#  Dangoron: Network Construction on Large-scale Time Series Data across
Sliding Windows

Yunlong Xu  0000-0003-2589-7232 University of Rochester<EMAIL_ADDRESS>,  Peizhen Yang  University of Rochester<EMAIL_ADDRESS>and  Zhengbin
Tao  University of Rochester<EMAIL_ADDRESS>
(2023)

time-series, climate data, correlation matrix, sliding query

††copyright: acmcopyright††journalyear: 2023††copyright:
rightsretained††conference: Companion of the 2023 International Conference on
Management of Data; June 18–23, 2023; Seattle, WA, USA††booktitle: Companion
of the 2023 International Conference on Management of Data (SIGMOD-Companion
’23), June 18–23, 2023, Seattle, WA, USA††doi: 10.1145/3555041.3589399††isbn:
978-1-4503-9507-6/23/06††ccs: Information systems††ccs: Information systems
Data management systems††ccs: Information systems Stream management

##  1\. Problem and Motivation

The monitoring and analysis of large-scale time series data has become a
significant area of interest within the database community. Constructing
dynamic correlation-based complex networks is a highly effective method for
extracting insights from extensive time series data across a range of
disciplines, such as neuroscience (Hutchison et al., 2013; Cabral et al.,
2017; Thirion et al., 2014), climate science (Kim et al., 2019; Gozolchiani et
al., 2008), and finance (Tilfani et al., 2021; Kenett et al., 2010). In a
correlation-based network, nodes are defined by their respective time series,
while edges represent the correlations between nodes within a specified time
range. Typically, in most analyses, a series of correlation matrices are
generated by calculating the pairwise correlation of all time series within
sliding windows, given a query time range, a query window size, and a sliding
step. Pearson’s correlation serves as one of the most prevalent measures
across various domains (Jayaweera and Aziz, 2018; Krendl and Betzel, 2022;
Betzel, 2022).

Figure 1. Network Construction on FMRI across Sliding Windows

Motivation Example. Recently, dynamic analysis of blood oxygenation level-
dependent functional magnetic resonance imaging (BOLD-fMRI) has provided
valuable insights into brain network properties by quantifying functional
connectivity metric changes over time (Hutchison et al., 2013). This analysis
relies on 4-D fMRI data, which contains temporal BOLD activity recordings
(Cabral et al., 2017). Each 3-D fMRI dataset comprises approximately 100K-10M
regions at each time point. Traditional 4-D fMRI data analysis involves brain
parcellation on region-based connectivity matrices, which refers to distinct,
discontinuous yet closely interacting brain partitions (Thirion et al., 2014).
A superior alternative includes building a graph on voxel-level data and
performing feature selection and graph embedding. However, constructing a
network with raw voxel-level data is computationally infeasible due to the
time-consuming pairwise correlation calculations required for voxel-based
functional connectivity matrices within sliding windows. This paper introduces
techniques to compute a sequence of correlation matrices on 3-D fMRI data for
all brain regions

Problem Definition. We are given a collection of time series represented by
matrix $\mathcal{X}$ of size $N\times L$. Here, we have $N$ time series, and
each row represents a time series of length $L$. Each element $x_{ij}$ in
$\mathcal{X}$ denotes the time-stamped value of a variable collected at
location $i$ at time $j$. We use $X_{i}$ to represent the $i$th row, which
corresponds to the $i$th time series. We assume all time series in
$\mathcal{X}$ are synchronized, meaning each time series has a value available
at every periodic time interval or time resolution. This can be achieved
through aggregation and interpolation on non-synchronized series. At query
time, a user defines the query range $r=(s,e)$, where $s$ and $e$ are the
starting and ending points, respectively, the query window size $l$, the
sliding step size $\eta$, and a threshold $\beta$. The problem requires
computing a series of correlation matrices,
$\mathcal{C}={C^{0},C^{1},\cdots,C^{\gamma}}$, where each $C^{k}$ is the
correlation matrix of $\mathcal{X}{k*\eta:k*\eta+l}$ (the submatrix of
$\mathcal{X}$ from the $k*\eta$th column to the $k*\eta+l$th column). In
$C^{k}$, each element $c{ij}^{k}(\geq\beta)$ denotes the Pearson correlation
between $X_{i}$ and $X_{j}$ within the range of the $k$th sliding window. If
$c_{ij}^{k}<\beta$, we replace it with $0$.

