# SpaceNet MVOI: a Multi-View Overhead Imagery Dataset

Nicholas Weir  In-Q-Tel CosmiQ Works, [nweir, avanetten<EMAIL_ADDRESS>David Lindenbaum  Accenture Federal Services,
<EMAIL_ADDRESS>Alexei Bastidas  Intel AI Lab,
[alexei.a.bastidas, sean.mcpherson, varun.v.kumar<EMAIL_ADDRESS>Adam Van Etten  In-Q-Tel CosmiQ Works, [nweir, avanetten<EMAIL_ADDRESS>Sean McPherson  Intel AI Lab, [alexei.a.bastidas, sean.mcpherson,
varun.v.kumar<EMAIL_ADDRESS>Jacob Shermeyer  In-Q-Tel CosmiQ Works,
[nweir, avanetten<EMAIL_ADDRESS>Varun Kumar  Intel AI Lab,
[alexei.a.bastidas, sean.mcpherson, varun.v.kumar<EMAIL_ADDRESS>Hanlin Tang  Intel AI Lab, [alexei.a.bastidas, sean.mcpherson, varun.v.kumar,
<EMAIL_ADDRESS>
###### Abstract

Detection and segmentation of objects in overheard imagery is a challenging
task. The variable density, random orientation, small size, and instance-to-
instance heterogeneity of objects in overhead imagery calls for approaches
distinct from existing models designed for natural scene datasets. Though new
overhead imagery datasets are being developed, they almost universally
comprise a single view taken from directly overhead (“at nadir”), failing to
address a critical variable: look angle. By contrast, views vary in real-world
overhead imagery, particularly in dynamic scenarios such as natural disasters
where first looks are often over $40^{\circ}$ off-nadir. This represents an
important challenge to computer vision methods, as changing view angle adds
distortions, alters resolution, and changes lighting. At present, the impact
of these perturbations for algorithmic detection and segmentation of objects
is untested. To address this problem, we present an open source Multi-View
Overhead Imagery dataset, termed SpaceNet MVOI, with 27 unique looks from a
broad range of viewing angles ($-32.5^{\circ}$ to $54.0^{\circ}$). Each of
these images cover the same 665 $km^{2}$ geographic extent and are annotated
with 126,747 building footprint labels, enabling direct assessment of the
impact of viewpoint perturbation on model performance. We benchmark multiple
leading segmentation and object detection models on: (1) building detection,
(2) generalization to unseen viewing angles and resolutions, and (3)
sensitivity of building footprint extraction to changes in resolution. We find
that state of the art segmentation and object detection models struggle to
identify buildings in off-nadir imagery and generalize poorly to unseen views,
presenting an important benchmark to explore the broadly relevant challenge of
detecting small, heterogeneous target objects in visually dynamic contexts.

##  1 Introduction

Recent years have seen increasing use of convolutional neural networks to
analyze overhead imagery collected by aerial vehicles or space-based sensors,
for applications ranging from agriculture [18] to surveillance [39, 32] to
land type classification [3]. Segmentation and object detection of overhead
imagery data requires identifying small, visually heterogeneous objects (e.g.
cars and buildings) with varying orientation and density in images, a task
ill-addressed by existing models developed for identification of comparatively
larger and lower-abundance objects in natural scene images. The density and
visual appearance of target objects change dramatically as look angle,
geographic location, time of day, and seasonality vary, further complicating
the problem. Addressing these challenges will provide broadly useful insights
for the computer vision community as a whole: for example, how to build
segmentation models to identify low-information objects in dense contexts.

|  | Urban | Industrial | Dense Residential | Sparse Residential
---|---|---|---|---|---
|  7 (NADIR) |  |  |  |
LOOK ANGLE (BIN) |  -32 (OFF) |  |  |  |
|  52 (VOFF) |  |  |  |

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 1: Sample imagery from SpaceNet MVOI. Four of the 2222 geographically
unique image chips in the dataset are shown (columns), with three of the 27
views of that chip (rows), one from each angle bin. Negative look angle
corresponds to South-facing views, whereas positive look angles correspond to
North-facing views (Figure 2). Chips are down-sampled from $900\times 900$
pixel high-resolution images. In addition to the RGB images shown, the dataset
comprises a high-resolution pan-chromatic (grayscale) band, a high-resolution
near-infrared band, and a lower-resolution 8-band multispectral image for each
geographic location/view combination. The dataset is available at
https://spacenet.ai under a CC-BY SA 4.0 License.

Though public overhead imagery datasets explore geographic and sensor
homogeneity [8, 12, 22, 34, 19], they generally comprise a single view of the
imaged location(s) taken nearly directly overhead (“at nadir”). Nadir imagery
is not representative of collections during disaster response or other urgent
situations: for example, the first public high-resolution cloud-free image of
San Juan, Puerto Rico following Hurricane Maria was taken at $51.9^{\circ}$
“off-nadir”, i.e., a $51.9^{\circ}$ angle between the nadir point directly
underneath the satellite and the center of the imaged scene [10]. The
disparity between looks in public training data and relevant use cases hinders
development of models applicable to real-world problems. More generally,
satellite and drone images rarely capture identical looks at objects in
different contexts, or even when repeatedly imaging the same geography.
Furthermore, no existing datasets or metrics permit assessment of model
robustness to different looks, prohibiting evaluation of performance. These
limitations extend to tasks outside of the geospatial domain: for example,
convolutional neural nets perform inconsistently in many natural scene video
frame classification tasks despite minimal pixel-level variation [1], and Xiao
et al. showed that spatial transformation of images, effectively altering
view, represents an effective adversarial attack against computer vision
models [36]. Addressing generalization across views both within and outside of
the geospatial domain requires two advancements: 1. A large multi-view dataset
with diversity in land usage, population density, and views, and 2. A metric
to assess model generalization.

To address the limitations detailed above, we introduce the SpaceNet Multi-
View Overhead Imagery (MVOI) dataset, which includes 62,000 overhead images
collected over Atlanta, Georgia USA and the surrounding areas. The dataset
comprises 27 distinct looks, including both North- and South-facing views,
taken during a single pass of a Maxar WorldView-2 satellite. The looks range
from almost directly overhead ($7.8^{\circ}$ off-nadir) to up to $54^{\circ}$
off-nadir, with the same $665\ km^{2}$ geographic area covered by each.
Alongside the imagery we open sourced an attendant 126,747 building footprints
created by expert labelers. To our knowledge, this is the first multi-
viewpoint dataset for overhead imagery with dense object annotations. The
dataset covers heterogeneous geographies, including highly treed rural areas,
suburbs, industrial areas, and high-density urban environments, resulting in
heterogeneous building size, density, context and appearance (Figure 1). At
the same time, the dataset abstracts away many other time-sensitive variables
(e.g. seasonality), enabling careful assessment of the impact of look angle on
model training and inference. The training imagery and labels and public test
images are available at https://spacenet.ai under the CC-BY SA 4.0
International License.

Though an ideal overhead imagery dataset would cover all the variables present
in overhead imagery, i.e. look angle, seasonality, geography, weather
condition, sensor, and light conditions, creating such a dataset is impossible
with existing imagery. To our knowledge, the 27 unique looks in SpaceNet MVOI
represent one of only two such imagery collections available in the commercial
realm, even behind imagery acquisition company paywalls. We thus chose to
focus SpaceNet MVOI on providing a diverse set of views with varying look
angle and direction, a variable that is not represented in any existing
overhead imagery dataset. SpaceNet MVOI could potentially be combined with
existing datasets to train models which generalize across more variables.

We benchmark state-of-the art models on three tasks:

  1. 1.

Building segmentation and detection.

  2. 2.

Generalization of segmentation and object detection models to previously
unseen angles.

  3. 3.

Consequences of changes in resolution for segmentation and object detection
models.

Our benchmarking reveals that state-of-the-art detectors are challenged by
SpaceNet MVOI, particularly in views left out during model training.
Segmentation and object detection models struggled to account for displacement
of building footprints, occlusion, shadows, and distortion in highly off-nadir
looks (Figure 3). The challenge of addressing footprint displacement is of
particular interest, as it requires models not only to learn visual features,
but to adjust footprint localization dependent upon the view context.
Addressing these challenges is relevant to a number of applications outside of
overhead imagery analysis, e.g. autonomous vehicle vision.

To assess model generalization to new looks we developed a generalization
metric $G$, which reports the relative performance of models when they are
applied to previously unseen looks. While specialized models designed for
overhead imagery out-perform general baseline models in building footprint
detection, we found that models developed for natural image computer vision
tasks have better $G$ scores on views absent during training. These
observations highlight the challenges associated with developing robust models
for multi-view object detection and semantic segmentation tasks. We therefore
expect that developments in computer vision models for multi-view analysis
made using SpaceNet MVOI, as well as analysis using our metric $G$, will be
broadly relevant for many computer vision tasks.

The dataset is available at www.spacenet.ai.

##  2 Related Work

Object detection and segmentation is a well-studied problem for natural scene
images, but those objects are generally much larger and suffer minimally from
distortions exacerbated in overhead imagery. Natural scene research is driven
by datasets such as MSCOCO [20] and PASCALVOC [13], but those datasets lack
multiple views of each object. PASCAL3D [35], autonomous driving datasets such
as KITTI [14], CityScapes [7], existing multi-view datasets [29, 30], and
tracking datasets such as MOT2017[24] or OBT [33] contains different views but
are confined to a narrow range of angles, lack sufficient heterogeneity to
test generalization between views, and are restricted to natural scene images.
Multiple viewpoints are found in 3D model datasets [5, 23], but those are not
photorealistic and lack the occlusion and visual distortion properties
encountered with real imagery.

Previous datasets for overhead imagery focus on classification [6], bounding
box object detection [34, 19, 25], instance-based segmentation [12], and
object tracking [26] tasks. None of these datasets comprise multiple images of
the same field of view from substantially different look angles, making it
difficult to assess model robustness to new views. Within segmentation
datasets, SpaceNet [12] represents the closest work, with dense building and
road annotations created by the same methodology. We summarize the key
characteristics of each dataset in Table 1. Our dataset matches or exceeds
existing datasets in terms of imagery size and annotation density, but
critically includes varying look direction and angle to better reflect the
visual heterogeneity of real-world imagery.

The effect of different views on segmentation or object detection in natural
scenes has not been thoroughly studied, as feature characteristics are
relatively preserved even under rotation of the object in that context.
Nonetheless, preliminary studies of classification model performance on video
frames suggests that minimal pixel-level changes can impact performance [1].
By contrast, substantial occlusion and distortion occurs in off-nadir overhead
imagery, complicating segmentation and placement of geospatially accurate
object footprints, as shown in Figure 3A-B. Furthermore, due to the
comparatively small size of target objects (e.g. buildings) in overhead
imagery, changing view substantially alters their appearance (Figure 3C-D). We
expect similar challenges to occur when detecting objects in natural scene
images at a distance or in crowded views. Existing solutions to occlusion are
often domain specific [37] or rely on attention mechanisms to identify common
elements [40] or landmarks [38]. The heterogeneity in building appearance in
overhead imagery, and the absence of landmark features to identify them, makes
their detection an ideal research task for developing domain-agnostic models
that are robust to occlusion.

