# Digital Reconstruction of Elmina Castle for Mobile Virtual Reality via
Point-based Detail Transfer

Sifan Ye, Computer Science Department, Stanford University, Palo Alto, CA USA
Ting Wu, eBay, San Jose, CA USA
Michael Jarvis, Department of History, University of Rochester, Rochester, NY
USA
Yuhao Zhu, Department of Computer Science, University of Rochester, Rochester,
NY USA

###### Abstract

Reconstructing 3D models from large, dense point clouds is critical to enable
Virtual Reality (VR) as a platform for entertainment, education, and heritage
preservation. Existing 3D reconstruction systems inevitably make trade-offs
between three conflicting goals: the efficiency of reconstruction (e.g., time
and memory requirements), the visual quality of the constructed scene, and the
rendering speed on the VR device. This paper proposes a reconstruction system
that simultaneously meets all three goals. The key idea is to avoid the
resource-demanding process of reconstructing a high-polygon mesh altogether.
Instead, we propose to directly transfer details from the original point cloud
to a low polygon mesh, which significantly reduces the reconstruction time and
cost, preserves the scene details, and enables real-time rendering on mobile
VR devices.

While our technique is general, we demonstrate it in reconstructing cultural
heritage sites. We for the first time digitally reconstruct the Elmina Castle,
a UNESCO world heritage site at Ghana, from billions of laser-scanned points.
The reconstruction process executes on low-end desktop systems without
requiring high processing power, making it accessible to the broad community.
The reconstructed scenes render on Oculus Go in 60 FPS, providing a real-time
VR experience with high visual quality.

## Introduction

Historic sites of cultural significance are being destroyed on a daily basis.
While it is impossible to fully prevent this, we can digitally preserve
important historic sites by reconstructing these sites in 3D before
destruction for later research. Apart from academic research, reconstructing
cultural heritage sites is also instrumental to educating and entertaining the
general public; websites such as CyArk [2], Open Heritage 3D [5], and Bermuda
100 [1] promise to let people virtually visit “the world’s most famous
monuments in immersive and accurate 3D.” Humanities and education researchers
have long advocated using digital interactive games/simulations to transport
students across time and space to better understand the past and other
cultures [16, 11, 18, 35].

This paper describes our effort to reconstruct Elmina Castle, a historical
slave trade castle and a UNESCO World Heritage Site in Ghana, from large-scale
laser-scanned point clouds, which we collected during field trips to Ghana
over the past three years. We target mobile Virtual Reality (VR) headsets,
e.g., Oculus Go, which provide an ideal platform for immersive experience but
also present the challenge of limited processing power.

Besides the historical and humanitarian significance of the reconstructed site
itself, we present a reconstruction workflow that addresses two technical
challenges: 1) the reconstruction algorithm must generate 3D models that can
be rendered in real-time on resource-limited mobile VR devices with high
quality, and 2) the reconstruction algorithm itself must be lightweight so as
to be accessible to the broad archaeology community. Let us elaborate.

First, rendering large-scale 3D models on mobile VR headsets is challenging
due to the limited amount of compute resources. A 3D model of even a single
room in Elmina Castle has millions of polygons, which render at merely 10 FPS
on Oculus Go. Existing workflows either simply use a low-polygon mesh or
decimate a high-polygon mesh [13, 15] (i.e., remeshing). Both yield smaller
meshes and improve the rendering speed but usually come at a cost of losing
critical scene details, which is disruptive when closely examining an
artifact.

Second, we also aim to develop a lightweight reconstruction workflow that
makes reconstructing heritage sites from huge point clouds more accessible to
historians and archaeologists, who may not have access to powerful computers
and could not afford long reconstruction time. For instance, the entire scan
of Elmina Castle contains 33 billions of points; reconstruct the castle on a
high-end Intel Xeon server with 256 GB RAM and 2 Nvidia GTX 1080Ti GPUs takes
three weeks, which in reality is even longer because the constructed mesh
could be iteratively edited.

Our reconstruction workflow simultaneously addresses the two challenges above.
We completely skip high-polygon meshes altogether. Instead, we use a low-
polygon mesh as the base representation of the scene, and then transfers the
details, e.g., color and normal, directly from the original point cloud to the
low-polygon mesh. Our reconstruction workflow is lightweight and fast since we
never reconstruct a high-polygon mesh throughout the whole process. The
reconstructed model can be rendered in real-time since the underlying
representation is a low-polygon mesh, which, critically, is associated with
high-quality texture information transferred from the original point clouds.

Fig. 1: Overlook of Elmina Castle.

Fig. 2: Scanning the exterior of the castle.

Overall, our reconstruction system achieves a wall-clock speedup of up to 3.1
times compared with an existing commercial reconstruction workflow. On
average, our reconstruction system converts scans with over 30 million points
to meshes with 200 thousand polygons, which render in 60 FPS on Oculus Go, a
representative mobile VR headset, while presenting a desirable visual quality,
evaluated both objectively and subjectively.

To encourage further digitization of the castle and provide immersive
learning, research, and education experience to people who could not visit the
castle in person, our system is available at https://github.com/horizon-
research/3D-Reconstruction-From-Point-Cloud.

## Context and Background

### Why Digitally Reconstructing Elmina Castle?

#### Historical Significance

Built in 1482 by the Portuguese, Elmina Castle (Figure 2) is the first
European trading base constructed in Sub-Saharan Africa. For more than five
hundred years, it was a commercial hub where European goods were exchanged for
gold, ivory, and slaves. During the Portuguese period (1482-1637) and after
the Dutch West India Company captured it in 1637, Elmina Castle was the
central administrative base for commercial operations at satellite forts
stretching hundreds along the Gold and Slave Coasts. For their central role in
the trans-Atlantic slave trade and as a unique collection of European
fortifications adapted to an African environment, Ghana’s slave trade forts
and castles were given UNESCO World Heritage Site status in 1979.

#### Humanity

Elmina holds personal relevance for tens of millions of people globally.
Elmina Castle is a major center for directing the African slave trade and the
oldest site of sustained Euro-African contact and commerce in Sub-Saharan
Africa. Under the Ghana Museums and Monuments Board, the well-preserved site
(along with nearby Cape Coast Castle) has become a major pilgrimage site of
African Diaspora heritage tourism, attracting tens of thousands of African
Americans annually seeking insights into their ancestors’ experiences of
enslavement. Creating a virtual experience through a digital surrogate has
immense humanity value for those who could not physically visit the castle,
whose importance is only heightened by COVID-19.

#### Research and Education

Virtually reconstructing Elmina has significant research value for
archaeologists, historians, and mechanical engineers. For instance,
archaeologists use the reconstructed fabric of the castle’s floors, walls, and
ceilings to date the castle’s various rooms and determine the structure’s
evolution. Early Portuguese parts of the castle would be indicated by the use
of mud mortar, thin red bricks, and roof tiles imported from Portugal. Later
Dutch repairs and expansions of the castle incorporated lime mortal, imported
yellow Dutch bricks, and angled ramparts designed to sustain cannon fire.
Mechanical engineers could view the reconstructed model to dynamically model
the monument’s structural integrity and recommend repairs.

### Scanning System

Over a three-year span, we use a FARO Focus3D X 130 laser scanner [3] to
generate colored point clouds for each of the castle’s 120 rooms and exterior
areas (Figure 2). For each scan, the time-of-flight ranging unit first
generates a raw (colorless) point cloud of the scene; on a separate pass, a
color camera built-in with the scanner captures the color images, which are
then later registered with the raw point cloud to generate a colored point
cloud.

Overlapped individual scans within rooms were first registered using FARO’s
Scene software; the outliers were then removed to create the final dense point
cloud, which covers every room. By combining a total of 427 individual scans,
the point cloud for the entire castle exceeds 33 billion points with an
average resolution (distance between neighboring points) of
$1.9\text{\,}\mathrm{mm}$. In tests conducted offline, our scans have a point
accuracy of $4.7\text{\,}\mathrm{mm}$ overall; most rooms have point
accuracies under $2.0\text{\,}\mathrm{mm}$.

Fig. 3: End-to-end reconstruction pipeline from the dense laser scanned point
to mobile VR rendering. We transfer scene details directly from the point
cloud to the UV map generated from a low-polygon mesh. The low-polygon mesh
ensures high rendering speed on mobile VR headsets while the detail transfer
ensures satisfactory visual quality.

### Point Cloud Reconstruction for Mobile VR

While it is possible to directly render the point clouds [34, 9], doing so has
two key disadvantages. First, pure point clouds are just not visually
appealing. Users can see “holes” in the scenes, which is particularly
disruptive when users want to closely examine a historical artifact. Second,
many VR use-cases such as virtual tours allows users to interact with the
scene, e.g., walking around the castle, or observing avatars of historical
figures living/working in the castle, which in turn requires a continuous
surface to simulate collision for realistic interactions. Thus, we reconstruct
polygon meshes from the colored point clouds.

## Design Objectives

Fig. 4: The design objective of our workflow is to simultaneously ensure high
reconstruction efficiency (speed and RAM requirement) as well as high
rendering speed and visual quality.

#### Real-Time Rendering Speed and Desirable Visual Quality

To faithfully represent the details of a scene a high-polygon dense mesh is
preferred. Constructing meshes from point clouds is a well-studied problem in
the literature [19, 20], and there are mature open-source and commercial
software and libraries such as PCL [33], CGAL [37], MeshLab [12], and
Metashape [7].

However, dense meshes are hard to render in real-time, especially on mobile VR
devices. Oculus Go suggests to keep the number of polygons under 100 thousand
in order to achieve real-time rendering (60 FPS) [4]111Our experiments show
that this is a conservative estimate. In general, a mesh with about 400
thousand faces renders in 60 FPS, but even a 400 thousand polygon mesh greatly
loses visual details.. On average, each room in the castle is scanned with 77
million points, which when simplified to under 100 thousand polygon loses much
of the surface details that are critical for education, virtual tourism, and
historical studies.

#### Reducing Reconstruction Overhead

To mitigate the loss of details while retaining high rendering speed, a common
solution would be to decimate a high-polygon mesh to generate a low-polygon
mesh and, optionally, transfer the details (e.g., texture and normal) from the
high-polygon mesh to the low-polygon mesh. For each polygon in the low-polygon
mesh, one could find its corresponding polygon in the high-polygon mesh and
use the color and normal details there to generate the texture and normal for
the low-polygon mesh. The decimation and transfer are readily supported in
existing 3D modeling software such as Blender [14] and MeshLab [12].

However, constructing high-polygon meshes (tens of millions of faces) and the
texture map, either for direct rendering or for transferring details to low-
polygon meshes, is extremely slow and requires high-end computers that are
inaccessible to the broad archaeology and history communities. Using a high-
end Intel Xeon server with 256 GB RAM and 2 Nvidia GTX 1080Ti GPUs, we
estimate that reconstructing the entire castle would take three weeks using a
typical workflow that consists of professional software such as Blender [14]
and MeshLab [12] well-optimized for 3D modeling. In addition, since
reconstruction processes huge amount of points and polygons, it consumes
$\sim$20 GB RAM even for processing one single room. We thus had to
reconstruct each room individually and stitch rooms later, further
complicating the reconstruction process.