The core task involves addressing the problem of large-scale, all-pair time-
series correlation calculation across sliding windows. The key challenges
include: 1) ensuring the efficiency of network construction and updates for
large-scale data to achieve interactivity, 2) maintaining the robustness of
the methods on datasets with varying distributions, and 3) optimizing the
effectiveness of sketching techniques for subsequent network analysis.

##  2\. Background and Related Work

Correlation calculation across sliding windows has been explored through
frequency-based transform (Zhong et al., 2020; Sakurai et al., 2005; Zhu and
Shasha, 2002; Mueen et al., 2010) and random projection (Yagoubi et al., 2018;
Zhang et al., 2009) approaches. Both aim to reduce time series dimensionality
and identify highly-correlated pairs in low-dimensional spaces. Basic windows
(Zhu and Shasha, 2002) were first proposed for time-series pairs with high
correlation, later improved in parallel (Mueen et al., 2010). Parcorr (Yagoubi
et al., 2018) optimized incremental computation, achieving state-of-the-art
efficiency. TSUBASA (Xu et al., 2022), the latest work, computes exact
pairwise correlations on arbitrary time windows using a novel sketch
framework, but lacks efficiency for sliding queries.

Existing techniques, except TSUBASA, face robustness issues due to data-
dependency. Frequency-based transform methods struggle with dimensionality
reduction for certain time series, only succeeding when energy concentrates in
a few domains (Zhang et al., 2009). To date, no benchmark has been proposed to
systematically test solution robustness.

##  3\. Approach and Uniqueness

In this work, we introduce Dangoron, a framework for efficiently computing
dynamic correlation matrices with high accuracy, and Tomborg, a benchmark for
generating time series datasets to test framework robustness.

The core concept of Dangoron is the relatively stable correlation when
transitioning to the next sliding window. By computing future upper and lower
bounds of correlation, we can optimize calculations. For instance, if the
current correlation is below the threshold and the next upper bound is also
below the threshold, we can skip the next time step computation. Estimating
bounds for the next $k$th window allows us to determine the number of
skippable steps and directly jump to the desired window.

Figure 2. Jumping Structure of Dangoron

We employ the basic window framework by dividing a series into smaller basic
windows, enabling data processing in smaller batches. Referring to the Problem
Definition, let’s consider a $C_{k}$ from $\mathcal{C}$ and denote it as $C$.
We select the $i$th and $j$th rows from $\mathcal{X}$ within the time range
specified by $C$, and label them as
$x=[{\textbf{x}}_{1},\ldots,{\textbf{x}}_{m}]$ and
$y=[{\textbf{y}}_{1},\ldots,{\textbf{y}}_{m}]$. We define the sizes of basic
windows as $\textbf{B}=[B_{1},B_{2},\ldots,B_{m}]$, where $B_{i}$ represents
the size of the $i$-th basic window. The exact Pearson’s correlation between
$x$ and $y$ is given by:

(1) $\displaystyle\vspace{-0.8cm}Corr$
$\displaystyle(x,y)=\frac{\sum_{j=1}^{n_{s}}B_{j}(\sigma_{x_{j}}\sigma_{y_{j}}c_{j}+\delta_{x_{j}}\delta_{y_{j}})}{\sqrt{\sum_{i=1}^{n_{s}}B_{i}({\sigma_{x_{i}}}^{2}+{\delta_{x_{i}}^{2}})}\sqrt{\sum_{i=1}^{n_{s}}B_{i}({\sigma_{y_{i}}}^{2}+{\delta_{y_{i}}^{2}})}}$
$\displaystyle\delta_{x_{i}}=\overline{x_{i}}-\frac{\sum_{k=1}^{n_{s}}\overline{x_{k}}}{n_{s}},~{}~{}\delta_{y_{i}}=\overline{y_{i}}-\frac{\sum_{k=1}^{n_{s}}\overline{y_{k}}}{n_{s}}$

$\sigma_{x_{i}}$ ($\sigma_{y_{i}}$) represents the standard deviation of the
basic window for $x_{i}$ ($y_{i}$), $c_{i}$ is the correlation of basic
windows $x_{i}$ and $y_{i}$, $\overline{x_{i}}$ ($\overline{y_{i}}$) is the
mean of the basic window $x_{i}$ ($y_{i}$), and $n_{s}$ denotes the number of
basic windows in a query window. Using Equation 1, we can pre-compute and
store basic window statistics and calculate correlations for arbitrary query
windows and sizes.