Dataset | Gigapixels | # Images | Resolution (m) | Nadir Angles | # Objects | Annotation
---|---|---|---|---|---|---
SpaceNet [12, 8] | 10.3 | 24586 | 0.31 | On-Nadir | 302701 | Polygons
DOTA [34] | 44.9 | 2806 | Google Earth* | On-Nadir | 188282 | Oriented Bbox
3K Vehicle Detection [21] | N/A | 20 | 0.20 | Aerial | 14235 | Oriented Bbox
UCAS-AOD [41] | N/A | 1510 | Google Earth* | On-Nadir | 3651 | Oriented Bbox
NWPU VHR-10 [4] | N/A | 800 | Google Earth* | On-Nadir | 3651 | Bbox
MVS [2] | 111 | 50 | 0.31-0.58 | [5.3, 43.3] | 0 | None
FMoW [6] | 1,084.0 | 523846 | 0.31-1.60 | [0.22, 57.5] | 132716 | Classification
xView [19] | 56.0 | 1400 | 0.31 | On-Nadir | $1000000$ | Bbox
SpaceNet MVOI (Ours) | 50.2 | 60000 | 0.46-1.67 | [-32.5, +54.0] | 126747 | Polygons
PascalVOC [13] | - | 21503 | - | - | 62199 | Bbox
MSCOCO [20] | - | 123287 | - | - | 886266 | Bbox
ImageNet [9] | - | 349319 | - | - | 478806 | Bbox
Table 1: Comparison with other computer vision and overhead imagery datasets.
Our dataset has a similar scale as modern computer vision datasets, but to our
knowledge is the first multi-view overhead imagery dataset designed for
segmentation and object detection tasks. *Google Earth imagery is a mosaic
from a variety of aerial and satellite sources and ranges from 15 cm to 12 m
resolution [15].

##  3 Dataset Creation

Figure 2: Collect views. Location of collection points during the WorldView-2
satellite pass over Atlanta, GA USA.

SpaceNet MVOI contains images of Atlanta, GA USA and surrounding geography
collected by Maxar’s WorldView-2 Satellite on December 22, 2009 [22]. The
satellite collected 27 distinct views of the same 665 km2 ground area during a
single pass over a 5 minute span. This produced 27 views with look angles
(angular distance between the nadir point directly underneath the satellite
and the center of the scene) from $7.8^{\circ}$ to $54^{\circ}$ off-nadir and
with a target azimuth angle (compass direction of image acquisition) of
$17^{\circ}$ to $182.8^{\circ}$ from true North (see Figure 2). See the
Supplementary Material and Tables S1 and S2 for further details regarding the
collections. The 27 views in a narrow temporal band provide a dense set of
visually distinct perspectives of static objects (buildings, roads, trees,
utilities, etc.) while limiting complicating factors common to remote sensing
datasets such as changes in cloud cover, sun angle, or land-use change. The
imaged area is geographically diverse, including urban areas, industrial
zones, forested suburbs, and undeveloped areas (Figure 1).

| Challenges in off-nadir imagery
---|---
Footprint offset and occlusion |  |
Shadows |  |

(a) 7 degrees

(b) 53 degrees

(c) 30 degrees

(d) -32 degrees

Figure 3: Challenges with off-nadir look angles. Though geospatially accurate
building footprints (blue) perfectly match building roofs at nadir (A), this
is not the case off-nadir (B), and many buildings are obscured by skyscrapers.
(C-D): Visibility of some buildings changes at different look angles due to
variation in reflected sunlight.

###  3.1 Preprocessing

Multi-view satellite imagery datasets are distinct from related natural image
datasets in several interesting ways. First, as look angle increases in
satellite imagery, the native resolution of the image decreases because
greater distortion is required to project the image onto a flat grid (Figure
1). Second, each view contains images with multiple spectral bands. For the
purposes our baselines, we used 3-channel images (RGB: red, green, blue), but
also examined the contributions of the near-infrared (NIR) channel (see
Supplementary Material). These images were enhanced with a separate, higher
resolution panchromatic (grayscale) channel to double the original resolution
of the multispectral imagery (i.e., “pan-sharpened”). The entire dataset was
tiled into $900px\times 900px$ tiles and resampled to simulate a consistent
resolution across all viewing angles of $0.5m\times 0.5m$ ground sample
distance. The dataset also includes lower-resolution 8-band multispectral
imagery with additional color channels, as well as panchromatic images, both
of which are common overhead imagery data types.

The 16-bit pan-sharpened RGB-NIR pixel intensities were truncated at 3000 and
then rescaled to an 8-bit range before normalizing to $[0,1]$. We also trained
models directly using Z-score normalized 16 bit images with no appreciable
difference in the results.

###  3.2 Annotations

We undertook professional labeling to produce high-quality annotations. An
expert geospatial team exhaustively labeled building footprints across the
imaged area using the most on-nadir image ($7.8^{\circ}$ off-nadir).
Importantly, the building footprint polygons represent geospatially accurate
ground truth, and therefore are shared across all views. For structures
occluded by trees, only the visible portion was labeled. Finally, one
independent validator and one remote sensing expert evaluated the quality of
each label.

Figure 4: Dataset statistics. Distribution of (A) building footprint areas and
(B) number of objects per $450m\times 450m$ geographic tile in the dataset.

###  3.3 Dataset statistics

Our dataset labels comprise a broad distribution of building sizes, as shown
in Figure 4A. Compared to natural image datasets, our dataset more heavily
emphasizes small objects, with the majority of objects less than 700 pixels in
area, or $\sim 25$ pixels across. By contrast, objects in the PASCALVOC [13]
or MSCOCO [20] datasets usually comprise 50-300 pixels along the major axis
[34].

An additional challenge presented by this dataset, consistent with many real-
world computer vision tasks, is the heterogeneity in target object density
(Figure 4B). Images contained between zero and 300 footprints, with
substantial coverage throughout that range. This variability presents a
challenge to object detection algorithms, which often require estimation of
the number of features per image [16]. Segmentation and object detection of
dense or variable density objects is challenging, making this an ideal dataset
to test the limits of algorithms’ performance.

Task | Baseline models
---|---
Semantic Segmentation | TernausNet [17] , U-NET [27]
Instance Segmentation | Mask R-CNN [16]
Object Detection | Mask R-CNN [16], YOLT [11]
Table 2: Benchmark model selections for dataset baselines. TernausNet and YOLT
are overhead imagery-specific models, whereas Mask R-CNN and U-Net are popular
natural scene analysis models.

##  4 Building Detection Experiments

###  4.1 Dataset preparation for analysis

We split the training and test sets 80/20 by randomly selecting geographic
locations and including all views for that location in one split, ensuring
that each type of geography was represented in both splits. We group each
angle into one of three categories: Nadir (NADIR), $\theta\leq 25^{\circ}$;
Off-nadir (OFF), $25^{\circ}<\theta<40^{\circ}$; and Very off-nadir (VOFF),
$\theta\geq 40^{\circ}$. In all experiments, we trained baselines using all
viewing angles (ALL) or one of the three subsets. These trained models were
then evaluated on the test set of each of the 27 viewing angles individually.

###  4.2 Models

We measured several state of the art baselines for semantic or instance
segmentation and object detection (Table 2). Where possible, we selected
overhead imagery-specific models as well as models for natural scenes to
compare their performance. Object detection baselines were trained using
rectangular boundaries extracted from the building footprints. To fairly
compare with semantic segmentation studies, the resulting bounding boxes were
compared against the ground truth building polygons for scoring (see Metrics).

###  4.3 Segmentation Loss

Due to the class imbalance of the training data – only 9.5% of the pixels in
the training set correspond to buildings – segmentation models trained with
binary cross-entropy (BCE) loss failed to identify building pixels, a problem
observed previously for overhead imagery segmentation models [31]. For the
semantic segmentation models, we therefore utilized a hybrid loss function
that combines the binary cross entropy loss and intersection over union (IoU)
loss with a weight factor $\alpha$ [31]:

$\mathcal{L}=\alpha\mathcal{L}_{\text{BCE}}+(1-\alpha)\mathcal{L}_{\text{IoU}}$
(1)

The details of model training and evaluation, including augmentation,
optimizers, and evaluation schemes can be found in the Supplementary Material.

###  4.4 Metrics

We measured performance using the building IoU-$F_{1}$ score defined in Van
Etten et al. [12]. Briefly, building footprint polygons were extracted from
segmentation masks (or taken directly from object detection bounding box
outputs) and compared to ground truth polygons. Predictions were labeled True
Positive if they had an IoU with a ground truth polygon above 0.5 and all
other predictions were deemed False Positives. Using these statistics and the
number of undetected ground truth polygons (False Negatives), we calculated
the precision $P$ and recall $R$ of the model predictions in aggregate. We
then report the $F_{1}$ score as

$F_{1}=\frac{2\times P\times R}{P+R}$ (2)

$F_{1}$ score was calculated within each angle bin (NADIR, OFF, or VOFF) and
then averaged for an aggregate score.

|  | $\boldsymbol{F_{1}}$
---|---|---
Task | Model | NADIR | OFF | VOFF | Avg.
Seg | TernausNet | 0.62 | 0.43 | 0.22 | 0.43
Seg | U-Net | 0.39 | 0.27 | 0.08 | 0.24
Seg | Mask R-CNN | 0.47 | 0.34 | 0.07 | 0.29
Det | Mask R-CNN | 0.40 | 0.30 | 0.07 | 0.25
Det | YOLT | 0.49 | 0.37 | 0.20 | 0.36
Table 3: Overall task difficulty. As a measure of overall task difficulty, the
performance ($F_{1}$ score) is assessed for the baseline models trained on all
angles, and tested on the three different viewing angle bins: nadir (NADIR),
off-nadir (OFF), and very off-nadir (VOFF). Avg. is the linear mean of the
three bins. Seg, segmentation; Det, object detection.