#### Summary

Figure 4 compares the existing methods along four dimensions: the speed and
memory consumption of reconstruction and the speed and visual quality of
mobile VR rendering. Directly reconstructing and rendering a high-polygon mesh
is slow in both reconstruction and rendering, albeit providing the best visual
quality. In contrast, directly reconstructing and rendering a low-polygon mesh
is efficient in both reconstruction and rendering, but suffers from low visual
quality. Decimating a high-polygon mesh to a low-polygon mesh requires a high-
polygon mesh to begin, and thus introduces high reconstruction overhead.

This paper proposes a reconstruction system that achieves high reconstruction
efficiency (speed and RAM requirement) as well as high rendering speed and
visual quality simultaneously. Specifically, our goal is to combine the best
of both worlds: delivering a reconstruction workflow that has a similar
efficiency as using low-polygon meshes only while meeting (and even out-
performing) the rendering quality of transferring details from a high-polygon
mesh to a low-polygon mesh.

## Point-based Detail Transfer

We propose a new reconstruction pipeline that directly transfer information
from point cloud obtained from the laser scanner to low-polygon meshes without
reconstructing high-polygon meshes altogether. This achieves both high
reconstruction and rendering efficiency while preserving the visual details in
the laser scan. We first provide an overview of the pipeline followed by
describing our point-based detail transfer algorithm.

### Pipeline Overview

Figure 3 shows the end-to-end reconstruction pipeline, which takes a laser-
scanned colored point cloud as the input, and generates the mesh along with
the texture and normal maps, which are then used for rendering on mobile VR
platforms.

The input point cloud is first triangulated to generate a low-polygon mesh. We
control the mesh to have a small amount of polygons to enable real-time
rendering. For instance, when targeting Oculus Go, the mesh has at most 100
thousand polygons, which are about the threshold to deliver a 60 FPS rendering
as suggested by Oculus [4]. We unwrap the mesh to generate the UV map, from
which we bake the texture map and the normal map. Note that directly using
these texture/normal maps for rendering would lead to low visual quality,
because the texture and normal maps are generated from a low-polygon mesh. To
increase the details in the texture/normal maps, one could increase the
polygon count in a mesh at the cost of higher mesh reconstruction time and
lower rendering speed. The left and middle panels in Figure 5 below illustrate
these two approaches.

Instead, our pipeline retains the same low-polygon UV map, but generates the
textures and normals using information directly from the point cloud — hence
point-based detail transfer. The right panel in Figure 5 illustrates this
idea. We first map points from the point cloud to the low-polygon mesh (single
triangle in this illustration); each pixel in the texture map is then
interpolated using both the triangle vertices and the mapped points.

Fig. 5: Three ways to generate texture (and normal). Left: interpolate from a
low-polygon UV map (single triangle here); Middle: interpolate from a high-
polygon UV map; Right (our method): map points from the point cloud to a low-
polygon UV map, and then interpolate using the vertices and points. Fig. 6:
Pseudocode of our details transfer algorithm, which transfers details from a
colored point cloud to a UV map, which is unwrapped from a mesh that is
constructed from the point cloud.

Compared to the low-polygon approach (left panel in Figure 5), our method
provides more visual details because the textures/normals are generated using
more information (triangle vertices + points vs. just triangle vertices).
Compared to the high-polygon approach (middle panel in Figure 5), our method
uses only a low-polygon mesh, and thus is faster.

### Detail Transfer Algorithm

Figure 6 shows the pseudo-code of the point-based detail transfer algorithm.
The goal is to map points from the point cloud to the (low-polygon) UV map so
that we can generate the texture and normal maps with rich information to
retain visual details. For each triangle in the mesh, we identify all the
points in the point cloud that are likely on the triangle (Line 5). This is
done by filtering points using their normals and distances to the triangle
(Line 20 – 28): points that are close to the triangle and whose normals are
similar to the triangle normal are likely on the triangle and thus will be
mapped to the UV map. The thresholds use for the distance and the angle
between point and triangle are 4.0 and $120^{\circ}$, respectively. These
parameters can be easily adjusted, and we empirically find that this setting
works the best.

We map a point $p$ from the point cloud to the triangle in the UV map in a way
that keeps the barycentric coordinates of $p$ constant with respect to the
triangle (Line 6 – 10), i.e., without changing the relative position of $p$ in
the triangle. After point mapping, each triangle in the UV map could contain
several points mapped from the point cloud. Each point is associated with its
color and normal as in the point cloud. The right panel in Figure 5 shows an
example of a triangle with 7 points mapped from the point cloud.

With the transferred points, we can now generate the texture and normal maps
(Line 11 – 17). This is done by iterating over each triangle in the UV map and
calculating the color and normal values of each pixel in the triangle. To
leverage the transferred points in each triangle $T$, we first triangulate
{$T.V,T.P$}, where $T.V$ and $T.P$ denote the vertices of $T$ and the
transferred points in $T$, respectively. For each pixel $p$ in $T$, we find
its bounding triangle from the triangulation results and use triangular
interpolation to calculate the pixel’s texture and normal. The same transfer
process is applied to both the texture map and normal map, which along with
the low-polygon mesh are fed into a VR rendering engine, e.g., Unity in our
case, finishing our end-to-end pipeline.

## Evaluation

We first describe our experimental methodology. We then show that the
rendering quality of our system is competitive or better than today’s typical
reconstruction workflow, both objectively and subjectively, followed up by
showing that our system is much faster and more resource efficient than
today’s workflow.

### Experimental Setup

#### Implementation Details

We on average capture 8 scans of each room in the castle. We use FARO’s Scene
software to register point clouds of individual scans. The register point
cloud becomes the input to our pipeline. We construct meshes using Meshlab
[12], a popular open-source mesh manipulation system, which implements the
Screened Poisson Surface Reconstruction algorithm to build a triangulated mesh
out of a point cloud [21]. We experimented with other mesh reconstruction
tools, such as the commercial Agisoft Metashape software [7]. The choice of
reconstruction algorithm does not qualitatively change our conclusion. We use
Meshlab in this paper because it allows us to build a completely open-source
workflow that we will make available.

We use Blender [14] for UV unwrapping to generate a UV map, from which Blender
is also used to generate the texture and normal map. We implement our point-
based detail transfer in C++ based on the widely-used CGAL [37]. The code is
accelerated using OpenMP [6].

#### Hardware

The reconstruction is done on a Dell Precision Workstation, which is equipped
with a 4-core Intel Xeon W-2123 processor operating at 3.6 GHz, an Nvidia
Quadro RTX4000 GPU, and 48 GB of RAM. This machine costs about $2,100 to
build, which we believe is a reasonable and representative machine
specification for historical and archaeological researchers. We use Unity on
Oculus Go for VR rendering. Oculus Go is a standalone VR headset that does not
require tethering to a PC.

#### Dataset

We focus on three scenes in the Elmina castle: D11v3, NETower, and D31. They
represent different indoor and outdoor scenes in Elmina Castle and are deemed
representative by the archaeologists and historians in our team. For each
scene, the camera viewpoint is chosen so that different geometries and
textures are visible, e.g., including corners, windows, doors, uneven grounds,
and walls with non-monotone colors.

For each scene, we generate a high-resolution mesh and a low-resolution mesh.
Table. 1 shows the number of faces in each mesh along with the corresponding
point cloud size (measured in the number of points). We validate that the low-
polygon meshes are able to be rendered on Oculus Go in 60 FPS.

Table 1: Point cloud size (number of points) and mesh size (number of polygons) of the three evaluated scenes. | D11v3 | NETower | D31
---|---|---|---
Point cloud | 28,368,767 | 8,652,472 | 31,431,682
High-polygon mesh | 4,352,296 | 6,857,430 | 3,690,788
Low-polygon mesh | 240,000 | 442,900 | 224,476

#### Baseline

We compare with three baselines (Figure 4):

  * •

HPM: constructing and rendering a high-polygon mesh.

  * •

LPM: constructing and rendering a low-polygon mesh.

  * •

Remeshing: constructing a high-polygon mesh, simplifying/decimating it to a
low-polygon mesh, and transferring color/normal details from the high-polygon
mesh to the low-polygon mesh. The transfer is done in Blender, a professional
3D modeling software.

Meshlabs use the Octree to control the resolution of the point cloud. We use
two different octree depths in the surface reconstruction algorithm to obtain
the HPM and LPM. Our pipeline uses a low-polygon mesh throughout the workflow
and never requires a high-polygon mesh.

It is worth noting again that our goal is not to directly render points in the
point cloud. As discussed in Section Point Cloud Reconstruction for Mobile VR,
doing so would result in “holes” in the structure upon close examinations,
which researchers in archaeology are strongly against. Therefore, we use HPM
as the “ground-truth” for quality comparison, as it constructs a high-quality,
water-tight model.

We also have experimented with level-of-detail (LOD) rendering [38] in Unity
by manually providing meshes with different polygons and specifying when to
use which mesh depending on the camera pose. While LOD is efficient when
objects are viewed at distance, it still requires a high-polygon mesh, and
thus does not provide real-time rendering, when viewing objects up close. In
contrast, we address the speed issue by always rendering only a low-polygon
mesh.

#### Metrics

We evaluate our system using three metrics:

  * •

Reconstruction time: the time it takes to generate the mesh and texture/normal
maps required for VR rendering.

  * •

Rendering speed: the FPS of rendering on Oculus Go.

  * •

Visual quality: the mean-opinion-score in the HDR-VDP-2 metric [27], which is
calibrated with user experience.

### Rendering Quality Comparison

We show that our reconstruction system generates 3D models that, when rendered
in mobile VR headsets, achieve higher visual quality than simply downsampling
the mesh (i.e., LPM) and match, sometimes outperform, the visual quality of
transferring details from high-polygon mesh (i.e., Remeshing).

#### Objective Comparison

To quantitatively measure user experience, we export the rendering frames from
the user perspective from Unity Player, and then compare our system with the
baselines using the HDR-VDP-2 metric on the rendered frames. We use the frames
from HPM as the reference frames to calculate the HDR-VDP-2 metric for the
other three systems. Table. 2 shows the results on the three scenes.

Table 2: Rendering quality comparison in HDR-VDP-2. | D11v3 | NETower | D31
---|---|---|---
LPM | 54.01 | 51.34 | 50.50
Remeshing | 52.90 | 52.31 | 53.60
Ours | 55.58 | 52.24 | 54.00

Our reconstructed 3D models consistently outperform LPM, indicating the
benefit of transferring details. Compared to Remeshing, our reconstructed
models have better visual quality in 2 out of the 3 scenes and come very close
on the other ($<$ 0.1 in HDR-VDP-2 score). Considering that our reconstruction
system delivers much faster speed (2.3 $\times$ speedup as we will show next),
we believe our workflow provides a desirable design point.