Figure 2 illustrates how Dangoron operates. Each yellow block represents a
pair of basic windows. Initially, we compute the exact correlation using
Equation 1; if it is lower than the threshold $\beta$, it is indicated by a
blue block. We then determine $k$ from the upper bound estimation, such as
Equation 2, by solving for $k$ such that $Corr_{i+k-1}<\beta$ and
$Corr_{i+k}\geq\beta$ using binary search. In Figure 2, $k=3$, and we use a
red block to signify an upper bound higher than $\beta$. After jumping $k$
steps (skipping the green ones), we recompute the exact correlation value.

(2) $\displaystyle\vspace{-1cm}Corr_{1+k}$ $\displaystyle\leqslant
Corr_{1}+\frac{1}{n_{s}}(k-\sum_{i}^{k}c_{i})=Corr_{i+k}*$

We have proved Equation 2 under the assumption that each basic window comes
from a sample distribution, as detailed in our technical report. For more
complex cases, we have derived general bounds. Another feature proposed in
Dangoron enables horizontal computation pruning. Given time-series $x$, $y$,
and $z$, if we know the pairwise correlations $c_{xz}$ and $c_{yz}$, we can
determine the range of $c_{xz}$ as
$c_{xz}c_{yz}-\sqrt{(1-c_{xz}^{2})(1-c_{yz}^{2})}\leq c_{xy}\leq
c_{xz}c_{yz}+\sqrt{(1-c_{xz}^{2})(1-c_{yz}^{2})}$. The Tomborg framework
involves: (1) generating $\mathcal{C}$ from a user-specified distribution, (2)
generating $\mathcal{x}$ in frequency space, and (3) transforming them to
$\mathcal{X}$ using inverse Fourier Transform. Step (2) is based on the fact
that Discrete Fourier Transform (DFT) preserves the distance between
coefficients and the original time series. We developed a real-value variant
of the inverse-DFT, transitioning from a complex space to a real space (unlike
the original inverse-DFT, which moves from a complex to complex space), and
provided proof.

##  4\. Results and Contributions

We selected TSUBASA (Xu et al., 2022) as our baseline. On the NCEA Data
Set111https://www.ncei.noaa.gov/pub/data/uscrn/products/hourly02/2020/,
Dangoron is an order of magnitude faster than TSUBASA in terms of pure query
time and achieves an accuracy above 90 percent, comparable to Parcorr. We plan
to conduct more large-scale experiments upon completing the implementation of
Tomborg. In this work, we make the following contributions: 1) We introduce
Dangoron, a framework that solves the problem of correlation matrix
computation across sliding windows, and is at least one order of magnitude
faster than the baseline; 2) We propose Tomborg, the first benchmark for the
problem of correlation matrix computation.