###  4.5 Results

|  | Image | Mask R-CNN | TernausNet | YOLT
---|---|---|---|---|---
|  10 (NADIR) |  |  |  |
LOOK ANGLE (BIN) |  -29 (OFF) |  |  |  |
|  53 (VOFF) |  |  |  |

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 5: Sample imagery (left) with ground truth building footprints and Mask
R-CNN bounding boxes (middle left), TernausNet segmentation masks (middle
right), and YOLT bounding boxes (right). Ground truth masks (light blue) are
shown under Mask R-CNN and TernausNet predictions (yellow). YOLT bounding
boxes shown in blue. Sign of the look angle represents look direction
(negative=south-facing, positive=north-facing). Predictions from models
trained on on all angles (see Table 3).  Figure 6: Performance by look angle
for various training subsets. TernausNet (left), Mask R-CNN (middle), and YOLT
(right) models, trained on ALL, NADIR, OFF, or VOFF, were evaluated in the
building detection task and $F_{1}$ scores are displayed for each evaluation
look angle. Imagery acquired facing South is represented as a negative number,
whereas looks facing North are represented by a positive angle value.
Additionally, TernausNet models trained only on North-facing OFF imagery
(positive OFF) and South-facing OFF imagery (negative OFF) were evaluated on
each angle to explore the importance of look direction.

The state-of-the-art segmentation and object detection models we measured were
challenged by this task. As shown in Table 3, TernausNet trained on all angles
achieves $F_{1}=0.62$ on the nadir angles, which is on par with previous
building segmentation results and competitions [12, 8]. However, performance
drops significantly for off-nadir ($F_{1}=0.43$) and very off-nadir
($F_{1}=0.22$) images. Other models display a similar degradation in
performance. Example results are shown in Figure 5.

Directional asymmetry. Figure 6 illustrates performance per angle for both
segmentation and object detection models. Note that models trained on positive
(north-facing) angles, such as Positive OFF (Red), fair particularly poorly
when tested on negative (south-facing) angles. This may be due to the smaller
dataset size, but we hypothesize that the very different lighting conditions
and shadows make some directions intrinsically more difficult (Figure 3C-D).
This observation reinforces that developing models and datasets that can
handle the diversity of conditions seen in overhead imagery in the wild
remains an important challenge.

Model architectures. Interestingly, segmentation models designed specifically
for overhead imagery (TernausNet and YOLT) significantly outperform general-
purpose segmentation models for computer vision (U-Net, Mask R-CNN). These
experiments demonstrate the value of specializing computer vision models to
the target domain of overhead imagery, which has different visual, object
density, size, and orientation characteristics.

Effects of resolution. OFF and VOFF images have lower base resolutions,
potentially confounding analyses of effects due exclusively to look angle. To
test whether resolution might explain the observed performance drop, we ran a
control study with normalized resolution. We trained TernausNet on images from
all look angles artificially reduced to the same resolution of $1.67m$, the
lowest base resolution from the dataset. This model showed negligible change
in performance versus the model trained on original resolution data (original
resolution: $F_{1}=0.43$, resolution equalized: $F_{1}=0.41$) (Table 4). This
experiment indicates that viewing angle-specific effects, not resolution,
drive the decline in segmentation performance as viewing angle changes.

| Training Resolution
---|---
| Original | Equalized
Test Angles | (0.46-1.67 m) | 1.67 m
NADIR | 0.62 | 0.59
OFF | 0.43 | 0.41
VOFF | 0.22 | 0.22
Summary | 0.43 | 0.41
Table 4: TernausNet model trained on different resolution imagery. Building
footprint extraction performance for a TernausNet model trained on ALL
original-resolution imagery (0.46 m ground sample distance (GSD) for
$7.8^{\circ}$ to 1.67 m GSD at $54^{\circ}$), left, compared to the same model
trained and tested on ALL imagery where every view is down-sampled to 1.67 m
GSD (right). Rows display performance ($F_{1}$ score) on different angle bins.
The original resolution imagery represents the same data as in Table 3.
Training set imagery resolution had only negligible impact on model
performance.

Generalization to unseen angles. Beyond exploring performance of models
trained with many views, we also explored how effectively models could
identify building footprints on look angles absent during training. We found
that the TernausNet model trained only on NADIR performed worse on evaluation
images from OFF (0.32) than models trained directly on OFF (0.44), as shown in
Table 5. Similar trends are observed for object detection (Figure 6). To
measure performance on unseen angles, we introduce a generalization score $G$,
which measures the performance of a model trained on $X$ and tested on $Y$,
normalized by the performance of a model trained on $Y$ and tested on $Y$:

$G_{Y}=\frac{1}{N}\sum_{X}\,\frac{F_{1}(\text{train}=X,\text{test}=Y)}{F_{1}(\text{train}=Y,\text{test}=Y)}$
(3)

This metric measures relative performance across viewing angles, normalized by
the task difficulty of the test set. We measured $G$ for all our model/dataset
combinations, as reported in Table 6. Even though the Mask R-CNN model has
worse overall performance, the model achieved a higher generalization score
($G=0.78$) compared to TernausNet ($G=0.42$) as its performance did not
decline as rapidly when look angle increased. Overall however, generalization
scores to unseen angles were low, highlighting the importance of future study
in this challenging task.

| Training Angles
---|---
Test Angles | All | NADIR | OFF | VOFF
NADIR | 0.62 | 0.59 | 0.23 | 0.13
OFF | 0.43 | 0.32 | 0.44 | 0.23
VOFF | 0.22 | 0.04 | 0.13 | 0.27
Summary | 0.43 | 0.32 | 0.26 | 0.21
Table 5: TernausNet model tested on unseen angles. Performance ($F_{1}$ score)
of the TernausNet model when trained on one angle bin (columns), and then
tested on each of the three bins (rows). The model trained on NADIR performs
worse on unseen OFF and VOFF views compared to models trained directly on
imagery from those views.

###  4.6 Effects of geography

We broke down geographic tiles into Industrial, Sparse Residential, Dense
Residential, and Urban bins, and examined how look angle influenced
performance in each. We observed greater effects on residential areas than
other types (Table S3). Testing models trained on MVOI with unseen cities[12]
showed almost no generalization (Table S4). Additional datasets with more
diverse geographies are needed.

##  5 Conclusion

We present a new dataset that is critical for extending object detection to
real-world applications, but also presents challenges to existing computer
vision algorithms. Our benchmark found that segmenting building footprints
from very off-nadir views was exceedingly difficult, even for state-of-the-art
segmentation and object detection models tuned specifically for overhead
imagery (Table 3). The relatively low $F_{1}$ scores for these tasks (maximum
VOFF $F_{1}$ score of 0.22) emphasize the amount of improvement that further
research could enable in this realm.

Furthermore, on all benchmark tasks we concluded that model generalization to
unseen views represents a significant challenge. We quantify the performance
degradation from nadir ($F_{1}$ = 0.62) to very off-nadir ($F_{1}$ = 0.22),
and note an asymmetry between performance on well-lit north-facing imagery and
south-facing imagery cloaked in shadows (Figure 3C-D and Figure 6). We
speculate that distortions in objects, occlusion, and variable lighting in
off-nadir imagery (Figure 3), as well as the small size of buildings in
general (Figure 4), pose an unusual challenge for segmentation and object
detection of overhead imagery.

The off-nadir imagery has a lower resolution than nadir imagery (due to simple
geometry), which theoretically complicates building extraction for high off-
nadir angles. However, by experimenting with imagery degraded to the same low
$1.67m$ resolution, we show that resolution has an insignificant impact on
performance (Table 4). Rather, variations in illumination and viewing angle
are the dominant factors. This runs contrary to recent observations [28],
which found that object detection models identify small cars and other
vehicles better in super-resolved imagery.

|  |  Generalization Score $\boldsymbol{G}$
---|---|---
Task | Model | NADIR | OFF | VOFF
Segmentation | TernausNet | 0.45 | 0.43 | 0.37
Segmentation | U-Net | 0.64 | 0.40 | 0.37
Segmentation | Mask R-CNN | 0.60 | 0.90 | 0.84
Detection | Mask R-CNN | 0.64 | 0.92 | 0.76
Detection | YOLT | 0.57 | 0.68 | 0.44
Table 6: Generalization scores. To measure segmentation model performance on
unseen views, we compute a generalization score $G$ (Equation 3), which
quantifies performance on unseen views normalized by task difficulty. Each
column corresponds to a model trained on one angle bin.

The generalization score $G$ is low for the highest-performing, overhead
imagery-specific models in these tasks (Table 6), suggesting that these models
may be over-fitting to view-specific properties. This challenge is not
specific to overhead imagery: for example, accounting for distortion of
objects due to imagery perspective is an essential component of 3-dimensional
scene modeling, or rotation prediction tasks [23]. Taken together, this
dataset and the $G$ metric provide an exciting opportunity for future research
on algorithmic generalization to unseen views.

Our aim for future work is to expose problems of interest to the larger
computer vision community with the help of overhead imagery datasets. While
only one specific application, advances in enabling analysis of overhead
imagery in the wild can concurrently solve broader tasks. For example, we had
anecdotally observed that image translation and domain transfer models failed
to convert off-nadir images to nadir images, potentially due to the spatial
shifts in the image. Exploring these tasks as well as other novel research
avenues will enable advancement of a variety of current computer vision
challenges.

## References

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SpaceNet MVOI: a Multi-View Overhead Imagery Dataset
Supplementary Material

##  Appendix A Dataset

###  A.1 Imagery details

The images from our dataset were obtained from DigitalGlobe, with 27 different
viewing angles collected over the same geographical region of Atlanta, GA.
Each viewing angle is characterized as both an off-nadir angle and a target
azimuth. We binned each angle into one of three categories (Nadir, Off-Nadir,
and Very Off-Nadir) based on the angle (see Table 8). Collects were also
separated into South- or North-facing based on the target azimuth angle.

The imagery dataset comprises Panchromatic, Multi-Spectral, and Pan-Sharpened
Red-Green-Blue-near IR (RGB-NIR) images The ground resolution of image varied
depending on the viewing angle and the type of image (Panchromatic, Multi-
spectral, Pan-sharpened). See Table 7 for more details. All experiments in
this study were performed using the Pan-Sharpened RGB-NIR image (with the NIR
band removed, except for the U-Net model).

The imagery was uploaded into the spacenet-dataset AWS S3 bucket, which is
publicly readable with no cost to download. Download instructions can be found
at www.spacenet.ai/off-nadir-building-detection/.