Fig. 7: Visual results on three scenes in the castle and their HDR-VDP-2
scores. Our system generates frames that are visually competitive or better
than Remeshing, a typical 3D reconstruction workflow.

#### Subjective Comparison

Our rendering quality is subjectively competitive or better than that of
Remeshing. Figure 7 shows the rendered frames of the three scenes under LPM
and ours; each frame has two zoom-in regions, for which we show the results of
LPM, ours, Remeshing, and HPM.

Not surprisingly, HPM consistently delivers the highest visual quality, due to
the much larger meshes it constructs. Ours and Remeshing generate much better
visual quality than LPM. Comparing ours and Remeshing, our result is visually
slightly worse in the NETower scene, but competitive or better in D11v3 and
D31 scenes, which is also confirmed by the HDR-VDP-2 metric. In particular for
D31 (the last scene), Remeshing generates significant blocking artifacts,
which are invisible in our system.

### Efficiency Results

#### Reconstruction Speed

We find that our workflow significantly reduces the end-to-end time to
reconstruct a 3D model. Figure 8 compares the reconstruction time of our
workflow with the three baselines across the three evaluated scenes. The
stacked bar charts dissect the reconstruction time into five components: high-
polygon mesh construction time, low-polygon mesh construction time, UV
unwrapping time, I/O time (reading and writing meshes, texture/normal maps,
and point clouds), and the texture/normal map generation (baking) time, which
in our pipeline is dominated by the point-based transfer algorithm.

HPM has the highest reconstruction time, which is dominated by UV unwrapping.
LPM has the lower reconstruction time by significantly reducing the UV
unwrapping time, as it unwraps much smaller meshes. LPM, however, has the
lowest visual quality as we will show later. Remeshing requires constructing a
high-polygon mesh and introduces significant I/O overhead as it must
manipulate high-polygon meshes throughout the workflow. Our point-based
transfer is significantly faster than Remeshing by avoiding constructing and
manipulating a high-polygon mesh. Across the three scenes, we achieve a 2.3
$\times$ and 33.8 $\times$ speedup over Remeshing and HPM, respectively.

(a) D11v3.

(b) NETower.

(c) D31.

Fig. 8: End-to-end reconstruction time comparison. Our system is consistently
faster (2.3 $\times$ speedup) than the typical Remeshing-based workflow, and
is 33.8 $\times$ faster than HPM.

#### RAM Consumption

Figure 9 shows that our reconstruction workflow requires much less CPU RAM
compared to HPM and Remeshing since we avoid reconstructing and processing
high-polygon meshes. On average, we reduce the peak RAM consumption by 51.8%
and 22.7% compared to HPM and Remeshing. Note that the peak memory consumption
of our workflow is slightly higher on D31 compared to Remeshing. This is
because D31 has the largest point cloud (Table. 1), on which our point
transfer algorithm consumes high RAM.

Fig. 9: CPU RAM consumption comparison. Our system typically uses much less
RAM than Remeshing and HPM. Fig. 10: End-to-end reconstruction time on a
mobile system.

#### Rendering Speed

Using low-polygon meshes for VR rendering achieves 60 FPS, whereas using high-
polygon meshes generally leads to an FPS below of 20. Table. 3 compares the
FPS across the four workflows. Note that all except HPM use low-polygon meshes
and thus have the same 60 FPS.

Table 3: FPS comparison. | D11v3 | NETower | D31
---|---|---|---
HPM | 16 | 10 | 17
Ours/LPM/Remeshing | 60 | 60 | 60

#### Evaluation on Low-end Systems

To further demonstrate that our workflow is resource-efficient and can be
easily deployed even on a laptop, we also evaluate our end-to-end system on a
2015 MacBook Pro, which has 16 GB of CPU RAM and a four-core Intel i7-4870HQ
processor running at 2.8 GHz.

Figure 10 shows the reconstruction time on the laptop for the three scenes.
While in generally it is slower on the laptop than on the workstation, the
reconstruction time is consistently below 200 seconds for the three scenes,
indicating that it is feasible to deploy our workflow on a mobile, low-end
system. On average, running our reconstruction system on the mobile laptop is
10.6% slower than on the Xeon workstation. Our system is also 33.1 $\times$
and 1.7 $\times$ faster than HPM and Remeshing on the laptop.

## Related Work

#### Digitizing Cultural Heritage

Digital modeling and reconstruction has helped preserve and archive many
cultural artifacts and heritage sites before. Perhaps the most well-known
example is the efforts to digitally model the sculptures of Michelangelo by
Levoy et. al. [24] and Rushmeier et. al. [8, 30, 31] two decades ago. The
latter uses a structured-light scanner to capture range, color cameras to
capture color, and a photometric stereo system to capture surface reflectance
and normals. The former uses a laser-stripe scanner and a color camera to
model the statues of Michelangelo and a time-of-flight rangefinder to scan
building interiors. Our project relies on a time-of-flight ranging unit and a
color camera (built in the FARO Focus3D X 130 scanner). As mentioned in
Section Scanning System, some rooms in the castle have to remain pitch-black,
which excludes any device using visible light.

Compared to art objects, scanning and modeling large heritage sites tends to
produce larger datasets with lower resolution, because heritage sites are much
bigger. For instance, Michelangelo’s statues are typically several meters
tall; both of the prior modeling efforts produced geometry at (sub)-millimeter
scale. In contrast, the footprint of the entire Elmina Castle is over
$8000\text{\,}{\mathrm{m}}^{2}$; in our scans points are about 1.9 millimeters
apart. Such a resolution is empirically sufficient for general entertainment,
but higher resolution is preferred for historical education and research that
require close-up examinations. It took us 500+ working hours to scan the
castle at the current resolution.

#### Mobile VR Rendering

Our reconstruction targets mobile VR, which is resource-constrained and
computationally weak. Much of the recent work focuses improving the VR/$None$
video rendering speed without sacrificing, using techniques ranging from
memoization [10], remote/offloaded rendering [28, 22, 26], specialized
hardware support [36, 23].

Fundamentally, these techniques focus only on VR rendering without considering
the inputs to the render: mesh, texture/normal maps. This paper optimizes the
reconstruction phase, which generates low-complexity mesh but highly detailed
texture and normal maps by directly transferring details from the point cloud.

#### Point-based Graphics

Our point-based detail transfer algorithm draws inspiration from classic
point-based graphics/rendering ligerature [17, 25, 32, 29], which skips meshes
altogether and renders images from points. While our algorithm does not
completely avoid reconstructing a mesh, it avoids reconstructing high-polygon
meshes by directly transferring details from points to low-polygon meshes in
order to reduce the mesh reconstruction overhead.

In the future, it would be interesting to study if we could directly apply
classic point-based rendering techniques by skipping mesh altogether. This is
especially relevant as 1) point cloud acquisition devices (e.g., time-of-
flight laser scanners such as the one we used for our project) become more
accessible, and 2) modeling high-resolution heritage sites requires huge point
clouds, for which mesh reconstruction is prohibitively expensive.

## Conclusion

We propose a system that efficiently reconstructs large heritage sites from
laser-scan point clouds while enabling real-time mobile VR rendering. We
demonstrate the system on Elmina Castle in Ghana. We show that it is possible
to deliver real-time rendering on mobile VR devices with high visual quality
of a complicated architecture without requiring a high-polygon mesh. The key
is to directly transfer details from the point cloud, which also significantly
speeds up the reconstruction time and makes it more accessible for
archaeologists and historians to reconstruct large heritage sites from huge
point clouds using everyday computers.