## References

  * (1)
  * Betzel (2022) Richard F Betzel. 2022\.  Network neuroscience and the connectomics revolution.  In _Connectomic Deep Brain Stimulation_. Elsevier, 25–58.
  * Cabral et al. (2017) Joana Cabral, Morten L Kringelbach, and Gustavo Deco. 2017\.  Functional connectivity dynamically evolves on multiple time-scales over a static structural connectome: Models and mechanisms.  _NeuroImage_ 160 (2017), 84–96.
  * Gozolchiani et al. (2008) Avi Gozolchiani, Kazuko Yamasaki, Oz Gazit, and Shlomo Havlin. 2008.  Pattern of climate network blinking links follows El Niño events.  _EPL (Europhysics Letters)_ 83, 2 (2008), 28005\.
  * Hutchison et al. (2013) R Matthew Hutchison, Thilo Womelsdorf, Elena A Allen, Peter A Bandettini, Vince D Calhoun, Maurizio Corbetta, Stefania Della Penna, Jeff H Duyn, Gary H Glover, Javier Gonzalez-Castillo, et al. 2013\.  Dynamic functional connectivity: promise, issues, and interpretations.  _Neuroimage_ 80 (2013), 360–378.
  * Jayaweera and Aziz (2018) CD Jayaweera and N Aziz. 2018.  Reliability of principal component analysis and Pearson correlation coefficient, for application in artificial neural network model development, for water treatment plants. In _IOP Conference Series: Materials Science and Engineering_ , Vol. 458. IOP Publishing, 012076\.
  * Kenett et al. (2010) Dror Y Kenett, Yoash Shapira, Asaf Madi, Sharron Bransburg-Zabary, Gitit Gur-Gershgoren, and Eshel Ben-Jacob. 2010.  Dynamics of stock market correlations.  _AUCO Czech Economic Review_ 4, 3 (2010).
  * Kim et al. (2019) Kyunghun Kim, Hongjun Joo, Daegun Han, Soojun Kim, Taewoo Lee, and Hung Soo Kim. 2019\.  On complex network construction of rain gauge stations considering nonlinearity of observed daily rainfall data.  _Water_ 11, 8 (2019), 1578.
  * Krendl and Betzel (2022) Anne C Krendl and Richard F Betzel. 2022.  Social cognitive network neuroscience.  _Social Cognitive and Affective Neuroscience_ 17, 5 (2022), 510–529.
  * Mueen et al. (2010) Abdullah Mueen, Suman Nath, and Jie Liu. 2010.  Fast approximate correlation for massive time-series data. In _Proceedings of the 2010 ACM SIGMOD International Conference on Management of data_. 171–182.
  * Sakurai et al. (2005) Yasushi Sakurai, Spiros Papadimitriou, and Christos Faloutsos. 2005\.  Braid: Stream mining through group lag correlations. In _Proceedings of the 2005 ACM SIGMOD international conference on Management of data_. 599–610.
  * Thirion et al. (2014) Bertrand Thirion, Gaël Varoquaux, Elvis Dohmatob, and Jean-Baptiste Poline. 2014.  Which fMRI clustering gives good brain parcellations?  _Frontiers in neuroscience_ 8 (2014), 167.
  * Tilfani et al. (2021) Oussama Tilfani, Paulo Ferreira, El Boukfaoui, and My Youssef. 2021.  Dynamic cross-correlation and dynamic contagion of stock markets: A sliding windows approach with the DCCA correlation coefficient.  _Empirical Economics_ 60, 3 (2021), 1127–1156.
  * Xu et al. (2022) Yunlong Xu, Jinshu Liu, and Fatemeh Nargesian. 2022.  TSUBASA: Climate Network Construction on Historical and Real-Time Data. In _Proceedings of the 2022 International Conference on Management of Data_. 286–295.
  * Yagoubi et al. (2018) Djamel Edine Yagoubi, Reza Akbarinia, Boyan Kolev, Oleksandra Levchenko, Florent Masseglia, Patrick Valduriez, and Dennis Shasha. 2018\.  ParCorr: efficient parallel methods to identify similar time series pairs across sliding windows.  _Data Mining and Knowledge Discovery_ 32 (2018), 1481–1507.
  * Zhang et al. (2009) Tiancheng Zhang, Dejun Yue, Yu Gu, Yi Wang, and Ge Yu. 2009.  Adaptive correlation analysis in stream time series with sliding windows.  _Computers & Mathematics with Applications_ 57, 6 (2009), 937–948.
  * Zhong et al. (2020) Sheng Zhong, Vinicius MA Souza, and Abdullah Mueen. 2020\.  FilCorr: Filtered and lagged correlation on streaming time series. In _2020 IEEE International Conference on Data Mining (ICDM)_. IEEE, 1436–1441.
  * Zhu and Shasha (2002) Yunyue Zhu and Dennis Shasha. 2002.  Statstream: Statistical monitoring of thousands of data streams in real time. In _VLDB’02: Proceedings of the 28th International Conference on Very Large Databases_. Elsevier, 358–369.


Pseudo-Riemannian Lie groups
of modified $H$-type

Justin M. Ryan

> Abstract. We define a class of Riemannian and pseudo-Riemannian 2-step
> nilpotent Lie groups with nondegenerate centers (Definition 2) that
> generalize the $H$-type groups of Kaplan [8, 9, 10]. Examples are given and
> geometric properties are investigated.

##  1 Introduction

Let $N$ be a 2-step nilpotent Lie group with Lie algebra $\mathfrak{n}$. Let
$\left[\ ,\,\right]$ denote the Lie bracket on $\mathfrak{n}$, and write
$\mathfrak{n}=\mathfrak{z}\oplus\mathfrak{v}$, where $\mathfrak{z}$ is the
center of $\mathfrak{n}$. Let $\left\langle\ ,\,\right\rangle$ denote both a
pseudo-Riemannian inner product on $\mathfrak{n}$ making $\mathfrak{z}$
nondegenerate, and the induced left-invariant metric tensor on $N$.