Image | Resolution at 7.8∘ | Resolution at 54∘
---|---|---
Panchromatic | 0.46m/px | 1.67m/px
Multi-spectral | 1.8m/px | 7.0m/px
Pan-sharpened | 0.46m/px | 1.67m/px
Table 7: Resolution across different image types for two nadir angles. Catalog ID | Pan-sharpened Resolution | Look Angle | Target Azimuth Angle | Angle Bin | Look Direction
---|---|---|---|---|---
1030010003D22F00 | 0.48 | 7.8 | 118.4 | Nadir | South
10300100023BC100 | 0.49 | 8.3 | 78.4 | Nadir | North
1030010003993E00 | 0.49 | 10.5 | 148.6 | Nadir | South
1030010003CAF100 | 0.48 | 10.6 | 57.6 | Nadir | North
1030010002B7D800 | 0.49 | 13.9 | 162 | Nadir | South
10300100039AB000 | 0.49 | 14.8 | 43 | Nadir | North
1030010002649200 | 0.52 | 16.9 | 168.7 | Nadir | South
1030010003C92000 | 0.52 | 19.3 | 35.1 | Nadir | North
1030010003127500 | 0.54 | 21.3 | 174.7 | Nadir | South
103001000352C200 | 0.54 | 23.5 | 30.7 | Nadir | North
103001000307D800 | 0.57 | 25.4 | 178.4 | Nadir | South
1030010003472200 | 0.58 | 27.4 | 27.7 | Off-Nadir | North
1030010003315300 | 0.61 | 29.1 | 181 | Off-Nadir | South
10300100036D5200 | 0.62 | 31 | 25.5 | Off-Nadir | North
103001000392F600 | 0.65 | 32.5 | 182.8 | Off-Nadir | South
1030010003697400 | 0.68 | 34 | 23.8 | Off-Nadir | North
1030010003895500 | 0.74 | 37 | 22.6 | Off-Nadir | North
1030010003832800 | 0.8 | 39.6 | 21.5 | Off-Nadir | North
10300100035D1B00 | 0.87 | 42 | 20.7 | Very Off-Nadir | North
1030010003CCD700 | 0.95 | 44.2 | 20 | Very Off-Nadir | North
1030010003713C00 | 1.03 | 46.1 | 19.5 | Very Off-Nadir | North
10300100033C5200 | 1.13 | 47.8 | 19 | Very Off-Nadir | North
1030010003492700 | 1.23 | 49.3 | 18.5 | Very Off-Nadir | North
10300100039E6200 | 1.36 | 50.9 | 18 | Very Off-Nadir | North
1030010003BDDC00 | 1.48 | 52.2 | 17.7 | Very Off-Nadir | North
1030010003193D00 | 1.63 | 53.4 | 17.4 | Very Off-Nadir | North
1030010003CD4300 | 1.67 | 54 | 17.4 | Very Off-Nadir | North
Table 8: DigitalGlobe Catalog IDs and the resolution of each image based upon
off-nadir angle and target azimuth angle.

###  A.2 Dataset breakdown

The imagery described above was split into three folds: 50% in a training set,
25% in a validation set, and 25% in a final test set. $900\times 900$-pixel
geographic tiles were randomly placed in one of the three categories, with all
of the look angles for a given geography assigned to the same subset to avoid
geographic leakage. The full training set and building footprint labels as
well as the validation set imagery were open sourced, and the validation set
labels and final test imagery and labels were withheld as scoring sets for
public coding challenges.

##  Appendix B Model Training

###  B.1 TernausNet

The TernausNet model was trained without pre-trained weights roughly as
described previously [17], with modifications. Firstly, only the Pan-sharpened
RGB channels were used for training, and were re-scaled to 8-bit. $90^{\circ}$
rotations, X and Y flips, imagery zooming of up to 25%, and linear brightness
adjustments of up to 50% were applied randomly to training images. After
augmentations, a $512\times 512$ crop was randomly selected from within each
$900\times 900$ training chip, with one crop used per chip per training epoch.
Secondly, as described in the Models section of the main text, a combination
loss function was used with a weight parameter $\alpha=0.8$. Secondly, a
variant of Adam incorporating Nesterov momentum [nadam] with default
parameters was used as the optimizer. The model was trained for 25-40 epochs,
and learning rate was decreased 5-fold when validation loss failed to improve
for 5 epochs. Model training was halted when validation loss failed to improve
for 10 epochs.

Type | NADIR | OFF - NADIR | VOFF - NADIR
---|---|---|---
Industrial | $0.51$ | $-0.13$ | $-0.28$
Sparse Res | $0.57$ | $-0.19$ | $-0.37$
Dense Res | $0.66$ | $-0.21$ | $-0.41$
Urban | $0.64$ | $-0.13$ | $-0.30$
Table 9: $F_{1}$ score for the model trained on all angles and evaluated
evaluated on the nadir bins (NADIR), then the relative decrease in $F_{1}$ for
the off-nadir and very off-nadir bins.

###  B.2 U-Net

The original U-Net [27] architecture was trained for 30 epochs with Pan-
Sharpened RGB+NIR 16-bit imagery, on a binary segmentation mask with a
combination loss as described in the main text with $\alpha=0.5$. Dropout and
batch normalization were used at each layer, with dropout with $p=0.33$. The
same augmentation pipeline was used as with TernausNet. An Adam Optimizer
[adam] was used with learning rate of 0.0001 was used for training.

###  B.3 YOLT

The You Only Look Twice (YOLT) model was trained as described previously [11].
Bounding box training targets were generated by converting polygon building
footprints into the minimal un-oriented bounding box that enclosed each
polygon.

###  B.4 Mask R-CNN

The Mask R-CNN model with the ResNet50-C4 backbone was trained as described
previously [16] using the same augmentation pipeline as TernausNet. Bounding
boxes were created as described above for YOLT.

##  Appendix C Geography-specific performance

###  C.1 Distinct geographies within SpaceNet MVOI

We asked how well the TernausNet model trained on SpaceNet MVOI performed both
within and outside of the dataset. First, we broke down the test dataset into
the four bins represented in main text Figure 1: Industrial, Sparse
Residential, Dense Residential, and Urban, and scored models within those bins
(Table 9). We observed slightly worse performance in Industrials areas than
elsewhere at nadir, but markedly stronger drops in performance in residential
areas as look angle increased.

###  C.2 Generalization to unseen geographies

We also explored how models trained on SpaceNet MVOI performed on building
footprint extraction from imagery from other geographies, in this case, the
Las Vegas imagery from SpaceNet [12]. After normalizing the Las Vegas (LV)
imagery for consistent pixel intensities and channel order with SpaceNet MVOI,
we predicted building footprints in LV imagery and scored prediction quality
as described in Metrics. We also re-trained TernausNet on the LV imagery and
examined building footprint extraction quality on the SpaceNet MVOI test set.
Strikingly, neither model was able to identify building footprints in the
unseen geographies, highlighting that adding novel looks angles does not
necessarily enable generalization to new geographic areas.

|  | Test Set
---|---|---
|  | MVOI 7∘ | SN LV
Training Set | MVOI ALL | 0.68 | 0.01
SN LV | 0.00 | 0.62
Table 10: Cross-dataset $F_{1}$. Models trained on MVOI or SpaceNet Las Vegas
[12] were inferenced on held out imagery from one of those two geographies,
and building footprint quality was assessed as described in Metrics.

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††thanks: The first author is the corresponding author and this work was
financially supported by UGC Senior Research Fellowship of India, Sr. No.
2061540940. Ref. No:21/06/2015(i)EU-V.

# Ricci solitons and certain related metrics on 3-dimensional trans-Sasakian
manifold

Sumanjit Sarkar  Department of Mathematics
Jadavpur University
Kolkata-700032, India<EMAIL_ADDRESS>Santu Dey  Department of
Mathematics
Bidhan Chandra College
Asansol, Burdwan, West Bengal-713304, India<EMAIL_ADDRESS>Arindam
Bhattacharyya  Department of Mathematics
Jadavpur University
Kolkata-700032, India<EMAIL_ADDRESS>
###### Abstract.

In this paper we study certain types of metrics such as Ricci soliton,
$*$-conformal Ricci soliton in 3-dimensional trans-Sasakian manifold. First we
have shown that a 3-dimensional trans-Sasakian manifold of type
$(\alpha,\beta)$ admits a Ricci soliton where the covariant derivative of
potential vector field in the direction of unit vector field $\xi$ is
orthogonal to $\xi$. It is also shown that if the structure functions satisfy
$\alpha^{2}=\beta^{2}$ then the covariant derivative of the potential vector
field in the direction of $\xi$ is a constant multiple of $\xi$. Further, we
have evolved the nature of scalar curvature when the manifold satisfies
$*$-conformal Ricci soliton of type $(\alpha,\beta)$, provided $\alpha\neq 0$.
Finally, we present an example to verify our findings.

###### Key words and phrases:

Ricci flow, Ricci soliton, $*$-conformal Ricci soliton, trans-Sasakian
manifold.

###### 1991 Mathematics Subject Classification:

53C25, 53C44, 53D15

##  1\. Introduction

Richard S. Hamilton introduced the concept of Ricci flow (for details see
[15]) which was named after great Italian mathematician Gregorio Ricci-
Curbastro. Later Grigori Perelman found it very useful to solve Poincare
conjecture. If we take a smooth closed (compact without boundary) Riemannian
manifold $M$ equipped with a smooth Riemannian metric $g$ then the Ricci flow
is defined by the geometric evolution equation,

$\frac{\partial g(t)}{\partial t}=-2S(g(t))$ (1.1)

where $S$ is the Ricci curvature tensor of the manifold and $g(t)$ is a one-
parameter family of metrices on M.

A Riemannian manifold $(M,g)$ is called a Ricci soliton if there exists a
vector field $V$ and a constant $\lambda$ such that the following equation
holds,

$\frac{1}{2}\mathcal{L}_{V}g+S+\lambda g=0$ (1.2)

where $\mathcal{L}_{V}$ denotes Lie derivative along the direction $V$ and
$\lambda$ is a non-zero constant. The vector field $V$ is called potential
vector field and $\lambda$ is called soliton constant. Ricci soliton which is
a natural extension of Einstein manifold is a self-similar solution of Ricci
flow. The potential vector field $V$ and soliton constant $\lambda$ play vital
roles while determining the nature of the soliton. A soliton is said to be
shrinking, steady or expanding according as $\lambda<0$, $\lambda=0$ or
$\lambda>0$. Now if $V$ is Killing then the Ricci soliton reduces to Einstein
manifold. Compact Ricci solitons are the fixed points of the Ricci flow (1.1)
projected from the space of metrics onto its quotient modulo diffeomorphisms
and scalings, and often arise as blow-up limits for the Ricci flow on compact
manifolds.