## References

  * [1] Bermuda 100.  http://bermuda100.ucsd.edu/.
  * [2] Cyark.  https://www.cyark.org/.
  * [3] Faro laser scanner focus3d x 130 manual.  https://faro.app.box.com/s/r45cyjqengcts8vnh5kawemgsvdfxt81.
  * [4] Oculus testing and performance analysis.  https://developer.oculus.com/documentation/unity/unity-perf/.
  * [5] Open heritage 3d.  https://openheritage3d.org/.
  * [6] Openmp.  https://www.openmp.org/.
  * [7] Agisoft, LLC.  Agisoft metashape.  https://www.agisoft.com.
  * [8] Fausto Bernardini, Holly Rushmeier, Ioana M Martin, Joshua Mittleman, and Gabriel Taubin.  Building a digital model of michelangelo’s florentine pieta.  IEEE Computer Graphics and Applications, 22(1):59–67, 2002.
  * [9] Daniele Bonatto, Ségolène Rogge, Arnaud Schenkel, Rudy Ercek, and Gauthier Lafruit.  Explorations for real-time point cloud rendering of natural scenes in virtual reality.  In 2016 International Conference on 3D Imaging (IC3D), pages 1–7. IEEE, 2016.
  * [10] Kevin Boos, David Chu, and Eduardo Cuervo.  Flashback: Immersive virtual reality on mobile devices via rendering memoization.  In Proceedings of the 14th Annual International Conference on Mobile Systems, Applications, and Services (MobiSys’16), pages 291–304. ACM, 2016.
  * [11] Erik Champion.  Applying game design theory to virtual heritage environments.  In Proceedings of the 1st international conference on Computer graphics and interactive techniques in Australasia and South East Asia, pages 273–274, 2003.
  * [12] Paolo Cignoni, Marco Callieri, Massimiliano Corsini, Matteo Dellepiane, Fabio Ganovelli, and Guido Ranzuglia.  MeshLab: an Open-Source Mesh Processing Tool.  In Vittorio Scarano, Rosario De Chiara, and Ugo Erra, editors, Eurographics Italian Chapter Conference. The Eurographics Association, 2008.
  * [13] Jonathan Cohen, Marc Olano, and Dinesh Manocha.  Appearance-preserving simplification.  In Proceedings of the 25th annual conference on Computer graphics and interactive techniques, pages 115–122, 1998.
  * [14] Blender Online Community.  Blender - a 3d modelling and rendering package, 2018.
  * [15] Michael Garland and Paul S Heckbert.  Surface simplification using quadric error metrics.  In Proceedings of the 24th annual conference on Computer graphics and interactive techniques, pages 209–216, 1997.
  * [16] James Paul Gee.  What video games have to teach us about learning and literacy.  Computers in Entertainment (CIE), 1(1):20–20, 2003.
  * [17] Markus Gross and Hanspeter Pfister.  Point-based graphics.  Elsevier, 2011.
  * [18] Matthew Wilhelm Kapell and Andrew BR Elliott.  Playing with the past: Digital games and the simulation of history.  Bloomsbury Publishing USA, 2013.
  * [19] Michael Kazhdan, Matthew Bolitho, and Hugues Hoppe.  Poisson surface reconstruction.  In Proceedings of the Fourth Eurographics Symposium on Geometry Processing, SGP ’06, page 61–70, Goslar, DEU, 2006. Eurographics Association.
  * [20] Michael Kazhdan and Hugues Hoppe.  Screened poisson surface reconstruction.  ACM Trans. Graph., 32(3), July 2013.
  * [21] Michael Kazhdan and Hugues Hoppe.  Screened poisson surface reconstruction.  ACM Transactions on Graphics (ToG), 32(3):1–13, 2013.
  * [22] Zeqi Lai, Y Charlie Hu, Yong Cui, Linhui Sun, Ningwei Dai, and Hung-Sheng Lee.  Furion: Engineering high-quality immersive virtual reality on today’s mobile devices.  IEEE Transactions on Mobile Computing, 2019.
  * [23] Yue Leng, Chi-Chun Chen, Qiuyue Sun, Jian Huang, and Yuhao Zhu.  Energy-efficient video processing for virtual reality.  In Proceedings of the 46th International Symposium on Computer Architecture, pages 91–103, 2019.
  * [24] Marc Levoy, Kari Pulli, Brian Curless, Szymon Rusinkiewicz, David Koller, Lucas Pereira, Matt Ginzton, Sean Anderson, James Davis, Jeremy Ginsberg, et al.  The digital michelangelo project: 3d scanning of large statues.  In Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pages 131–144, 2000.
  * [25] Marc Levoy and Turner Whitted.  The use of points as a display primitive.  Citeseer, 1985.
  * [26] Luyang Liu, Ruiguang Zhong, Wuyang Zhang, Yunxin Liu, Jiansong Zhang, Lintao Zhang, and Marco Gruteser.  Cutting the cord: Designing a high-quality untethered vr system with low latency remote rendering.  In Proceedings of the 16th Annual International Conference on Mobile Systems, Applications, and Services, pages 68–80, 2018.
  * [27] Rafat Mantiuk, Kil Joong Kim, Allan G Rempel, and Wolfgang Heidrich.  Hdr-vdp-2: A calibrated visual metric for visibility and quality predictions in all luminance conditions.  ACM Transactions on graphics (TOG), 30(4):1–14, 2011.
  * [28] Jiayi Meng, Sibendu Paul, and Y Charlie Hu.  Coterie: Exploiting frame similarity to enable high-quality multiplayer vr on commodity mobile devices.  In Proceedings of the Twenty-Fifth International Conference on Architectural Support for Programming Languages and Operating Systems, pages 923–937, 2020.
  * [29] Hanspeter Pfister, Matthias Zwicker, Jeroen Van Baar, and Markus Gross.  Surfels: Surface elements as rendering primitives.  In Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pages 335–342, 2000.
  * [30] Holly Rushmeier, Fausto Bernardini, Joshua Mittleman, and Gabriel Taubin.  Acquiring input for rendering at appropriate levels of detail: Digitizing a pieta.  In Rendering Techniques’ 98, pages 81–92. Springer, 1998.
  * [31] Holly Rushmeier, Gabriel Taubin, and André Guéziec.  Applying shape from lighting variation to bump map capture.  In Rendering Techniques’ 97, pages 35–44. Springer, 1997.
  * [32] Szymon Rusinkiewicz and Marc Levoy.  Qsplat: A multiresolution point rendering system for large meshes.  In Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pages 343–352, 2000.
  * [33] Radu B Rusu and S Cousins.  Point cloud library (pcl).  In 2011 IEEE International Conference on Robotics and Automation, pages 1–4, 2011.
  * [34] Markus Schütz, Katharina Krösl, and Michael Wimmer.  Real-time continuous level of detail rendering of point clouds.  In 2019 IEEE Conference on Virtual Reality and 3D User Interfaces (VR), pages 103–110. IEEE, 2019.
  * [35] Kurt Squire.  Video games and learning.  Teaching and participatory culture in the digital age, 2011.
  * [36] Qiuyue Sun, Amir Taherin, Yawo Siatitse, and Yuhao Zhu.  Energy-efficient 360-degree video rendering on fpga via algorithm-architecture co-design.  In The 2020 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays, pages 97–103, 2020.
  * [37] The CGAL Project.  CGAL User and Reference Manual.  CGAL Editorial Board, 5.0 edition, 2019.
  * [38] Julie C. Xia, Jihad El-Sana, and Amitabh Varshney.  Adaptive real-time level-of-detail based rendering for polygonal models.  IEEE Transactions on Visualization and Computer graphics, 3(2):171–183, 1997.

## Author Biography

Sifan Ye is a graduate student at Stanford University. He obtained his BS in
Computer Science from University of Rochester. His work was done in part while
at University of Rochester.

Ting Wu is a software engineer at eBay. She obtained Bachelor’s degree in
Vehicle Engineering from Tongji University (2017) and Master’s degree in
Computer Science from University of Rochester (2020). Her work was done
entirely while at University of Rochester.

Michael Jarvis is an Associate Professor of History, Director of Digital Media
Studies Program, and Director of Smiths Island Archaeology Project at the
University of Rochester. He works onEarly American, Atlantic, Maritime, Public
and Digital history and historical archaeology. He obtained his Ph.D. from
College of William and Mary (1998).

Yuhao Zhu received his BS in Computer Science from Beihang University (2010)
and his Ph.D. in Electrical and Computer Engineering from The University of
Texas at Austin (2017). He is an Assistant Professor of Computer Science at
University of Rochester. His work focuses on applications, algorithms, and
systems for visual computing.


# Some results on harmonic metrics

Fatima Zohra Kadi  Laboratory of Quantum Physics and Mathematical Modeling
(LPQ3M)
University of Mascara
Faculty of Exact Sciences
Department of Mathematics
BP 305 Route de Mamounia
29000 Mascara
Algeria<EMAIL_ADDRESS>Bouazza Kacimi  University of Mascara
Faculty of Exact Sciences
Department of Mathematics
BP 305 Route de Mamounia
29000 Mascara
Algeria<EMAIL_ADDRESS>Mustafa Özkan  Gazi University
Faculty of Sciences
Department of Mathematics
06500 Ankara
Turkey<EMAIL_ADDRESS>
(Date: January 1, 2004)

###### Abstract.

In this paper, we prove that the harmonicity of a metric on a pseudo-
Riemannian manifold is equivalent to the harmonicity of both its Sasaki (resp.
horizontal and complete) lift metric on the tangent bundle and its Sasaki lift
metric on the cotangent bundle. Some applications are included.

###### Key words and phrases:

Harmonic maps, Sasaki lift, Horizontal lift, Complete lift.

###### 1991 Mathematics Subject Classification:

Primary 53C30; Secondary 53C43

##  1\. Introduction

The notion of harmonic maps between (pseudo-)Riemannian manifolds grew out of
essential notions in differential geometry, such as geodesics, minimal
surfaces and harmonic functions (i.e., real $C^{2}$ functions belonging to the
kernel of the Laplace operator). Harmonic maps are solutions to a natural
geometrical variational problem, this means that they satisfy the Euler-
Lagrange systems see [9, 10], they are closely related to analysis, to the
general relativity, and to nonlinear field theory in theoretical physics see
[14, 16]. This subject has been developed extensively see for example [1, 10,
11, 16] and references therein. In [8], B. Y. Chen and T. Nagano introduced
the notions of harmonic metrics and harmonic tensors and obtain a necessary
and sufficient condition for an immersion between two Riemannian manifolds to
be harmonic, for recent developments of this topic see [7]. C. L. Bejan and S.
L. Druţă-Romaniuc in [3] proved that the harmonicity of a metric on four-
dimensional Walker manifold of signature (2,2) and that of its Sasaki (resp.
horizontal) lift metric on the tangent bundle are equivalent, the same
conclusion is obtained by A. Zaeim and P. Atashpeykarin in [18] and by A.
Zaeim, M. Jafari and M. Yaghoubi in [19] on non-reductive homogeneous manifold
of dimension four and on Gödel-type space time respectively but, in both cases
they proved also the equivalence for the complete lift metric on the tangent
bundle.

The aim of this note is twofold. The first is to study the relationship
between the harmonicity of a metric on a pseudo-Riemannian manifold and that
of its Sasaki (resp. horizontal and complete) lift metric on the tangent
bundle in more general case. The second is to highlight the relationship
between the harmonicity of a metric on a pseudo-Riemannian manifold and that
of its Sasaki lift metric on the cotangent bundle. This paper is structured as
follows. In section $2$, we give some definitions, properties and results
which will be used in the sequel. In section $3$, we study harmonic metrics
lifted to the tangent bundle of a pseudo-Riemannian manifold and we prove that
there is an equivalence between the harmonicity of a metric on a pseudo-
Riemannian manifold and that of its Sasaki (resp. horizontal and complete)
lift metric on the tangent bundle (see Theorem 3.1). As consequence, we
recover the results of [3, 18] and [19]. In section $4$, we deal with harmonic
metrics on Egorov spaces (see Theorem 4.1) and give an example (see Example
Example). The section $5$ is devoted to the study of harmonic metrics lifted
to the cotangent bundle of a pseudo-Riemannian manifold, we prove also that
the harmonicity of a metric and that of its Sasaki lift metric on the
cotangent bundle are equivalent (see Theorem 5.1). As consequence, we obtain
the main results of this note (see Theorem 5.2). In the last section, we apply
Theorem 5.1 on four-dimensional Walker manifold of signature (2,2), on Gödel-
type spacetime, on non-reductive homogeneous manifold of dimension four and on
Egorov space.

##  2\. Preliminaries

###  2.1. Harmonic maps and harmonic metrics

Let $(M,g)$ and $(N,h)$ be two pseudo-Riemannian manifolds of dimensions $m$
and $n$ respectively, and let $\varphi:(M,g)\rightarrow(N,h)$ a smooth map
between them. The map $\varphi$ is said to be harmonic if it is a critical
point of the energy functional defined by

$E(\varphi,D)=\frac{1}{2}\underset{D}{\int}|d\varphi|^{2}v_{g},$

for any compact domain $D$ of $M$, where $v_{g}$ is the volume element of
$(M,g)$. If $M$ is compact, we write

$E(\varphi)=E(\varphi,M).$

Let $\nabla d\varphi$ be the second fundamental form of $\varphi.$ The trace
of $\nabla d\varphi$ with respect to $g$ is called the tension field of
$\varphi$, and is denoted by $\tau(\varphi)$. The Euler-Lagrange equation of
$E(\varphi,D)$ is [11]

$\tau(\varphi)=tr(\nabla d\varphi)=0.$

Let $U\subset M$ be an open set with coordinates $(x^{1},\ldots,x^{m})$ and
$V\subset N$ be an open set with coordinates $(y^{1},\ldots,y^{n})$, such that
$\varphi(U)\subset V$ and suppose that $\varphi$ is locally represented by

$y^{\alpha}=\varphi^{\alpha}(x^{1},\ldots,x^{m}),\;\;\alpha=1,\ldots,n.$

Then we have:

$(\nabla d\varphi)^{\gamma}_{ij}=\frac{\partial^{2}\varphi^{\gamma}}{\partial
x^{i}\partial x^{j}}-\Gamma^{k}_{ij}\frac{\partial\varphi^{\gamma}}{\partial
x^{k}}+{}^{N}\Gamma^{\gamma}_{\alpha\beta}(\varphi)\frac{\partial\varphi^{\alpha}}{\partial
x^{i}}\frac{\partial\varphi^{\beta}}{\partial x^{j}},$ (2.1)

here $\Gamma^{k}_{ij}$ and ${}^{N}\Gamma^{\gamma}_{\alpha\beta}$ denote the
Christoffel symbols of $(M,g)$ and $(N,h)$ respectively. So, $\varphi$ is
harmonic if and only if

$\tau(\varphi)=tr(\nabla d\varphi)=g^{ij}(\nabla
d\varphi)^{\gamma}_{ij}\frac{\partial}{\partial y^{\gamma}}=0.$ (2.2)

######  Definition 2.1 ([8]).

Let $(M,g)$ be a pseudo-Riemannian manifold of dimension $m$ and $\widehat{g}$
be another pseudo-Riemannian metric on $M$. We say that $\widehat{g}$ is a
harmonic metric with respect to $g$ if and only if the identity map
$I:(M,g)\rightarrow(M,\widehat{g})$ is harmonic.