The adjoint representation of $\mathfrak{n}$ on itself,
$\mathop{\operator@font
ad}\nolimits:\mathfrak{n}\times\mathfrak{n}\to\mathfrak{n}$, is given by
$\mathop{\operator@font ad}\nolimits_{x}=\left[x,\ \right]$. For every
$z\in\mathfrak{z}$ define a skew-symmetric linear transformation $j(z)$ on
$\mathfrak{v}$ by

$j(z)=\mathop{\operator@font ad}^{*}\nolimits_{\bullet}z,$

where $\mathop{\operator@font ad}^{*}\nolimits$ is the adjoint of
$\mathop{\operator@font ad}\nolimits$ with respect to $\left\langle\
,\,\right\rangle$. That is,

$\left\langle\mathop{\operator@font
ad}\nolimits_{x}y,z\right\rangle=\left\langle y,j(z)x\right\rangle$

for all $x,y\in\mathfrak{v}$ and all $z\in\mathfrak{z}$.

These $j$-maps simultaneously encode the algebraic and geometric information
about the (pseudo-) Riemannian 2-step nilpotent Lie group $(N,\left\langle\
,\,\right\rangle)$. There is a special family of groups, originally studied by
Kaplan [9], for which these maps act like scaled anti-involutions.

  * Definition 1.definition

A 2-step nilpotent Lie group $N$ with left-invariant Riemannian metric
$\left\langle\ ,\,\right\rangle$ is said to be of _$H$ -type_ if and only if
for each $z\in\mathfrak{z}$,

$\left\langle j(z)x,j(z)y\right\rangle=\left\|z\right\|^{2}\left\langle
x,y\right\rangle$

for all $x,y\in\mathfrak{v}$. Alternatively, for each $z\in\mathfrak{z}$,
$j(z)^{2}=-\left\|z\right\|^{2}\mathrm{Id}|_{\mathfrak{v}}$.  $\Diamond$

  * Definition 1.definition

A pseudo-Riemannian 2-step nilpotent Lie group $N$ with nondegenerate center
is said to be of _pseudo- $H$-type_ if and only if for each
$z\in\mathfrak{z}$,

$\left\langle j(z)x,j(z)y\right\rangle=\left\|z\right\|^{2}\left\langle
x,y\right\rangle$

for all $x,y\in\mathfrak{v}$. Alternatively, for each $z\in\mathfrak{z}$,
$j(z)^{2}=-\left\|z\right\|^{2}\mathrm{Id}|_{\mathfrak{v}}$.  $\Diamond$

##  2 Definition and Examples

Definitions 1 and 1 can be generalized by replacing the quadratic form
determined by $\left\langle\ ,\,\right\rangle_{\mathfrak{z}}$ with another
quadratic form on $\mathfrak{z}$. J. Lauret [11] had this idea in the
Riemannian case, and gave the following definition.

  * Definition 2.definition

A Riemannian 2-step nilpotent Lie group is said to be a _modified $H$-type
group_ if for any nonzero $a\in\mathfrak{z}$,
$j(a)^{2}=\lambda(a)\mathrm{Id}|_{\mathfrak{v}}$ for some $\lambda(a)<0$.
$\Diamond$

_A priori_ , the $\lambda$ in this definition is just a function on
$\mathfrak{z}$. However, the properties of the $j$-maps actually imply that
$\lambda$ is a nondegenerate, negative-definite quadratic form on
$\mathfrak{z}$.

We generalize this definition by not only considering pseudo-Riemannian
metrics on $N$, but also allowing quadratic forms on $\mathfrak{z}$ of the
most general type. That is, we allow that the associated symmetric bilinear
form be indefinite and/or degenerate.

  * Definition 2.definition

Let $(N,\left\langle\ ,\,\right\rangle)$ be a 2-step nilpotent Lie group with
left-invariant metric $\left\langle\ ,\,\right\rangle=\left\langle\
,\,\right\rangle_{\mathfrak{z}}+\left\langle\ ,\,\right\rangle_{\mathfrak{v}}$
making $\mathfrak{z}$ nondegenerate, and let $\varphi$ be a quadratic form on
$\mathfrak{z}$. We shall say that $(N,\left\langle\ ,\,\right\rangle,\varphi)$
is of _modified $H$-type_ if and only if for each $z\in\mathfrak{z}$, the map
$j(z)\in\mathrm{End}(\mathfrak{v})$ satisfies

$\left\langle j(z)x,j(z)y\right\rangle_{\mathfrak{v}}=\varphi(z)\left\langle
x,y\right\rangle_{\mathfrak{v}}$ (1)

for all $x,y\in\mathfrak{v}$. Alternatively, since the $j$-maps are skew-
adjoint this property is characterized by