In 2005, A. E. Fischer [11] has introduced conformal Ricci flow which is a
mere generalisation of the classical Ricci flow equation that modifies the
unit volume constraint to a scalar curvature constraint. The conformal Ricci
flow equation was given by,

$\displaystyle\frac{\partial g}{\partial t}+2(S+\frac{g}{n})$ $\displaystyle=$
$\displaystyle-pg,$ $\displaystyle r(g)$ $\displaystyle=$ $\displaystyle-1,$

where $r(g)$ is the scalar curvature of the manifold, $p$ is scalar non-
dynamical field and $n$ is the dimension of the manifold. Corresponding to the
conformal Ricci flow equation in 2015, N. Basu and A. Bhattacharyya [3]
introduced the notion of conformal Ricci soliton equation as a generalization
of Ricci soliton equation given by,

$\mathcal{L}_{V}g+2S+[2\lambda-(p+\frac{2}{n})]g=0.$ (1.3)

In 2014, Kaimakamis and Panagiotidou [18] modified the definition of Ricci
soliton where they have used $*$-Ricci tensor $S^{*}$ which was introduced by
Tachibana [25] and Hamada [14] respectively, in place of Ricci tensor $S$. The
$*$-Ricci tensor $S^{*}$ is defined by

$S^{*}(X,Y)=\frac{1}{2}(trace\\{\phi.R(X,\phi Y)\\})$

for all vector fields $X$ and $Y$ on $M$. They have used the concept of
$*$-Ricci soliton within the framework of real hypersurfaces of a complex
space form. A pseudo-Riemannian metric $g$ is called a $*$-Ricci soliton if
there exists a constant $\lambda$ and a vector field $V$ such that

$\mathcal{L}_{V}g+2S^{*}+2\lambda g=0.$

Note that, $*$-soliton is trivial if the vector field $V$ is Killing, and in
this case, the manifold becomes $*$-Einstein. Thus, it is considered as a
natural generalization of $*$-Einstein metric. A $*$-Ricci soliton is said to
be almost $*$-Ricci soliton if $\lambda$ is a smooth function on $M$.
Moreover, an almost $*$-Ricci soliton is called shrinking, steady and
expanding according to as $\lambda$ is positive, zero and negative,
respectively. In [18], it was studied a real hypersurfaces of a non-flat
complex space form admitting a $*$-Ricci soliton whose potential vector field
is the structure vector field and proved that a real hypersurface in a complex
projective space does not admit a $*$-Ricci soliton. They have also shown that
a real hypersurface of complex hyperbolic space admitting a $*$-Ricci soltion
is locally congruent to a geodesic hypersphere.

Further P. Majhi and D. Dey [19] in 2020 revised the aforementioned definition
of $*$-Ricci soliton with the help of (1.3) and defined $*$-Conformal Ricci
soliton as

$\mathcal{L}_{V}g+2S^{*}+[2\lambda-(p+\frac{2}{n})]g=0.$ (1.4)

Ricci solitons have been studied in many contexts: on K$\ddot{a}$hler
manifolds[6], on contact and Lorentzian manifolds [1], [2], on K-contact
manifolds [24] etc. by many authors. Later, Nagaraja and Premalatha [21]
studied the nature of Ricci soliton on Kenmotsu manifold; C$\check{a}$lin and
Crasmareanu [5] on f-Kenmotsu manifold; He and Zhu [16] on Sasakian manifold
and Ingalahalli and Bagewadi [17] on $\alpha$-Sasakian manifold. Recently, in
2017, Yaning Wang [27] proved that if a 3-dimensional cosymplectic manifold
$M^{3}$ admits a Ricci soliton, then either $M^{3}$ is locally flat or the
potential vector field is an infinitesimal contact transformation. Also, S.
Pahan and A. Bhattacharyya gave some insight on trans-Sasakian manifold in
[22]. In 2016, T. Dutta et al. studied conformal Ricci soliton on
3-dimensional trans-Sasakian manifold[10].

As follows in the literature, $*$-Ricci soliton on contact geometry studied by
many authors: on Sasakian and $(\kappa,\mu)$-contact manifold by Ghosh and
Patra [12], on $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifolds by Dai, Zhao
and De[7], on contact 3-manifolds by Wang [28] etc. It is worthy to mention
that in [8] Dey and Majhi considered $*$-Ricci soliton on 3-dimensional trans-
Sasakian manifold and proved that if the metric of the manifold represents
$*$-Ricci soliton and if it satisfies a certain condition then the manifold
reduces to a $\beta$-Kenmotsu manifold. Also, very recently generalizations of
$*$-Ricci soliton on contact geometry have been studied by [9, 23].

Motivated from above mentioned well praised works we have studied behaviour of
Ricci soliton and $*$-conformal Ricci soliton on 3-dimensional trans-Sasakian
manifold. In the later sections we have revisited some definitions and
important properties of 3-dimensional trans-Sasakian manifold and after that
the main result of this paper, containing Theorem 3.1 and Theorem 3.2 has been
described. We also provide an example to justify our findings.

##  2\. Preliminaries

By [4], a differentiable manifold $M$ of dimension $2n+1$ is said to be an
almost contact structure or $(\phi,\xi,\eta)$ structure if $M$ admits a
$(1,1)$ tensor field $\phi$, a vector field $\xi$, a 1-form $\eta$ satisfying:

$\displaystyle\phi^{2}$ $\displaystyle=$ $\displaystyle-I+\eta\otimes\xi$
(2.1) $\displaystyle\eta(\xi)$ $\displaystyle=$ $\displaystyle 1$ (2.2)

where $I$ is the identity mapping. A Riemannian metric $g$ is said to be
_compatible metric_ if it satisfies:

$g(\phi X,\phi Y)=g(X,Y)-\eta(X)\eta(Y)$ (2.3)

A manifold having almost contact structure along with compatible Riemannian
metric is called _almost contact metric manifold_.

In an almost contact metric manifold the following conditions are satisfied:

$\displaystyle\phi\xi$ $\displaystyle=$ $\displaystyle 0$ (2.4)
$\displaystyle\eta\circ\phi$ $\displaystyle=$ $\displaystyle 0$ (2.5)
$\displaystyle g(X,\xi)$ $\displaystyle=$ $\displaystyle\eta(X)$ (2.6)
$\displaystyle g(\phi X,Y)$ $\displaystyle=$ $\displaystyle-g(X,\phi Y)$ (2.7)

for arbitrary $X,Y\in\chi(M)$.

Let $M$ be a $2n+1$ dimensional almost contact manifold. Then we define an
almost complex structure $J$ on $M\times\mathbb{R}$ by
$J(X,f\frac{d}{dt})=(\phi X-f\xi,\eta(X)\frac{d}{dt})$, where $X$ is a tangent
to $M$, $t$ is the coordinate on $\mathbb{R}$ and $f$ a $C^{\infty}$ function
on $M\times\mathbb{R}$. Clearly $J^{2}=-I$. If $J$ is integrable then the
almost contact structure is said to be normal. The normality of an almost
contact metric manifold is equivalent with vanishing of the tensor field
$[\phi,\phi]+2d\eta\otimes\xi$, where $[\phi,\phi]$ is the Nijenhuis torsion
tensor of $\phi$(for more details see [4]).

An almost contact metric manifold $M$ is called a trans-Sasakian manifold if
$(M\times\mathbb{R},J,G)$, where $G$ is the product metric on
$M\times\mathbb{R}$, belongs to the class $W_{4}$ (see [13]). If there are
smooth functions $\alpha,\beta$ on an almost contact metric manifold
$(M,\phi,\xi,\eta,g)$ satisfying:

$\displaystyle(\nabla_{X}\phi)Y$ $\displaystyle=$
$\displaystyle\alpha[g(X,Y)\xi-\eta(Y)X]+\beta[g(\phi X,Y)\xi-\eta(Y)\phi X]$
(2.8) $\displaystyle(\nabla_{X}\eta)(Y)$ $\displaystyle=$
$\displaystyle-\alpha g(\phi X,Y)+\beta g(\phi X,\phi Y)$ (2.9)

where $X,Y\in\chi(M)$ are arbitrary and $\nabla$ is the Levi-Civita connection
of $g$ on $M$, then the manifold is called trans-Sasakian manifold of type
$(\alpha,\beta)$. $\alpha,\beta$ are called structure functions of the
manifold. Trans-Sasakian manifolds of type $(0,0),(\alpha,0),(0,\beta)$ are
called cosymplectic, $\alpha$-Sasakian, $\beta$-Kenmotsu manifolds
respectively. Then form (2.9) we can deduce that:

$\nabla_{X}\xi=-\alpha\phi X+\beta(X-\eta(X)\xi)$ (2.10)

Marrero [20] showed that a trans-Sasakian manifold of dimension $\geq 5$ is
either cosymplectic or $\alpha$-Sasakian or $\beta$-Kenmotsu. So proper trans-
Sasakian manifold exists for dimension 3. In a 3 dimensional trans-Sasakian
manifold the following relations hold:

$\displaystyle R(X,Y)Z$ $\displaystyle=$
$\displaystyle(\frac{r}{2}+2\xi\beta-2(\alpha^{2}-\beta^{2}))(g(Y,Z)X-g(X,Z)Y)-g(Y,Z)$
(2.11)
$\displaystyle[(\frac{r}{2}+\xi\beta-3(\alpha^{2}-\beta^{2}))\eta(X)\xi-\eta(X)(\phi
D\alpha-D\beta)+(X\beta$ $\displaystyle+(\phi
X)\alpha)\xi]+g(X,Z)[(\frac{r}{2}+\xi\beta-3(\alpha^{2}-\beta^{2}))\eta(Y)\xi-\eta(Y)$
$\displaystyle(\phi D\alpha-D\beta)+(Y\beta+(\phi
Y)\alpha)\xi)]-[(Z\beta+(\phi Z)\alpha)\eta(Y)+$ $\displaystyle(Y\beta+(\phi
Y)\alpha)\eta(Z)+(\frac{r}{2}+\xi\beta-3(\alpha^{2}-\beta^{2}))\eta(Y)\eta(Z)]X$
$\displaystyle+[(Z\beta+(\phi Z)\alpha)\eta(X)+(X\beta+(\phi
X)\alpha)\eta(Z)+(\frac{r}{2}+\xi\beta-$ $\displaystyle
3(\alpha^{2}-\beta^{2}))\eta(X)\eta(Z)]Y$

where $Df$ denotes the gradient of the smooth function $f$ defined on $M$.

$\displaystyle S(X,Y)$ $\displaystyle=$
$\displaystyle(\frac{r}{2}+\xi\beta-(\alpha^{2}-\beta^{2}))g(X,Y)-(\frac{r}{2}+\xi\beta-3(\alpha^{2}-\beta^{2}))\eta(X)$
(2.12) $\displaystyle\eta(Y)-(Y\beta+(\phi Y)\alpha)\eta(X)-(X\beta+(\phi
X)\alpha)\eta(Y)$ $\displaystyle S(X,\xi)$ $\displaystyle=$
$\displaystyle(2(\alpha^{2}-\beta^{2})-\xi\beta)\eta(X)-X\beta-(\phi X)\alpha$
(2.13)

where $R,S,r$ are Riemannian curvature tensor, Ricci tensor of type (0,2) and
scalar curvature of the manifold respectively and $\alpha,\beta$ are smooth
functions on the manifold (for details see [22]).