By taking into account (2.1) and (2.2), $\widehat{g}$ is harmonic with respect
to $g$ if and only if

$g^{ij}(\widehat{\Gamma}^{k}_{ij}-\Gamma^{k}_{ij})=0,$

or, equivalently

$tr(G^{-1}(\widehat{\Gamma}^{k}-\Gamma^{k}))=0,$ (2.3)

where $G$ is the matrix presentation of $g$ and
$\Gamma^{k},\widehat{\Gamma}^{k}$ are the matrices of the Christoffel symbols
of $g$ and $\widehat{g}$ respectively.

###  2.2. Geometry of tangent bundle

Let $(M,g)$ be an $m-$dimensional pseudo-Riemannian manifold and $(TM,\pi,M)$
be its tangent bundle. We recall some facts on the geometry of the tangent
bundle, following [17]. A system of local coordinates
$(U,x^{i}),i=1,\ldots,m$, in $M$ induces on $TM$ a system of local coordinates
$(\pi^{-1}(U),x^{i},x^{\overline{i}}=u^{i}),$
$\overline{i}=m+i=m+1,\ldots,2m,$ where $(x^{\overline{i}}=u^{i})$ are the
components of vector fields on $M$ with respect to the coordinate vector
fields $\\{\partial_{x^{1}},\ldots,\partial_{x^{m}}\\}$, here
$\partial_{x^{i}}=\frac{\partial}{\partial{x^{i}}}.$

Denote the Levi-Civita connection of $g$ by $\nabla$. The tangent space
$T_{(x,u)}TM$ at a point $(x,u)$ in $TM$ is the direct sum of the vertical
subspace $\mathcal{V}_{(x,u)}=\ker(d\pi\mid_{(x,u)})$ and the horizontal
subspace $\mathcal{H}_{(x,u)}$, with respect to $\nabla$:

$T_{(x,u)}TM=\mathcal{H}_{(x,u)}\oplus\mathcal{V}_{(x,u)}.$

We define the local vector fields on $TM$ as

$\delta_{x^{i}}=\partial_{x^{i}}-\Gamma^{k}_{0i}\partial_{x^{\overline{i}}},\quad\text{with}\quad\Gamma^{k}_{0i}=u^{h}\Gamma^{k}_{hi}$

The local frame field $\\{\delta_{x^{i}},\partial_{x^{\overline{i}}}\\}_{1\leq
i\leq m}$ is called the adapted local frame field on $M$, we easily see that
the set $\\{dx^{i},\partial_{x^{\overline{i}}}^{\ast}\\}_{1\leq i\leq m}$ is
the coframe dual to the adapted local frame field
$\\{\delta_{x^{i}},\partial_{x^{\overline{i}}}\\}_{1\leq i\leq m}$, where
$\partial_{x^{\overline{i}}}^{\ast}=dx^{\overline{i}}+x^{\overline{h}}\Gamma^{i}_{hj}dx^{j}$.
Let $X|_{U}=X^{i}\partial_{x^{i}}$ be a local vector field on $M$. The
vertical and the horizontal lifts of $X$ are defined with respect to the
adapted frame field by:

$X^{V}=X^{i}\partial_{x^{\overline{i}}},\qquad X^{H}=X^{i}\delta_{x^{i}}.$

The Sasaki lift metric $g^{S}$ and the horizontal lift metric $g^{H}$ of the
metric $g$ on $TM$ are defined by

$\left\\{\begin{array}[]{c}g^{S}(X^{H},Y^{H})=g(X,Y)\circ\pi,\\\
g^{S}(X^{V},Y^{H})=0,\\\
g^{S}(X^{V},Y^{V})=g(X,Y)\circ\pi,\end{array}\right.\qquad\left\\{\begin{array}[]{c}g^{H}(X^{H},Y^{H})=g^{H}(X^{V},Y^{V})=0,\\\
g^{H}(X^{V},Y^{H})=g(X,Y)\circ\pi,\end{array}\right.$

for all vector fields $X,Y\in\Gamma(TM)$. The matrix representations of
$g^{S}$ and $g^{H}$, respectively are given by

$G^{S}=\left(\begin{array}[]{cc}g_{ij}&0\\\ 0&g_{ij}\\\
\end{array}\right),\qquad G^{H}=\left(\begin{array}[]{cc}0&g_{ij}\\\
g_{ij}&0\\\ \end{array}\right).$ (2.4)

Also the inverses of the matrices of $g^{S}$ and $g^{H}$ are given by

$(G^{S})^{-1}=\left(\begin{array}[]{cc}g^{ij}&0\\\ 0&g^{ij}\\\
\end{array}\right),\qquad(G^{H})^{-1}=\left(\begin{array}[]{cc}0&g^{ij}\\\
g^{ij}&0\\\ \end{array}\right).$ (2.5)

The matrices of the Christoffel symbols of $g^{S}$ and $g^{H}$ are determined
respectively as follows

${}^{S}\Gamma^{k}$
$\displaystyle=\left(\begin{array}[]{cc}\Gamma^{k}_{ij}&\frac{1}{2}R^{k}_{hji}u^{h}\\\
\\\ \frac{1}{2}R^{k}_{hij}u^{h}&0\\\
\end{array}\right),\quad^{S}\Gamma^{\overline{k}}=\left(\begin{array}[]{cc}-\frac{1}{2}R^{k}_{ijh}u^{h}&\Gamma^{k}_{ij}\\\
\\\ \Gamma^{k}_{ij}&0\end{array}\right),$ (2.12) ${}^{H}\Gamma^{k}$
$\displaystyle=\left(\begin{array}[]{cc}\Gamma^{k}_{ij}&\Gamma^{k}_{ij}\\\
\Gamma^{k}_{ij}&0\\\ \end{array}\right),\qquad^{H}\Gamma^{\overline{k}}=0,$
(2.15)

where $R$ is the curvature tensor of $(M,g)$ and
$R(\partial_{x_{i}},\partial_{x_{j}})\partial_{x_{h}}=R^{k}_{ijh}\partial_{x_{k}}$.
Now, let $\omega$ be a $1-$form on $M$. The evaluation map is a function
$i\omega:TM\rightarrow\mathbb{R}$, defined by $i\omega(p,u)=\omega_{p}(u)$. As
a special case, for each function $f$, $f^{C}=i(df)$ is called the complete
lift of the function $f$. These functions on $TM$ characterize vector fields
on $TM$, since, for vector fields $\widetilde{X}$ and $\widetilde{Y}$ on $TM$,
$\widetilde{X}(f^{C})=\widetilde{Y}(f^{C})$ if and only if
$\widetilde{X}=\widetilde{Y}$, for all functions $f$. For each vector field
$X$ on $M$, its complete lift $X^{C}$ is the vector field on $TM$ defined by

$X^{C}(f^{C})=(Xf)^{C}.$

For a pseudo-Riemannian manifold $(M,g)$, the tangent bundle $TM$ is naturally
equipped with the complete lift metric

$g^{C}(X^{C},Y^{C})=(g(X,Y))^{C},$

for any $X,Y\in\Gamma(TM)$. The matrices presentations of $g^{C}$ and its
inverse are given by (see [6], [17])

$G^{C}=\left(\begin{array}[]{cc}x^{\overline{k}}\cfrac{\partial
g_{ij}}{\partial x^{k}}&g_{ij}\\\ \\\
g_{ij}&0\end{array}\right),\qquad(G^{C})^{-1}=\left(\begin{array}[]{cc}0&g^{ij}\\\
\\\ g^{ij}&x^{\overline{k}}\cfrac{\partial g^{ij}}{\partial
x^{k}}\end{array}\right).$ (2.16)

Note that $g^{C}$ is a pseudo-Riemannian metric of neutral signature. The
Levi-Civita connections of $g^{C}$ and $g$ are related as follows

${}^{C}\Gamma^{k}=\left(\begin{array}[]{cc}\Gamma^{k}_{ij}&0\\\
0&0\end{array}\right),\qquad^{C}\Gamma^{\overline{k}}=\left(\begin{array}[]{cc}x^{\overline{l}}\cfrac{\partial}{\partial
x^{l}}\Gamma^{k}_{ij}&\Gamma^{k}_{ij}\\\ \\\
\Gamma^{k}_{ij}&0\end{array}\right).$ (2.17)

##  3\. Harmonic metrics lifted to the tangent bundle

######  Theorem 3.1.

Let $(M^{m},g)$ be a pseudo-Riemannian manifold and $\widehat{g}$ is a pseudo-
Riemannian metric on $M.$ The following statements are equivalent:

  * (i)

$\widehat{g}$ is harmonic with respect to $g$.

  * (ii)

$\widehat{g}^{S}$ is harmonic with respect to $g^{S}$.

  * (iii)

$\widehat{g}^{H}$ is harmonic with respect to $g^{H}$.

  * (iv)

$\widehat{g}^{C}$ is harmonic with respect to $g^{C}$.