$j(z)^{2}=-\varphi(z)\,\mathrm{Id}|_{\mathfrak{v}}$ (2)

for all $z\in\mathfrak{z}$.  $\Diamond$

  * Remark 2.remark

Even when the metric is Riemannian, Definition 2 is more general (although not
necessarily more interesting) than Definition 2. This is illustrated below by
Example 2.  $\Diamond$

  * Example 2.example

If $\left\langle\ ,\,\right\rangle$ is a left-invariant Riemannian metric on
$N$ and $\varphi=\left\|\cdot\right\|_{\mathfrak{z}}^{2}$, then
$(N,\left\langle\ ,\,\right\rangle)$ is an $H$-type group in the sense of
Kaplan [9]. If $\left\langle\ ,\,\right\rangle$ is a left-invariant pseudo-
Riemannian metric on $N$ and
$\varphi=\left\|\cdot\right\|_{\mathfrak{z}}^{2}$, then $(N,\left\langle\
,\,\right\rangle)$ is a pseudo-$H$-type group in the sense of Ciatti [1].
$\Diamond$

  * Example 2.example

Consider the Heisenberg algebra $\mathfrak{h}_{3}$ spanned by the vectors
$\\{e_{1},e_{2},e_{3}\\}$ with nontrivial bracket
$\left[e_{1},e_{2}\right]=e_{3}$. Clearly the center is
$\mathfrak{z}=\text{span}\left\\{e_{3}\right\\}$. Consider the Lorentzian
inner product on $\mathfrak{h}_{3}$ given on this basis by

$-\left\langle e_{1},e_{1}\right\rangle=\left\langle
e_{2},e_{2}\right\rangle=\left\langle e_{3},e_{3}\right\rangle=1.$

Now let $(H_{3},\left\langle\ ,\,\right\rangle)$ be the 3-dimensional
Heisenberg group with the left-invariant Lorentz metric induced by this inner
product on $\mathfrak{h}_{3}$. The $j$-maps on this group satisfy

$j(z)^{2}=\left\|z\right\|_{\mathfrak{z}}^{2}\mathrm{Id}|_{\mathfrak{v}},$

so this group is not of pseudo-$H$-type. However, taking
$\varphi=-\left\|\cdot\right\|_{\mathfrak{z}}^{2}$ one sees that it is of
modified $H$-type. Therefore the class of modified $H$-type metrics is
strictly larger than the class of pseudo-$H$-type metrics.  $\Diamond$

  * Example 2.example

In the Riemannian case, the _generalized Heisenberg groups_ satisfy

$j(z)^{2}=-4c\left\|z\right\|_{\mathfrak{z}}^{2}\mathrm{Id}|_{\mathfrak{v}}$

for all $z\in\mathfrak{z}$, $c\in(0,\infty)$. See [14, 6], for example. These
groups are of modified $H$-type with
$\varphi=4c\left\|\cdot\right\|_{\mathfrak{z}}^{2}$, but they are only of
$H$-type when $c=\tfrac{1}{4}$. Thus the class of modified $H$-type metrics is
strictly larger than the class of $H$-type metrics.

This example may be extended straightforwardly to pseudo-Riemannian groups to
obtain the pseudo-Riemannian generalized Heisenberg groups.  $\Diamond$

Each of the previous examples constructs modified $H$-type metrics on groups
that are already known to admit $H$-type or pseudo-$H$-type metrics. The next
examples show that the class of groups admitting modified $H$-type metrics is
strictly larger than those admitting (pseudo-) $H$-type metrics.

Consider the 4-dimensional group obtained by a trivial central extension of
the Heisenberg group, $N=H_{3}(\mathbb{R})\times\mathbb{R}$. The Lie algebra
$\mathfrak{n}=\mathfrak{h}_{3}\oplus\mathbb{R}$ has basis
$\\{e_{0},e_{1},e_{2},e_{3}\\}$ with nontrivial bracket
$\left[e_{1},e_{2}\right]=e_{3}$. Thus
$\mathfrak{z}=\text{span}\left\\{e_{0},e_{3}\right\\}$ and
$\mathfrak{v}=\text{span}\left\\{e_{1},e_{2}\right\\}$. The following examples
show that there are modified $H$-type metrics on $N$ of each possible
signature.

Moreover, this group does not admit an $H$-type or pseudo-$H$-type metric.
Indeed, it does not admit an $H$-type metric because it is singular, and it
does not admit a pseudo-$H$-type metric because the minimum dimension of a
pseudo-$H$-type group with 2-dimensional center is six; _viz._ [1].

