Indian J. Pure Appl. Math., (2018).
  * [10] A. A Shaikh and P. R. Ghosh Curves on a smooth surface with position vectors lie in the tangent plane, Submited, (2018).