###### Proof.

The equivalence between the items (i) and (ii) arises as follows:
For the identity map $I:(TM,g^{S})\rightarrow(TM,\widehat{g}^{S})$ the
harmonicity condition (2.3) can be expressed by

$\left\\{\begin{array}[]{c}tr((G^{S})^{-1}(^{S}\widehat{\Gamma}^{k}-{}^{S}\Gamma^{k}))=0,\\\
tr((G^{S})^{-1}(^{S}\widehat{\Gamma}^{\overline{k}}-{}^{S}\Gamma^{\overline{k}}))=0,\\\
\end{array}\right.$ (3.1)

for all indices $k=1,\ldots,m$, where
${}^{S}\Gamma^{k},{}^{S}\Gamma^{\overline{k}}$ are the matrices given by
(2.12) and ${}^{S}\widehat{\Gamma}^{k},{}^{S}\widehat{\Gamma}^{\overline{k}}$
the similar matrices for ${}^{S}\widehat{g}$. By using (2.5) and (2.12), the
first equation of (3.1) reduces to

$tr(G^{-1}(\widehat{\Gamma}^{k}-\Gamma^{k}))=0,\qquad\forall k=1,\ldots,m.$
(3.2)

Now, denoting the curvature tensor of $g$ and $\widehat{g}$ by $R$ and
$\widehat{R}$, respectively, and taking into account (2.5) and (2.12), it
follows that the second equation in (3.1) yields

$tr(g^{ij}(\widehat{R}^{k}_{ij0}-R^{k}_{ij0}))=0,$ (3.3)

where $R^{k}_{ij0}=R^{k}_{ijh}u^{h}$ and
$\widehat{R}^{k}_{ij0}=\widehat{R}^{k}_{ijh}u^{h}$. Equivalently, (3.3) can be
written as follows

$tr(G^{-1}(\widehat{R}^{k}-R^{k}))=0,\qquad\forall k=1,\ldots,m,$ (3.4)

where $R^{k}$ and $\widehat{R}^{k}$ are the matrices given by
$R^{k}=(R^{k}_{ij0})$, $\widehat{R}^{k}=(\widehat{R}^{k}_{ij0})$,
$i,j=1,\ldots,m.$ Since $G^{-1}$ is symmetric and $(\widehat{R}^{k}-R^{k})$ is
antisymmetric, then by a result of linear algebra (3.4) is always satisfied.
The relations (3.2) expresses the harmonicity condition of the identity map
$I:(M,g)\rightarrow(M,\widehat{g})$. Hence, the metric $\widehat{g}$ is
harmonic with respect to $g$, which yields the equivalence between (i) and
(ii).

Using (2.3), the harmonicity condition of $\widehat{g}^{H}$ with respect to
$g^{H}$ can be expressed by:

$\left\\{\begin{array}[]{c}tr((G^{S})^{-1}(^{H}\widehat{\Gamma}^{k}-{}^{H}\Gamma^{k}))=0,\\\
tr((G^{S})^{-1}(^{H}\widehat{\Gamma}^{\overline{k}}-{}^{H}\Gamma^{\overline{k}}))=0,\\\
\end{array}\right.$ (3.5)

for all indices $k=1,\ldots,m$, where
${}^{H}\Gamma^{k},{}^{H}\Gamma^{\overline{k}}$ are the matrices given by
(2.15) and ${}^{H}\widehat{\Gamma}^{k},{}^{H}\widehat{\Gamma}^{\overline{k}}$
the similar matrices for ${}^{H}\widehat{g}$. So by (2.15) the system (3.5)
gives

$tr(G^{-1}(\widehat{\Gamma}^{k}-\Gamma^{k}))=0\qquad\forall k=1,\ldots,m.$

This equation shows exactly the same conditions of harmonicity of the identity
map $I:(M,g)\rightarrow(M,\widehat{g})$.

The third equivalence between (i) and (iv) follows by the same method as above
by using (2.16) and (2.17). Thus the proof is complete. ∎

######  Remark 3.2.

Theorem 3.1, recovers the results due to [3, 18] and [19].

##  4\. Harmonic metrics on Egorov spaces

The Egorov spaces are Lorentzian manifolds $(\mathbb{R}^{m},g_{f})$, $m\geq
3$, where $f$ is a positive smooth function of a real variable and

$g_{f}=f(x^{m})\sum^{m-2}_{i=1}(dx^{i})^{2}+2dx^{m-1}dx^{m}.$ (4.1)

These manifolds are named after I. P. Egorov, who first introduced and studied
them in [12]. In [2], W. Batat, G. Calvaruso and B. De Leo described some
curvature properties of Egorov spaces. Now, we study the harmonicity condition
for Lorentzian metrics which are raised from Egorov space.

######  Theorem 4.1.

Let $(\mathbb{R}^{m},g_{f})$, $m\geq 3$ be an Egorov space with the metric
defined by (4.1). Then, the metric

$\widehat{g}_{\widehat{f}}=\widehat{f}(x^{m})\sum^{m-2}_{i=1}(dx^{i})^{2}+2dx^{m-1}dx^{m},$
(4.2)

is harmonic with respect to $g_{f}$ if and only if

$\widehat{f}^{\prime}=f^{\prime}.$

###### Proof.

Let $(\mathbb{R}^{m},g_{f})$, $m\geq 3$ be an Egorov space with the metric
defined by (4.1), and $\widehat{g}_{\widehat{f}}$ be the metric

$\widehat{g}_{\widehat{f}}=\widehat{f}(x^{m})\sum^{m-2}_{i=1}(dx^{i})^{2}+2dx^{m-1}dx^{m},$

which is harmonic with respect to the metric $g_{f}$. Put $f(x^{m})=f$, the
matrix presentation of the metric $g_{f}$ is

$G_{f}=\left(\begin{array}[]{c|c}\begin{array}[]{ccc}f&&O\\\ &\ddots&\\\
0&&f\end{array}&O\\\ \hline\cr\\\ O&\begin{array}[]{cc}0&1\\\
1&0\end{array}\end{array}\right).$ (4.3)

It’s obvious that

$(G_{f})^{-1}=\left(\begin{array}[]{c|c}\begin{array}[]{ccc}\frac{1}{f}&&O\\\
&\ddots&\\\ O&&\frac{1}{f}\end{array}&O\\\ \hline\cr\\\
O&\begin{array}[]{cc}0&1\\\ 1&0\end{array}\end{array}\right).$ (4.4)

Put $\Gamma^{k}=(\Gamma^{k}_{ij})$, $i,j=1,\ldots,m$ and $k=1,\ldots,m,$ then
from [2], the only possible non-vanishing Christoffel symbols are the
following ones:

$\Gamma^{m-1}_{ii}=\frac{-f^{\prime}}{2},\;\;\;\Gamma^{i}_{im}=\frac{f^{\prime}}{2f},\;\;\;i=1,\cdots,m-2.$
(4.5)

Thus

$\Gamma^{1}=\left(\begin{array}[]{c|c}O&\begin{array}[]{c}\cfrac{f^{\prime}}{2f}\\\
0\\\ \vdots\\\ 0\end{array}\\\
\hline\cr\begin{array}[]{cccc}\cfrac{f^{\prime}}{2f}&0&\cdots&0\end{array}&0\end{array}\right),$
$\Gamma^{2}=\left(\begin{array}[]{c|c}O&\begin{array}[]{c}0\\\
\cfrac{f^{\prime}}{2f}\\\ 0\\\ \vdots\\\ 0\end{array}\\\
\hline\cr\begin{array}[]{ccccc}0&\cfrac{f^{\prime}}{2f}&0&\cdots&0\end{array}&0\end{array}\right).$

And so on until we arrive to

$\Gamma^{m-2}=\left(\begin{array}[]{c|c}O&\begin{array}[]{c}0\\\ \vdots\\\
0\\\ \cfrac{f^{\prime}}{2f}\\\ 0\end{array}\\\
\hline\cr\begin{array}[]{ccccc}0&\cdots&0&\cfrac{f^{\prime}}{2f}&0\end{array}&0\end{array}\right)$

and

$\Gamma^{m-1}=\left(\begin{array}[]{c|c}\begin{array}[]{ccc}-\cfrac{f^{\prime}}{2}&&O\\\
&\ddots&\\\ O&&-\cfrac{f^{\prime}}{2}\end{array}&O\\\ \hline\cr\\\
O&\begin{array}[]{cc}0&0\\\
0&0\end{array}\end{array}\right),\qquad\Gamma^{m}=0.$

The Levi-Civita components $\widehat{\Gamma}^{1},\ldots,\widehat{\Gamma}^{m}$
of the Lorentzian metric given in (4.2), are obviously deduced by replacing
$\widehat{f}$ instead of $f$ in $\Gamma^{1},\ldots,\Gamma^{m}$. By direct
computations, we find the only non-zero trace as follows:

$tr(G^{-1}_{f}(\widehat{\Gamma}^{m-1}-\Gamma^{m-1}))=\cfrac{(m-2)(f^{\prime}-\widehat{f}^{\prime})}{2f}.$

Thus, by virtue of (2.3), the identity map
$I:(\mathbb{R}^{m},g_{f})\rightarrow(\mathbb{R}^{m},\widehat{g}_{\widehat{f}})$
is harmonic if and only if $\widehat{f}^{\prime}=f^{\prime}$, which completes
the proof. ∎

######  Example.

Let $(M,g)$ be an $m-$dimensional pseudo-Riemannian manifold and $TM$ be its
tangent bundle. The Sasaki and horizontal lift metrics of $g$ with respect to
the local coordinates $(x^{i},x^{\overline{i}})$ of $TM$ are respectively [17]

$\displaystyle g^{S}$
$\displaystyle=g_{ij}dx^{i}dx^{j}+g_{ij}\partial_{x^{\overline{i}}}^{\ast}\partial_{x^{\overline{j}}}^{\ast},$
(4.6) $\displaystyle g^{H}$
$\displaystyle=2g_{ij}dx^{i}\partial_{x^{\overline{j}}}^{\ast}.$ (4.7)

Then, by virtue of (2.16), (4.1), (4.5), (4.6) and (4.7), for an Egorov space
$(\mathbb{R}^{m},g_{f})$, $m\geq 3$, the Sasaki, horizontal and complete lift
metrics of $g_{f}$ with respect to the local coordinates
$(x^{i},x^{\overline{i}})$ of the tangent bundle of $\mathbb{R}^{m}$ are
respectively

$\displaystyle g^{S}_{f}$
$\displaystyle=f\sum^{m-2}_{i=1}(dx^{i})^{2}+2dx^{m-1}dx^{m}+f\sum^{m-2}_{i=1}(dx^{\overline{i}})^{2}+2dx^{\overline{m-1}}dx^{\overline{m}}$
$\displaystyle+f^{\prime}\sum^{m-2}_{i=1}x^{\overline{i}}dx^{m}dx^{\overline{i}}-f^{\prime}\sum^{m-2}_{i=1}x^{\overline{i}}dx^{\overline{m}}dx^{i}+\frac{(f^{\prime})^{2}}{4f}\sum^{m-2}_{i=1}(x^{\overline{i}})^{2}(dx^{m})^{2},$
$\displaystyle g^{H}_{f}$
$\displaystyle=2f(x^{m})\sum^{m-2}_{i=1}dx^{i}dx^{\overline{i}}+2dx^{m-1}dx^{\overline{m}}+2dx^{\overline{m-1}}dx^{m},$
$\displaystyle g^{C}_{f}$
$\displaystyle=2f(x^{m})\sum^{m-2}_{i=1}dx^{i}dx^{\overline{i}}+2dx^{m-1}dx^{\overline{m}}+2dx^{\overline{m-1}}dx^{m}+x^{\overline{m}}f^{\prime}(x^{m})\sum^{m-2}_{i=1}(dx^{i})^{2}.$

By using Theorem 3.1 and Theorem 4.1, the metric:

  1. (1)

$g_{e^{x^{m}}+K}$ is harmonic with respect to $g_{e^{x^{m}}}$,

  2. (2)

$g^{S}_{e^{x^{m}}+K}$ is harmonic with respect to $g^{S}_{e^{x^{m}}}$,

  3. (3)

$g^{H}_{e^{x^{m}}+K}$ is harmonic with respect to $g^{H}_{e^{x^{m}}}$,

  4. (4)

$g^{C}_{e^{x^{m}}+K}$ is harmonic with respect to $g^{C}_{e^{x^{m}}}$, where
$K$ is positive constant.