  * Example 2.example

Consider the Riemannian metric $g_{0}$ on
$N=H_{3}(\mathbb{R})\times\mathbb{R}$ making the basis
$\\{e_{0},e_{1},e_{2},e_{3}\\}$ orthonormal. That is,

$\left\langle e_{i},e_{j}\right\rangle=\begin{cases}1&i=j\\\ 0&i\neq
j\end{cases}.$

The $j$-maps on this group are given by

$j(e_{0})=\begin{pmatrix}0&0\\\ 0&0\end{pmatrix},\ \ \text{and}\ \
j(e_{3})=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},$

so that their squares satisfy $j(e_{0})^{2}=0$ and
$j(e_{3})^{2}=-\mathrm{Id}|_{\mathfrak{v}}$. The maps
$j(e_{0}+e_{3})=j(e_{3})$ and $j(e_{0}-e_{3})=-j(e_{3})$, so that the
$j(e_{0}+e_{3})^{2}=j(e_{0}-e_{3})^{2}=-\mathrm{Id}|_{\mathfrak{v}}$. Let
$\varphi_{0}$ be the quadratic form on $\mathfrak{z}$ satisfying

$\varphi_{0}(e_{0})=0,\ \
\varphi_{0}(e_{3})=\varphi_{0}(e_{0}+e_{3})=\varphi_{0}(e_{0}-e_{3})=1.$

The induced symmetric bilinear form on $\mathfrak{z}$ satisfies

$\left(e_{0},e_{3}\right)_{\varphi_{0}}=\tfrac{1}{4}\left(\varphi_{0}(e_{0}+e_{3})-\varphi_{0}(e_{0}-e_{3})\right)=\tfrac{1}{4}\left(1-1\right)=0,$

so that its matrix representation is

$\Phi_{0}=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}.$

This bilinear form is exactly what one should expect since $N$ is a trivial
extension of $H_{3}(\mathbb{R})$. We see that
$j(z)^{2}=-\varphi_{0}(z)\,\mathrm{Id}|_{\mathfrak{v}}$ for every
$z\in\mathfrak{z}$. Therefore $(N,g_{0},\varphi_{0})$ is a modified $H$-type
group with degenerate quadratic form $\varphi_{0}$.  $\Diamond$

  * Example 2.example

Now consider the left-invariant Lorentzian metric $g_{1}$ on $N$ given by

$\left\langle e_{0},e_{3}\right\rangle=\left\langle
e_{1},e_{1}\right\rangle=\left\langle e_{2},e_{2}\right\rangle=1,$

with all other combinations zero. The $j$-maps for this group are

$j(e_{0})=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},\text{\ \ and\ \
}j(e_{3})=\begin{pmatrix}0&0\\\ 0&0\end{pmatrix},$

so that $j(e_{0})^{2}=-\mathrm{Id}|_{\mathfrak{v}}$ and $j(e_{3})^{2}=0$.
Taking a pseudo-orthonormal basis of $\mathfrak{z}$,

$u_{1}=\tfrac{1}{\sqrt{2}}(e_{0}+e_{3}),\ \
u_{2}=\tfrac{1}{\sqrt{2}}(e_{0}-e_{3}),$

we obtain $j(u_{1})=j(u_{2})=\tfrac{1}{\sqrt{2}}j(e_{0})$, so that
$j(u_{1})^{2}=j(u_{2})^{2}=-\tfrac{1}{2}\,\mathrm{Id}|_{\mathfrak{v}}$.
Moreover, $u_{1}+u_{2}=\sqrt{2}e_{0}$ and $u_{1}-u_{2}=-\sqrt{2}e_{3}$, so
that $j(u_{1}+u_{2})^{2}=-2\,\mathrm{Id}|_{\mathfrak{v}}$ and
$j(u_{1}-u_{2})^{2}=0$. Let $\varphi_{1}$ be the quadratic form on
$\mathfrak{z}$ satisfying

$\varphi_{1}(u_{1})=\varphi_{1}(u_{2})=\tfrac{1}{2},\ \
\varphi_{1}(u_{1}+u_{2})=2,\ \ \text{and}\ \ \varphi(u_{1}-u_{2})=0.$

The induced symmetric bilinear form on $\mathfrak{z}$ satisfies

$\left(u_{1},u_{2}\right)_{\varphi_{1}}=\tfrac{1}{4}\left(\varphi_{1}(u_{1}+u_{2})-\varphi_{1}(u_{1}-u_{2})\right)=\tfrac{1}{4}(2-0)=\tfrac{1}{2}.$