Here in this paper we restricted the smooth functions $\alpha,\beta$ to be
constant functions. Then we got some special relations compatible to our
restrictions:

$\displaystyle R(X,Y)\xi$ $\displaystyle=$
$\displaystyle(\alpha^{2}-\beta^{2})(\eta(Y)X-\eta(X)Y)$ (2.14) $\displaystyle
S(X,Y)$ $\displaystyle=$
$\displaystyle(\frac{r}{2}-(\alpha^{2}-\beta^{2}))g(X,Y)-(\frac{r}{2}-3(\alpha^{2}-\beta^{2}))\eta(X)\eta(Y)$
$\displaystyle S(X,\xi)$ $\displaystyle=$ $\displaystyle
2(\alpha^{2}-\beta^{2})\eta(X)$ (2.16) $\displaystyle QX$ $\displaystyle=$
$\displaystyle(\frac{r}{2}-(\alpha^{2}-\beta^{2}))X-(\frac{r}{2}-3(\alpha^{2}-\beta^{2}))\eta(X)\xi$
(2.17)

where $Q$ is the Ricci operator given by $S(X,Y)=g(QX,Y)$. The expression of
$*$-Ricci tensor (for details see Lemma 3.1 of [8]) on a 3-dimensional trans-
Sasakian manifold is given by

$S^{*}(X,Y)=\frac{1}{2}(r-4(\alpha^{2}-\beta^{2}))[g(X,Y)-\eta(X)\eta(Y)]$
(2.18)

##  3\. Main Results

In this section we consider the metric of 3-dimensional trans-Sasakian
manifold as a Ricci soliton and a $*$-conformal Ricci soliton and proved the
following two results.

######  Theorem 3.1.

Let $M$ be a 3-dimensional trans-Sasakian manifold of type $(\alpha,\beta)$
admitting a Ricci soliton where the structure functions $\alpha$ and $\beta$
are constant. Then the following relations are satisfied:
(i) If $\nabla_{\xi}V$ is orthogonal to $\xi$, then the soliton is shrinking
for $\alpha^{2}<\beta^{2}$, steady for $\alpha^{2}=\beta^{2}$ and expanding
for $\alpha^{2}>\beta^{2}$.
(ii) If $\alpha^{2}=\beta^{2}$, then the covariant derivative of the potential
vector field $V$ in the direction of $\xi$ is a constant multiple of $\xi$.

###### Proof.

In a 3-dimensional trans-Sasakian manifold where $\alpha$ and $\beta$ are non-
zero constant, we know from (2.17) that the Ricci operator can be written as,

$QX=(\frac{r}{2}-(\alpha^{2}-\beta^{2}))X-(\frac{r}{2}-3(\alpha^{2}-\beta^{2}))\eta(X)\xi$
(3.1)

Where $r$ denotes the scalar curvature of the manifold and $X\in\chi(M)$ is
any vector field. The aforementioned equation implies that it is an
$\eta$-Einstein manifold. Now taking covariant derivative of (3.1) w.r.t
arbitrary $Y\in\chi(M)$ we have,

$\displaystyle(\nabla_{Y}Q)X$ $\displaystyle=$
$\displaystyle\frac{1}{2}(Yr)X-\frac{1}{2}(Yr)\eta(X)\xi-(\frac{r}{2}-3(\alpha^{2}-\beta^{2}))[-\alpha
g(\phi Y,X)\xi$ (3.2) $\displaystyle+\beta g(X,Y)\xi-\alpha\eta(X)\phi
Y+\beta\eta(X)Y-2\beta\eta(X)\eta(Y)\xi]$

Contracting $X$ and using the well-known formula
$trace\\{X\rightarrow(\nabla_{X}Q)Y\\}=\frac{1}{2}(Yr)$ in (3.2) we have,

$\xi r=-2r\beta+12(\alpha^{2}-\beta^{2})\beta$ (3.3)

Using (2.15) in the definition of Ricci soliton (1.2) we have for any vector
fields $Y,Z\in\chi(M)$,

$(\mathcal{L}_{V}g)(Y,Z)=(2\lambda-r+2(\alpha^{2}-\beta^{2}))g(Y,Z)+(r-6(\alpha^{2}-\beta^{2}))\eta(Y)\eta(Z)$
(3.4)

Now taking covariant derivative of (3.4) along an arbitrary vector field
$X\in\chi(M)$,

$\displaystyle(\nabla_{X}\mathcal{L}_{V}g)(Y,Z)$ $\displaystyle=$
$\displaystyle-(Xr)g(Y,Z)+(Xr)\eta(Y)\eta(Z)+(r-6(\alpha^{2}-\beta^{2}))$
(3.5) $\displaystyle\\{-\alpha g(\phi X,Y)\eta(Z)+\beta g(\phi X,\phi
Y)\eta(Z)-\alpha g(\phi X,Z)$ $\displaystyle\eta(Y)+\beta g(\phi X,\phi
Z)\eta(Y)\\}$

Again for any vector fields $X,Y,Z\in\chi(M)$ we know (for further information
please refer to [26]),

$\displaystyle(\mathcal{L}_{V}\nabla_{X}g-\nabla_{X}\mathcal{L}_{V}g-\nabla_{[V,X]}g)(Y,Z)$
$\displaystyle=$ $\displaystyle-g((\mathcal{L}_{V}\nabla)(X,Y),Z)$
$\displaystyle-g((\mathcal{L}_{V}\nabla)(X,Z),Y)$

Since $g$ is Riemannian metric connection, $\nabla g=0$. So the above equation
reduces to,

$\displaystyle(\nabla_{X}\mathcal{L}_{V}g)(Y,Z)=g((\mathcal{L}_{V}\nabla)(X,Y),Z)+g((\mathcal{L}_{V}\nabla)(X,Z),Y)$

Using symmetry of $\mathcal{L}_{V}\nabla$ i.e.,
$(\mathcal{L}_{V}\nabla)(X,Y)=(\mathcal{L}_{V}\nabla)(Y,X)$ we can have,

$\displaystyle
2g((\mathcal{L}_{V}\nabla)(X,Y),Z)=(\nabla_{X}\mathcal{L}_{V}g)(Y,Z)+(\nabla_{Y}\mathcal{L}_{V}g)(Z,X)-(\nabla_{Z}\mathcal{L}_{V}g)(X,Y)$

Using (3.5) in the above equation we get,

$\displaystyle(\mathcal{L}_{V}\nabla)(X,Y)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(Xr)Y-\frac{1}{2}(Yr)X+\frac{1}{2}g(\phi X,\phi
Y)Dr+\frac{1}{2}(Xr)\eta(Y)\xi$ (3.6)
$\displaystyle+\frac{1}{2}(Yr)\eta(X)\xi+(r-6(\alpha^{2}-\beta^{2}))\\{-\alpha\eta(Y)\phi
X-\alpha\eta(X)$ $\displaystyle\phi Y+\beta g(\phi X,\phi Y)\xi\\}$

Taking covariant derivative w.r.t arbitrary vector field we have,

$\displaystyle(\nabla_{X}\mathcal{L}_{V}\nabla)(Y,Z)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}g(Z,\nabla_{X}Dr)Y-\frac{1}{2}g(Y,\nabla_{X}Dr)Z+\frac{1}{2}g(\phi
Y,\phi Z)$ $\displaystyle(\nabla_{X}Dr)-\alpha\eta(Z)(Xr)\phi
Y-\alpha\eta(Y)(Xr)\phi Z+\frac{1}{2}\\{(Zr)$
$\displaystyle\eta(Y)+(Yr)\eta(Z)\\}(\nabla_{X}\xi)+\frac{1}{2}\\{g(Y,\nabla_{X}Dr)\eta(Z)-\alpha$
$\displaystyle(Yr)g(\phi X,Z)+\beta g(\phi X,\phi
Z)(Yr)+g(Z,\nabla_{X}Dr)\eta(Y)$ $\displaystyle-\alpha(Zr)g(\phi X,Y)+\beta
g(\phi X,\phi Y)(Zr)+2\beta g(\phi Y,\phi Z)$
$\displaystyle(Xr)\\}\xi+\frac{1}{2}\\{\alpha g(\phi X,Y)\eta(Z)-\beta g(\phi
X,\phi Y)\eta(Z)+\alpha$ $\displaystyle g(\phi X,Z)\eta(Y)-\beta g(\phi X,\phi
Z)\eta(Y)\\}Dr+(r-6(\alpha^{2}-$ $\displaystyle\beta^{2}))[\\{\alpha^{2}g(\phi
X,Z)-\alpha\beta g(\phi X,\phi Z)\\}\phi Y+\\{\alpha^{2}g(\phi X,Y)$
$\displaystyle-\alpha\beta g(\phi X,\phi Y)\\}\phi
Z-\alpha\eta(Z)((\nabla_{X}\phi)Y)-\alpha\eta(Y)$
$\displaystyle((\nabla_{X}\phi)Z)+\beta g(\phi Y,\phi
Z)(\nabla_{X}\xi)+\\{\alpha\beta g(\phi X,Y)\eta(Z)$
$\displaystyle-\beta^{2}g(\phi X,\phi Y)\eta(Z)+\alpha\beta g(\phi
X,Z)\eta(Y)$ $\displaystyle-\beta^{2}g(\phi X,\phi Z)\eta(Y)\\}\xi]$

From Yano (for more details see [26]) we know
$(\mathcal{L}_{V}R)(X,Y)Z=(\nabla_{X}\mathcal{L}_{V}\nabla)\\\
(Y,Z)-(\nabla_{Y}\mathcal{L}_{V}\nabla)(X,Z)$. Using this formula in the above
equation we get,