##  5\. Harmonic metrics lifted to the cotangent bundle

Let $(M,g)$ be an $m-$dimensional pseudo-Riemannian manifold, $T^{*}M$ its
cotangent bundle, and $\widetilde{\pi}$ the natural projection
$T^{*}M\rightarrow M$. We recall some basic properties on the geometry of the
cotangent bunde following [15, 17]. A system of local coordinates
$(U,x^{i}),i=1,\ldots,m$ in $M$ induces on $T^{*}M$ a system of local
coordinates $(\widetilde{\pi}^{-1}(U),x^{i},x^{\widetilde{i}}=p_{i}),$
$\widetilde{i}=m+i=m+1,\ldots,2m$, where $(x^{\widetilde{i}}=p_{i})$ is the
components of covector $p$ in each cotangent space $T^{*}_{x}M$, $x\in U$ with
respect to the natural coframe $\\{dx^{i}\\}$.

Let $X=X^{i}\frac{\partial}{\partial x^{i}}=X^{i}\partial_{x^{i}}$ and
$\omega=\omega_{i}dx^{i}$ be the local expressions in $U\subset M$ of a vector
and covector (1-form) fields $X$ and $\omega$, respectively. Then the
horizontal lift ${}^{H}\\!X$ of $X$ and the vertical lift ${}^{V}\\!\omega$ of
$\omega$ are given respectively, by

${}^{H}\\!X$
$\displaystyle=X^{i}\partial_{x^{i}}+p_{h}\Gamma^{h}_{ij}X^{j}\partial_{x^{\widetilde{i}}};$
${}^{V}\\!\omega$ $\displaystyle=\omega_{i}\partial_{x^{\widetilde{i}}}$

with respect to the natural frame
$\\{\partial_{x^{i}},\partial_{x^{\widetilde{i}}}\\}$, where $\Gamma^{h}_{ij}$
are components of the Levi-Civita connection $\nabla$ of $M$. Define the local
vector fields on $T^{*}M$ as

$\widetilde{\delta}_{x^{i}}=\partial_{x^{i}}+p_{a}\Gamma^{a}_{hi}\partial_{x^{\widetilde{i}}},$

we call the set
$\\{\widetilde{\delta}_{x^{i}},\partial_{x^{\widetilde{i}}}\\}$, the frame
adapted to $\nabla$, it’s clear that the set
$\\{dx^{i},\partial_{x^{\widetilde{i}}}^{\ast}\\}_{1\leq i\leq m}$ is the
coframe dual to frame
$\\{\widetilde{\delta}_{x^{i}},\partial_{x^{\widetilde{i}}}\\}$, where
$\partial_{x^{\widetilde{i}}}^{\ast}=dx^{\widetilde{i}}-p_{a}\Gamma^{a}_{hi}dx^{h}$.
For each $x\in M$ the scalar product $g^{-1}=(g^{ij})$ is defined on the
cotangent space $\widetilde{\pi}^{-1}(x)=T^{*}_{x}M$ by

$g^{-1}(\omega,\theta)=\omega_{i}\theta_{j}g^{ij},$

for all covectors $\omega$ and $\theta$ on $M$.

A Sasakian metric $g^{\widetilde{S}}$ is defined on $T^{*}M$ by the three
equations

$\left\\{\begin{array}[]{c}g^{\widetilde{S}}(^{V}\\!\omega,^{V}\\!\theta)={}^{V}\\!(g^{-1}(\omega,\theta))=g^{-1}(\omega,\theta)\circ\pi,\\\
g^{\widetilde{S}}(^{V}\\!\omega,^{H}\\!Y)=0,\\\
g^{\widetilde{S}}(^{H}\\!X,^{H}\\!Y)={}^{V}\\!(g(X,Y))=g(X,Y)\circ\pi,\\\
\end{array}\right.$

for any $X,Y\in\Gamma(TM)$ and any covectors $\omega$ and $\theta$ on $M$. The
matrix representations of $g^{\widetilde{S}}$ and its inverse are given by,
respectively

$G^{\widetilde{S}}=\left(\begin{array}[]{cc}g_{ij}&0\\\ 0&g^{ij}\\\
\end{array}\right),\qquad(G^{\widetilde{S}})^{-1}=\left(\begin{array}[]{cc}g^{ij}&0\\\
0&g_{ij}\\\ \end{array}\right),$ (5.1)

with respect to the adapted frame
$\\{\widetilde{\delta}_{x^{i}},\partial_{x^{\widetilde{i}}}\\}$. The matrices
of the Christoffel symbol of $g^{\widetilde{S}}$ are given as follows :

${}^{\widetilde{S}}\Gamma^{k}$
$\displaystyle=\left(\begin{array}[]{cc}\Gamma^{k}_{ij}&\frac{1}{2}p_{m}(R^{k}_{.i.})^{jm}\\\
\\\ \frac{1}{2}p_{m}(R^{k}_{.j.})^{im}&0\\\ \end{array}\right),$ (5.5)
${}^{\widetilde{S}}\Gamma^{\widetilde{k}}$
$\displaystyle=\left(\begin{array}[]{cc}\frac{1}{2}p_{m}R^{m}_{ijk}&-\Gamma^{j}_{ik}\\\
\\\ -\Gamma^{i}_{jk}&0\\\ \end{array}\right),$ (5.9)

where $(R^{k}_{.i.})^{jm}=g^{kt}g^{js}R^{m}_{tis}.$

######  Theorem 5.1.

Let $(M,g)$ be a pseudo-Riemannian manifold and $\widehat{g}$ is a pseudo-
Riemannian metric on $M$. The following statements are equivalent:

  * (i)

$\widehat{g}$ is harmonic with respect to $g$.

  * (ii)

$\widehat{g}^{\widetilde{S}}$ is harmonic with respect to $g^{\widetilde{S}}$.

###### Proof.

For the identity map
$I:(T^{*}M,g^{\widetilde{S}})\rightarrow(T^{*}M,\widehat{g}^{\widetilde{S}})$
the harmonicity condition (2.3) can be described by:

$\left\\{\begin{array}[]{c}tr((G^{\widetilde{S}})^{-1}(^{\widetilde{S}}\widehat{\Gamma}^{k}-{}^{\widetilde{S}}\Gamma^{k}))=0,\\\
tr((G^{\widetilde{S}})^{-1}(^{\widetilde{S}}\widehat{\Gamma}^{\widetilde{k}}-{}^{\widetilde{S}}\Gamma^{\widetilde{k}}))=0,\\\
\end{array}\right.$ (5.10)

for all indices $k=1,\ldots,m,$ by virtue of (5.1) and (5.5), the first
equation of (5.10) reduces to

$tr(G^{-1}(\widehat{\Gamma}^{k}-\Gamma^{k}))=0\qquad\forall k=1,\ldots,m.$

Now, taking into account (5.1) and (5.9), it follows that the second equation
in (5.10) yields

$tr(g^{ij}(\widehat{R}^{o}_{ijk}-R^{o}_{ijk}))=0,$ (5.11)

where $R^{o}_{ijk}=p_{h}R^{h}_{ijk}$ and
$\widehat{R}^{o}_{ijk}=p_{h}\widehat{R}^{h}_{ijk}$. Equivalently, (5.11) can
be written as follows

$tr(G^{-1}(\widehat{R}_{k}-R_{k}))=0,\qquad\forall k=1,\ldots,m,$ (5.12)

where $R_{k}$ and $\widehat{R}_{k}$ are the matrices given by
$R_{k}=(R^{o}_{ijk})$, $\widehat{R}_{k}=(\widehat{R}^{o}_{ijk})$ ,
$i,j=1,\ldots,m.$ Since $G^{-1}$ is symmetric and $(\widehat{R}_{k}-R_{k}))$
is antisymmetric, then (5.12) is always verified. Hence the metric
$\widehat{g}$ is harmonic with respect to $g$, which gives the equivalence
between (i) and (ii). ∎

From Theorem 3.1 and Theorem 5.1, we obtain

######  Theorem 5.2.

Let $(M^{m},g)$ be a pseudo-Riemannian manifold and $\widehat{g}$ is a pseudo-
Riemannian metric on $M.$ The following statements are equivalent:

  * (i)

$\widehat{g}$ is harmonic with respect to $g$.

  * (ii)

$\widehat{g}^{S}$ is harmonic with respect to $g^{S}$.

  * (iii)

$\widehat{g}^{H}$ is harmonic with respect to $g^{H}$.

  * (iv)

$\widehat{g}^{C}$ is harmonic with respect to $g^{C}$.

  * (v)

$\widehat{g}^{\widetilde{S}}$ is harmonic with respect to $g^{\widetilde{S}}$.

##  6\. Applications

In this section, we apply Theorem 5.1 on four-dimensional Walker manifolds of
signature (2,2), on Gödel-type spacetimes, on non-reductive homogeneous
manifolds of dimension four and on Egorov spaces.

###  6.1. Four-dimensional Walker manifolds of signature (2,2)

Let $(M,g)$ be a four-dimensional Walker manifold of signature (2,2). We know
from [4] that there exists local coordinates $(x^{1},x^{2},x^{3},x^{4})$ such
that $g$ is described as

$g_{a,b,c}=2dx^{1}dx^{4}+2dx^{2}dx^{3}+a(dx^{3})^{2}+b(dx^{4})^{2}+2cdx^{3}dx^{4}$
(6.1)

where $a,b,c$ are arbitrary smooth functions on $M$. Let
$\widehat{g}_{\widehat{a},\widehat{b},\widehat{c}}$ an arbitrary Walker
metric, the authors in [3] proved that
$\widehat{g}_{\widehat{a},\widehat{b},\widehat{c}}$ is harmonic with respect
to $g_{a,b,c}$ if and only if the following relations are established

$\left\\{\begin{array}[]{c}a_{t}+c_{x}=\widehat{a}_{t}+\widehat{c}_{x}\\\
c_{t}+b_{x}=\widehat{c}_{t}+\widehat{b}_{x}\\\ \end{array}\right..$ (6.2)

By virtue of Theorem 5.1 and (6.2), we get the following

######  Theorem 6.1.

Let $(M,g_{a,b,c})$ be a four-dimensional Walker manifold and
$\widehat{g}_{\widehat{a},\widehat{b},\widehat{c}}$ an arbitrary Walker metric
on $M$. Then the Sasaki lift metric
$\widehat{g}_{\widehat{a},\widehat{b},\widehat{c}}^{\widetilde{S}}$ is
harmonic with respect to $g_{a,b,c}^{\widetilde{S}}$ if and only if (6.2)
holds.

###  6.2. Gödel-type spacetimes

Let $(M,g)$ be a Gödel-type spacetime where $g$ is described by the following
Lorentzian metric

$g=[dt+H(r)d\varphi]^{2}-dr^{2}-P^{2}(r)d\varphi^{2}-dz^{2},$

here $t$ is the time variable and $(r,\varphi,z)$ are the usual cylindrical
coordinates and so $r\geq 0,$ $\varphi\in\mathbb{R}$ (undetermined for $r=0)$
and $z\in\mathbb{R}$. Moreover, $H(r)$ and $P(r)$ are arbitrary smooth
functions on $M$ and $g$ is non degenerate where $P(r)\neq 0$. We can rewrite
$g$ as following

$g=[dx^{1}+H(x^{2})dx^{3}]^{2}-(dx^{2})^{2}-P^{2}(x^{2})(dx^{3})^{2}-(dx^{4})^{2},$
(6.3)

such that $(x^{1},x^{2},x^{3},x^{4})=(t,r,\varphi,z)$. The authors in [19]
proved that the Gödel-type metric

$\widehat{g}=[dx^{1}+\widehat{H}(x^{2})dx^{3}]^{2}-(dx^{2})^{2}-\widehat{P}^{2}(x^{2})(dx^{3})^{2}-(dx^{4})^{2},$
(6.4)

is harmonic with respect to $g$ (6.3) if and only if

$\widehat{H}^{\prime}(x^{2})(\widehat{H}(x^{2})-H(x^{2}))-\widehat{P}(x^{2})\widehat{P}^{\prime}(x^{2})+P(x^{2})P^{\prime}(x^{2})=0.$
(6.5)

Using Theorem 5.1 and (6.5), we deduce

######  Theorem 6.2.

Let $(M,g)$ be a Gödel-type spacetime with the metric (6.3) and $\widehat{g}$
the Lorentzian metric given by (6.4). Then the Sasaki lift metric
$\widehat{g}^{\widetilde{S}}$ is harmonic with respect to $g^{\widetilde{S}}$
if and only if (6.5) is satisfied.

###  6.3. Non-reductive homogeneous manifolds of dimension four

A (connected) pseudo-Riemannian manifold $(M,g)$ is said to be homogeneous if
it admits a group $G$ of isometries, acting transitively on it. In this case,
$(M,g)$ can be identified with $(G/H,g)$, where $H$ is the isotropy group at a
fixed point of $M$ and $g$ is invariant pseudo-Riemannian metric. A
homogeneous pseudo-Riemannian manifold $(M,g)$ is reductive if the Lie algebra
$\mathfrak{g}$ of $G$ may be decomposed into a vector space direct sum
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ where $\mathfrak{m}$ is an
$Ad(H)$-invariant complement to $\mathfrak{h}$. Non-reductive homogeneous
manifolds of dimension four were classified in [13], in terms of the
corresponding non-reductive Lie algebra presentations. This classification
contains 8 classes $A_{1},A_{2},A_{3},A_{4},A_{5}$ and $B_{1},B_{2},B_{3}$.
The classes $A_{1},$ $A_{2},$ $A_{3}$ contain both Lorentzian and neutral
examples, classes $A_{4},A_{5}$ are just Lorentzian and $B_{1},B_{2},B_{3}$
are always of neutral signature. Also the description in coordinates system
for the invariant metrics on the spaces mentioned were obtained in [5]. Any
invariant metric on a non-reductive homogeneous four manifold is described by
some arbitrary coefficients $a,b,\ldots$, see [5, 18]. We take an arbitrary
invariant metric $\widehat{g}$, which is defined by arbitrary coefficients
$\widehat{a},\widehat{b},\ldots$. The authors in [18] obtained the conditions
under which the metric $\widehat{g}$ is harmonic with respect to the metric
$g$. By using Theorem 5.1 and Theorem 5.1 in [18], we obtain the following

######  Theorem 6.3.

Let $(M=G/H,g)$ be a non-reductive homogeneous manifold of dimension four,
equipped with invariant metric $g$. Then the Sasaki metric
$\widehat{g}^{\widetilde{S}}$ is harmonic with respect to $g^{\widetilde{S}}$
if and only if one of the following cases occurs

  1. (1)

$(G/H,g)$ is of type $A_{1}$ and $c\widehat{a}=a\widehat{c}$.

  2. (2)

$(G/H,g)$ is of type $A_{2}$ and one of the following cases occurs

    1. (a)

$\alpha=0,$ $c\widehat{a}=a\widehat{c},$

    2. (b)

$\alpha=\cfrac{1}{4},$ $a\widehat{d}=d\widehat{a},$

    3. (c)

$\alpha=arbitrary,$ $c\widehat{a}=a\widehat{c},$ $a\widehat{d}=d\widehat{a}.$

  3. (3)

$(G/H,g)$ is of type $A_{3}$ for $\varepsilon=\pm 1$ and
$b\widehat{a}=a\widehat{b},$ $b\widehat{c}=c\widehat{b}$.

  4. (4)

$(G/H,g)$ is of type $A_{4},A_{5},B_{2}$ or $B_{3}$.

  5. (5)

$(G/H,g)$ is of type $B_{1}$ and $c\widehat{a}=a\widehat{c},$
$a\widehat{d}=d\widehat{a}$.

###  6.4. Egorov spaces

Let $(\mathbb{R}^{m},g_{f})$, $m\geq 3$ be an Egorov space. The Sasaki lift
metric of $g_{f}$ with respect to the coordinates $(x^{i},x^{\widetilde{i}})$
in the cotangent bundle $T^{\ast}\mathbb{R}^{m}$ is given by

$g_{f}^{\widetilde{S}}=(g_{f})_{ij}dx^{i}dx^{j}+(g_{f})^{ij}\partial_{x^{\widetilde{i}}}^{\ast}\partial_{x^{\widetilde{j}}}^{\ast},$
(6.6)

Then, taking into account (4.1), (4.5) and (6.6), we get

$\displaystyle g_{f}^{\widetilde{S}}$
$\displaystyle=f\sum^{m-2}_{i=1}(dx^{i})^{2}+2dx^{m-1}dx^{m}+\frac{1}{f}\sum^{m-2}_{i=1}(dx^{\widetilde{i}})^{2}+2dx^{\widetilde{m-1}}dx^{\widetilde{m}},$
$\displaystyle+\frac{f^{\prime}}{f}\sum^{m-2}_{i=1}x^{\widetilde{m-1}}dx^{\widetilde{i}}dx^{i}-\frac{f^{\prime}}{f}\sum^{m-2}_{i=1}x^{\widetilde{i}}dx^{\widetilde{m-1}}dx^{i}+\frac{(f^{\prime})^{2}}{4f}(x^{\widetilde{m-1}})^{2}\sum^{m-2}_{i=1}(dx^{i})^{2}.$

Combining Theorem 4.1 and Theorem 5.1, we deduce

######  Theorem 6.4.

Let $(\mathbb{R}^{m},g_{f})$, $m\geq 3$ be an Egorov space. Then the Sasaki
metric $\widehat{g}_{\widehat{f}}^{\widetilde{S}}$ is harmonic with respect to
$g_{f}^{\widetilde{S}}$ if and only if

$\widehat{f}^{\prime}=f^{\prime}.$

## References

  * [1] Baird, P., Wood, J. C.: Harmonic morphisms between Riemannain manifolds, Clarendon Press Oxford (2003)
  * [2] Batat, W., Calvaruso, G., De Leo, B.: Curvature properties of Lorentzian manifolds with large isometry groups. Math. Phys. Anal. Geom., 12, 201–217 (2009)
  * [3] Bejan, C. L., Druţă-Romaniuc, S. L.: Structures which are harmonic with respect to Walker metrics. Mediterr. J. Math., 12, 481–496 (2015)
  * [4] Brozos-Vázquez, M., García-Río, E., Gilkey, P., Nikčeví, S., Vázquez-Lorenzo, R. : The Geometry of Walker Manifolds. Synthesis Lectures on Mathematics and Statistics 5, Morgan and Claypool Publishers, St. Louis (2009)
  * [5] Calvaruso, G., Fino, A., Zaeim, A.: Homogeneous geodesics of non-reductive homogeneous pseudo-Riemannian 4-manifolds. Bull. Braz. Math. Soc. (N.S.), 46, 23–64 (2015)
  * [6] Calviño-Louzao, E., García-Río, E., Sixto-Neira, M., Vázquez-Abal, M. E.: Biharmonic maps on tangent and cotangent bundles. J. Geom. Phys., 101, 1–10 (2016)
  * [7] Chen, B. Y.: Differential geometry of identity maps: A survey. Mathematics, 36(8), 1–33 (2020)
  * [8] Chen, B. Y., Nagano, T.: Harmonic metrics, harmonic tensors and Gauss maps. J. Math. Soc. Japan, 36, 295–313 (1981)
  * [9] Eells, J., Lemaire, L.: A report on harmonic maps. Bull. London Math. Soc., 10, 1–68 (1978)
  * [10] Eells, J., Lemaire, L.: Another report on harmonic maps. Bull. London Math. Soc., 20, 385–524 (1988)
  * [11] Eells, J., Sampson, J. H.: Harmonic mappings of Riemannian manifolds. Am. J. Math., 86, 109–160 (1964)
  * [12] Egorov, I. P.: Riemannian spaces of the first three lacunary types in the geometric sense (Russian). Dokl. Akad. Nauk. SSSR 150, 730–732 (1963)
  * [13] Fels, M. E., Renner, A. G.: Non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Can. J. Math., 58, 282–311 (2006)
  * [14] Gauduchon, P.: Harmonic Mappings, Twistors and Sigma Models. World Scientific Publishing Company, (1988)
  * [15] Salimov, A. A., Agca, F.: Some properties of Sasakian metrics in cotangent bundles. Mediterr. J. Math., 8, 243–255 (2011)
  * [16] Xin, Y.: Geometry of Harmonic Maps. Birkhäuser Boston, Boston, MA (1996)
  * [17] Yano, K., Ishihara, S.: Tangent and Cotangent Bundles. Dekker, New York (1973)
  * [18] Zaeim, A., Atashpeykar, P.: Harmonic metrics on four dimensional non-reductive homogeneous manifolds. Czechoslovak Math. J. 68(2), 475–490 (2018)
  * [19] Zaeim, A., Jafari, M., Yaghoubi, M.: Harmonic metrics on Gödel type spacetiemes. Int J. Geom. Methods Mod. Phys. 17(6), 2050092 (2020)