$\displaystyle(\mathcal{L}_{V}R)(X,Y)Z$ $\displaystyle=$
$\displaystyle\frac{1}{2}g(Z,\nabla_{Y}Dr)X-\frac{1}{2}g(Z,\nabla_{X}Dr)Y+\frac{1}{2}\\{-\alpha(Yr)g(\phi
X,$ (3.7) $\displaystyle Z)-\beta(Yr)g(\phi X,\phi
Z)+g(Z,\nabla_{X}Dr)\eta(Y)-\alpha(Zr)$ $\displaystyle g(\phi
X,Y)+\alpha(Xr)g(\phi Y,Z)+\beta g(\phi Y,\phi Z)(Xr)-g(Z,$
$\displaystyle\nabla_{Y}Dr)\eta(X)+\alpha(Zr)g(\phi
Y,X)\\}\xi+\alpha\eta(Z)(Yr)\phi X-$ $\displaystyle\alpha\eta(Z)(Xr)\phi
Y+\alpha\\{\eta(X)(Yr)-\eta(Y)(Xr)\\}\phi Z+$
$\displaystyle\frac{1}{2}\\{\alpha g(\phi X,Y)\eta(Z)+\alpha g(X,\phi
Y)\eta(Z)-\alpha g(\phi X,Z)$ $\displaystyle\eta(Y)-\beta g(\phi X,\phi
Z)\eta(Y)-\alpha g(\phi Y,Z)\eta(X)+\beta g(\phi Y,$ $\displaystyle\phi
Z)\eta(X)\\}Dr+\frac{1}{2}\\{(Yr)\eta(Z)+(Zr)\eta(Y)\\}(\nabla_{X}\xi)-$
$\displaystyle\frac{1}{2}\\{(Xr)\eta(Z)+(Zr)\eta(X)\\}(\nabla_{Y}\xi)+\frac{1}{2}g(\phi
Y,\phi Z)(\nabla_{X}Dr)$ $\displaystyle-\frac{1}{2}g(\phi X,\phi
Z)(\nabla_{Y}Dr)+(r-6(\alpha^{2}-\beta^{2}))[\\{\alpha\beta g(\phi Y,\phi Z)$
$\displaystyle-\alpha^{2}g(\phi Y,Z)\\}\phi X-\\{\alpha\beta g(\phi X,\phi
Z)-\alpha^{2}g(\phi X,Z)\\}\phi Y+$ $\displaystyle 2\alpha^{2}g(\phi X,Y)\phi
Z+\\{2\alpha\beta g(\phi X,Y)\eta(Z)+\alpha\beta g(\phi X,Z)$
$\displaystyle\eta(Y)-\beta^{2}g(\phi X,\phi Z)\eta(Y)-\alpha\beta g(\phi
Y,Z)\eta(X)+\beta^{2}$ $\displaystyle g(\phi Y,\phi Z)\eta(X)\\}\xi+\beta
g(\phi Y,\phi Z)(\nabla_{X}\xi)-\beta g(\phi X,\phi Z)$
$\displaystyle(\nabla_{Y}\xi)-\alpha\eta(Z)((\nabla_{X}\phi)Y)-\alpha\eta(Y)((\nabla_{X}\phi)Z)+\alpha\eta(Z)$
$\displaystyle((\nabla_{Y}\phi)X)+\alpha\eta(X)((\nabla_{Y}\phi)Z)$

This equation holds for any $X,Y,Z\in\chi(M)$. Contracting $X$ in (3.7) we
get,

$(\mathcal{L}_{V}S)(Y,Z)=(\frac{\Delta
r}{2}-6\alpha^{4}+12\alpha^{2}\beta^{2}-6\beta^{4}+r\alpha^{2}-r\beta^{2})g(\phi
Y,\phi Z)$ (3.8)

for any $Y,Z\in\chi(M)$. Again from (LABEL:2.15) we can have,

$\displaystyle(\mathcal{L}_{V}S)(Y,Z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}g(\phi Y,\phi
Z)(Vr)+(\frac{r}{2}-(\alpha^{2}-\beta^{2}))\\{g(\nabla_{Y}V,Z)$
$\displaystyle+g(Y,\nabla_{Z}V)\\}-(\frac{r}{2}-3(\alpha^{2}-\beta^{2}))\\{\eta(Z)((\nabla_{V}\eta)Y)$
$\displaystyle+\eta(Y)((\nabla_{V}\eta)Z)+\eta(Z)\eta(\nabla_{Y}V)+\eta(Y)\eta(\nabla_{Z}V)\\}$

Comparing (3.8) with (3.1) yields,

$\displaystyle(\frac{\Delta
r}{2}-6\alpha^{4}+12\alpha^{2}\beta^{2}-6\beta^{4}+r\alpha^{2}-r\beta^{2})g(\phi
Y,\phi Z)=\frac{1}{2}\\{g(\phi Y,\phi Z)(Vr)$
$\displaystyle+(\frac{r}{2}-(\alpha^{2}-\beta^{2}))\\{g(\nabla_{Y}V,Z)+g(Y,\nabla_{Z}V)\\}-(\frac{r}{2}-3(\alpha^{2}-\beta^{2}))\\{\eta(Z)$
$\displaystyle((\nabla_{V}\eta)Y)+\eta(Y)((\nabla_{V}\eta)Z)+\eta(Z)\eta(\nabla_{Y}V)+\eta(Y)\eta(\nabla_{Z}V)\\}$
(3.10)

Now, letting $Y=Z=\xi$ gives rise to
$(\alpha^{2}-\beta^{2})\eta(\nabla_{\xi}V)=0$. Now, there will arise two cases
either $\eta(\nabla_{\xi}V)=0$ or $(\alpha^{2}-\beta^{2})=0$. From the
definition of Ricci soliton (1.2) we have,

$\frac{1}{2}(g(\nabla_{X}V,Y)+g(\nabla_{Y}V,X))+S(X,Y)=\lambda g(X,Y)$ (3.11)

for any vector fields $X$ and $Y$.For first case $\eta(\nabla_{\xi}V)=0$ which
implies $\nabla_{\xi}V$ is orthogonal to $\xi$, putting $X=Y=\xi$ in (3.11)
gives $2(\alpha^{2}-\beta^{2})=\lambda$. It directly implies that the soliton
is shrinking if $\alpha^{2}<\beta^{2}$, steady if $\alpha^{2}=\beta^{2}$ and
expanding if $\alpha^{2}>\beta^{2}$.

For the second case where $\alpha^{2}=\beta^{2}$, then it follows directly
from (3.11) that $\nabla_{\xi}V=\lambda\xi$ i.e., the covariant derivative of
the potential vector field $V$ in the direction of $\xi$ is a $\lambda$
multiple of $\xi$. ∎

######  Theorem 3.2.

Let the $M$ be a 3-dimensional trans-Sasakian manifold of type
$(\alpha,\beta)$ where the structure functions $\alpha$ and $\beta$ are
constant with $\alpha\neq 0$. If the metric $g$ represents a $*$-conformal
Ricci soliton then the scalar curvature of the manifold is given by
$r=(1-\frac{\beta^{2}}{\alpha^{2}})(\frac{p}{2}+\frac{1}{3}-\lambda+4\alpha^{2})$.

###### Proof.

Since the metric $g$ represents a $*$-conformal Ricci solition, using (2.18)
in the definition of $*$-conformal Ricci solition (1.4) we get,

$(\mathcal{L}_{V}g)(X,Y)=(p+\frac{2}{3}+4(\alpha^{2}-\beta^{2})-r-2\lambda)g(X,Y)+(r-4(\alpha^{2}-\beta^{2}))\eta(X)\eta(Y)$
(3.12)

for all vector fields $X$ and $Y$ on $M$. If we consider covariant derivative
w.r.t. arbitrary vector field $Z$ then (3.12) reduces to,

$\displaystyle(\nabla_{Z}\mathcal{L}_{V}g)(X,Y)$ $\displaystyle=$
$\displaystyle(Zr)[\eta(X)\eta(Y)-g(X,Y)]-(r-4(\alpha^{2}-\beta^{2}))$ (3.13)
$\displaystyle[\alpha g(\phi Z,X)\eta(Y)-\beta g(\phi X,\phi Z)\eta(Y)+\alpha
g(\phi Z,Y)$ $\displaystyle\eta(X)-\beta g(\phi Y,\phi Z)\eta(X)]$

for all $X,Y,Z\in\chi(M)$. Again from Yano[26] we have the following
commutation formula,

$\displaystyle(\mathcal{L}_{V}\nabla_{Z}g-\nabla_{Z}\mathcal{L}_{V}g-\nabla_{[V,Z]}g)(X,Y)$
$\displaystyle=$ $\displaystyle-g((\mathcal{L}_{V}\nabla)(X,Z),Y)$
$\displaystyle-g((\mathcal{L}_{V}\nabla)(Y,Z),X),$

where $g$ is the metric connection i.e., $\nabla g=0$. So the above equation
reduces to,

$(\nabla_{Z}\mathcal{L}_{V}g)(X,Y)=g((\mathcal{L}_{V}\nabla)(X,Z),Y)+g((\mathcal{L}_{V}\nabla)(Y,Z),X),$
(3.14)

for all vector fields $X$, $Y$, $Z$ on $M$. Combining (3.13) and (3.14) and by
a straightforward combinatorial computation and using the symmetry of
$(\mathcal{L}_{V}\nabla)$ along with (2.7) the foregoing equation yields,

$\displaystyle(\mathcal{L}_{V}\nabla)(X,Y)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(Dr)[g(X,Y)-\eta(X)\eta(Y)]-\frac{1}{2}(Xr)[Y-\eta(Y)\xi]-$
(3.15)
$\displaystyle\frac{1}{2}(Yr)[X-\eta(X)\xi]+(r-4(\alpha^{2}-\beta^{2}))[\beta
g(\phi X,\phi Y)\xi$ $\displaystyle-\alpha\eta(Y)(\phi X)-\alpha\eta(X)(\phi
Y)]$

for arbitrary vector fields $X$ and $Y$ on $M$. Setting $Y=\xi$ in (3.15) we
have,

$(\mathcal{L}_{V}\nabla)(X,\xi)=-\frac{1}{2}(\xi
r)[X-\eta(X)\xi]-\alpha(r-4(\alpha^{2}-\beta^{2}))(\phi X).$ (3.16)

Applying covariant derivative w.r.t. arbitrary vector field $Y$ and making use
of (2.8), (2.9) and (2.10) we obtain,

$\displaystyle(\nabla_{Y}\mathcal{L}_{V}\nabla)(X,\xi)$ $\displaystyle=$
$\displaystyle\alpha(\mathcal{L}_{V}\nabla)(X,\phi
Y)-\beta(\mathcal{L}_{V}\nabla)(X,Y)-\frac{1}{2}(Y(\xi r))$ (3.17)
$\displaystyle[X-\eta(X)\xi]+\frac{1}{2}(\xi r)[\alpha g(\phi X,Y)\xi+\beta
g(\phi X,\phi Y)\xi$ $\displaystyle-\alpha\eta(X)(\phi
Y)+\beta\eta(X)Y-\beta\eta(Y)X]-\alpha(Yr)(\phi X)$
$\displaystyle-\alpha(r-4(\alpha^{2}-\beta^{2}))[\alpha
g(X,Y)\xi-\alpha\eta(X)Y+$ $\displaystyle\beta g(\phi
Y,X)\xi-\beta\eta(X)(\phi Y)+\beta\eta(Y)(\phi X)].$

From Yano (for more details see [26]) we know
$(\mathcal{L}_{V}R)(X,Y)Z=(\nabla_{X}\mathcal{L}_{V}\nabla)\\\
(Y,Z)-(\nabla_{Y}\mathcal{L}_{V}\nabla)(X,Z)$. Using (3.17) in this formula we
get,

$\displaystyle(\mathcal{L}_{V}R)(X,Y)\xi$ $\displaystyle=$
$\displaystyle\alpha(\mathcal{L}_{V}\nabla)(\phi
X,Y)-\alpha(\mathcal{L}_{V}\nabla)(X,\phi Y)-\frac{1}{2}(X(\xi r))[Y-$
$\displaystyle\eta(Y)\xi]+\frac{1}{2}(Y(\xi r))[X-\eta(X)\xi]+\frac{1}{2}(\xi
r)[2\alpha g(X,\phi Y)\xi$ $\displaystyle-\alpha\eta(Y)(\phi
X)+\alpha\eta(X)(\phi Y)+2\beta\eta(Y)X-2\beta\eta(X)Y]$
$\displaystyle-\alpha(Xr)(\phi Y)+\alpha(Yr)(\phi
X)-\alpha(r-4(\alpha^{2}-\beta^{2}))$
$\displaystyle[\alpha\eta(X)Y-\alpha\eta(Y)X+2\beta g(\phi
X,Y)\xi+2\beta\eta(X)(\phi Y)$ $\displaystyle-2\beta\eta(Y)(\phi X)]$

Setting $Y=\xi$ in the foregoing equation we get,

$\displaystyle(\mathcal{L}_{V}R)(X,\xi)\xi$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\xi(\xi r))[X-\eta(X)\xi]+\beta(\xi
r)[X-\eta(X)\xi]-$ $\displaystyle 2\alpha(r-4(\alpha^{2}-\beta^{2}))[-\alpha
X+\alpha\eta(X)\xi-\beta(\phi X)$

Again, Lie differentiation of the equation (2.14) w.r.t. soliton vector field
$V$ and use of (2.11) and (2.14) leads to,

$(\mathcal{L}_{V}R)(X,\xi)\xi=(\alpha^{2}-\beta^{2})[g(X,\mathcal{L}_{V}\xi)\xi-((\mathcal{L}_{V}\eta)X)\xi-2\eta(\mathcal{L}_{V}\xi)X]$
(3.19)

which holds good for arbitrary vector field $X$ on $M$. Setting $Y=\xi$ in
(3.12)implies,

$(\mathcal{L}_{V}\eta)X-g(X,\mathcal{L}_{V}\xi)=(p+\frac{2}{3}-2\lambda)\eta(X).$
(3.20)

Taking (3.20) into account, Lie derivative of $\eta(\xi)=1$ along the
direction of $V$ leads to

$2\eta(\mathcal{L}_{V}\xi)=-(p+\frac{2}{3}-2\lambda).$ (3.21)

After using (3.20) and (3.21), the equation (3.19) reduces to,

$(\mathcal{L}_{V}R)(X,\xi)\xi=(\alpha^{2}-\beta^{2})(p+\frac{2}{3}-2\lambda)[X-\eta(X)\xi]$
(3.22)

for all $X\in\chi(M)$ Comparing (3.2) with (3.22) leads to,

$\displaystyle(\alpha^{2}-\beta^{2})(p+\frac{2}{3}-2\lambda)[X-\eta(X)\xi]=\frac{1}{2}(\xi(\xi
r))[X-\eta(X)\xi]+\beta(\xi r)$ (3.23)
$\displaystyle[X-\eta(X)\xi]-2\alpha(r-4(\alpha^{2}-\beta^{2}))[-\alpha
X+\alpha\eta(X)\xi-\beta(\phi X)$

for $X\in\chi(M)$. Inner product w.r.t. arbitrary vector field $Y$ gives,

$\displaystyle[\frac{1}{2}(\xi(\xi r))+\beta(\xi
r)+2\alpha^{2}(r-4(\alpha^{2}-\beta^{2}))-(\alpha^{2}-\beta^{2})(p+\frac{2}{3}-2\lambda)]$
(3.24)
$\displaystyle[g(X,Y)-\eta(X)\eta(Y)]+2\alpha\beta(r-4(\alpha^{2}-\beta^{2}))g(\phi
X,Y)=0.$

Anti-symmetrizing the foregoing equation yields,

$[\frac{1}{2}(\xi(\xi r))+\beta(\xi
r)+2\alpha^{2}(r-4(\alpha^{2}-\beta^{2}))-(\alpha^{2}-\beta^{2})(p+\frac{2}{3}-2\lambda)]g(\phi
X,\phi Y)=0.$ (3.25)

Since this equation holds for arbitrary vector fields $\phi X$ and $\phi Y$
and as we know from (3.3) that $\xi r=-2r\beta+12\beta(\alpha^{2}-\beta^{2})$
holds in 3-dimensional trans-Sasakian manifold, we can easily conclude that
the scalar curvature of the manifold satisfies
$r=(1-\frac{\beta^{2}}{\alpha^{2}})(\frac{p}{2}+\frac{1}{3}-\lambda+4\alpha^{2})$.
∎

##  4\. Example of 3-dimensional trans-Sasakian manifold admitting Ricci
soliton

In this section we provide an example to verify our outcomes.

######  Example 4.1.

We consider the manifold as $M=\\{(x,y,z)\in\mathbb{R}^{3}:y\neq 0\\}$, where
$(x,y,z)$ are the standard coordinates in $\mathbb{R}^{3}$. The vector fields
as defined bellow:

$\displaystyle e_{1}$ $\displaystyle=e^{2z}\frac{\partial}{\partial x},$
$\displaystyle e_{2}$ $\displaystyle=e^{2z}\frac{\partial}{\partial y},$
$\displaystyle e_{3}=\frac{\partial}{\partial z}$

are linearly independent at each point of $M$. The Riemannian metric $g$ is
defined by:

$g_{ij}=\begin{cases}1,&\mbox{if }i=j~{}and~{}i,j\in\\{1,2,3\\}\\\
0,&\mbox{otherwise.}\end{cases}$

Let $\xi=e_{3}$. Then the 1-form $\eta$ is defined by $\eta(Z)=g(Z,e_{3})$,
for arbitrary $Z\in\chi(M)$, then we have the following relations:

$\displaystyle\eta(e_{1})=\eta(e_{2})$ $\displaystyle=0,$
$\displaystyle\eta(e_{3})$ $\displaystyle=1.$

Let us define the (1,1)-tensor field $\phi$ as

$\displaystyle\phi e_{1}$ $\displaystyle=e_{2},$ $\displaystyle\phi e_{2}$
$\displaystyle=-e_{1},$ $\displaystyle\phi e_{3}$ $\displaystyle=0,$

then it satisfies,

$\displaystyle\phi^{2}(Z)$ $\displaystyle=$ $\displaystyle-Z+\eta(Z)e_{3},$
$\displaystyle g(\phi Z,\phi W)$ $\displaystyle=$ $\displaystyle
g(Z,W)-\eta(Z)\eta(W)$

for arbitrary $Z,W\in\chi(M)$.

Thus $(\phi,\xi,\eta,g)$ defines an almost contact metric structure on $M$. We
can now easily conclude:

$\displaystyle[e_{1},e_{2}]$ $\displaystyle=0,$ $\displaystyle[e_{2},e_{3}]$
$\displaystyle=-2e_{2},$ $\displaystyle[e_{1},e_{3}]$ $\displaystyle=-2e_{1}.$

Let $\nabla$ be the Levi-Civita connection of $g$. Then from
$Koszul^{\prime}sformula$ for arbitrary $X,Y,Z\in\chi(M)$ given by:

$\displaystyle 2g(\nabla_{X}Y,Z)$ $\displaystyle=$ $\displaystyle
Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)-g(X,[Y,Z])$ $\displaystyle-$ $\displaystyle
g(Y,[X,Z])+g(Z,[X,Y]),$

we can have:

$\displaystyle\nabla_{e_{1}}e_{1}$ $\displaystyle=2e_{3},$
$\displaystyle\nabla_{e_{1}}e_{2}$ $\displaystyle=0,$
$\displaystyle\nabla_{e_{1}}e_{3}$ $\displaystyle=-2e_{1},$
$\displaystyle\nabla_{e_{2}}e_{1}$ $\displaystyle=0,$
$\displaystyle\nabla_{e_{2}}e_{2}$ $\displaystyle=2e_{3},$
$\displaystyle\nabla_{e_{2}}e_{3}$ $\displaystyle=-2e_{2},$
$\displaystyle\nabla_{e_{3}}e_{1}$ $\displaystyle=0,$
$\displaystyle\nabla_{e_{3}}e_{2}$ $\displaystyle=0,$
$\displaystyle\nabla_{e_{3}}e_{3}$ $\displaystyle=0.$

From here we can easily verify that the relations (2.8) and (2.9) are
satisfied. Hence the considered manifold is trans-Sasakian manifold of type
$(0,-2)$. The components of Riemannian curvature tensor are given by,

$\displaystyle R(e_{1},e_{2})e_{1}$ $\displaystyle=-4e_{3},$ $\displaystyle
R(e_{1},e_{2})e_{2}$ $\displaystyle=-4e_{1},$ $\displaystyle
R(e_{1},e_{2})e_{3}$ $\displaystyle=0,$ $\displaystyle R(e_{1},e_{3})e_{1}$
$\displaystyle=4e_{2},$ $\displaystyle R(e_{1},e_{3})e_{2}$ $\displaystyle=0,$
$\displaystyle R(e_{1},e_{3})e_{3}$ $\displaystyle=-4e_{1},$ $\displaystyle
R(e_{2},e_{3})e_{1}$ $\displaystyle=0,$ $\displaystyle R(e_{2},e_{3})e_{2}$
$\displaystyle=-4e_{2},$ $\displaystyle R(e_{2},e_{3})e_{3}$
$\displaystyle=-4e_{2}.$

And the components of Ricci tensor and $*$-Ricci tensor are given by:

$\displaystyle S(e_{1},e_{1})$ $\displaystyle=0,$ $\displaystyle
S(e_{2},e_{2})$ $\displaystyle=0,$ $\displaystyle S(e_{3},e_{3})$
$\displaystyle=-8.$

From here we can easily deduce that the scalar curvature of the manifold
$r=\sum_{i=1}^{3}S(e_{i},e_{i})=-8$. Let us define a vector field by, $V=\xi$.
Then we can obtain:

$\displaystyle(\mathcal{L}_{V}g)(e_{1},e_{1})$ $\displaystyle=-4,$
$\displaystyle(\mathcal{L}_{V}g)(e_{2},e_{2})$ $\displaystyle=-4,$
$\displaystyle(\mathcal{L}_{V}g)(e_{3},e_{3})$ $\displaystyle=0.$

Contracting (1.2) and using the result $r=-8$ we deduce $\lambda=4$. So $g$
defines a Ricci soliton on this trans-Sasakian manifold for $\lambda=4$.

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