Therefore its matrix representation is

$\Phi_{1}=\begin{pmatrix}\tfrac{1}{2}&\tfrac{1}{2}\\\\[1.07639pt]
\tfrac{1}{2}&\tfrac{1}{2}\end{pmatrix},$

and $\varphi_{1}$ is degenerate. However,
$j(z)^{2}=-\varphi_{1}(z)\,\mathrm{Id}|_{\mathfrak{v}}$ for all
$z\in\mathfrak{z}$, so $(N,g_{1},\varphi_{1})$ is indeed a modified $H$-type
group.  $\Diamond$

  * Example 2.example

Next consider the left-invariant neutral metric $g_{2}$ on $N$ given on the
Lie algebra $\mathfrak{n}$ by

$\left\langle e_{0},e_{3}\right\rangle=\left\langle
e_{1},e_{2}\right\rangle=1,$

with all other combinations zero. Before computing the $j$-maps, change the
basis of $\mathfrak{n}$ to

$\displaystyle u_{1}$ $\displaystyle=$
$\displaystyle\tfrac{1}{\sqrt{2}}\left(e_{0}+e_{3}\right),$ $\displaystyle
u_{2}$ $\displaystyle=$
$\displaystyle\tfrac{1}{\sqrt{2}}\left(e_{0}-e_{3}\right),$ $\displaystyle
v_{1}$ $\displaystyle=$
$\displaystyle\tfrac{1}{\sqrt{2}}\left(e_{1}+e_{2}\right),$ $\displaystyle
v_{2}$ $\displaystyle=$
$\displaystyle\tfrac{1}{\sqrt{2}}\left(e_{1}-e_{2}\right).$

The basis $\\{u_{1},u_{2},v_{1},v_{2}\\}$ is pseudo-orthonormal with

$\displaystyle\left\langle u_{1},u_{1}\right\rangle$ $\displaystyle=$
$\displaystyle-\left\langle u_{2},u_{2}\right\rangle=1,\text{\ and}$
$\displaystyle\left\langle v_{1},v_{1}\right\rangle$ $\displaystyle=$
$\displaystyle-\left\langle v_{2},v_{2}\right\rangle=1.$

The Lie bracket is now given by
$\left[v_{1},v_{2}\right]=\tfrac{1}{\sqrt{2}}\left(u_{1}-u_{2}\right)$,
$\mathfrak{z}=\text{span}\left\\{u_{1},u_{2}\right\\}$, and
$\mathfrak{v}=\text{span}\left\\{v_{1},v_{2}\right\\}$. The $j$-maps are given
with respect to these bases by

$j(u_{1})=j(u_{2})=\frac{1}{\sqrt{2}}\begin{pmatrix}0&-1\\\
-1&0\end{pmatrix},$

so that

$j(u_{1}+u_{2})=\sqrt{2}\begin{pmatrix}0&-1\\\ -1&0\end{pmatrix}.$

We see that
$j(u_{1})^{2}=j(u_{2})^{2}=\tfrac{1}{2}\,\mathrm{Id}|_{\mathfrak{v}}$, and
$j(u_{1}+u_{2})^{2}=2\,\mathrm{Id}|_{\mathfrak{v}}$. Let $\varphi_{2}$ be the
quadratic form on $\mathfrak{z}$ satisfying

$\varphi_{2}(u_{1})=\varphi_{2}(u_{2})=-\tfrac{1}{2},\ \ \text{and}\ \
\varphi_{2}(u_{1}+u_{2})=-2.$

Then the associated bilinear form $\left(\ ,\,\right)_{\varphi_{2}}$ satisfies

$\left(u_{1},u_{2}\right)_{\varphi_{2}}=\tfrac{1}{2}\left(\varphi_{2}(u_{1}+u_{2})-\varphi_{2}(u_{1})-\varphi_{2}(u_{2})\right)=\tfrac{1}{2}\left(-2+\tfrac{1}{2}+\tfrac{1}{2}\right)=-\tfrac{1}{2},$

and its matrix representation is

$\Phi_{2}=\begin{pmatrix}-\tfrac{1}{2}&-\tfrac{1}{2}\\\\[1.07639pt]
-\tfrac{1}{2}&-\tfrac{1}{2}\end{pmatrix}.$

We see that $j(z)^{2}=-\varphi_{2}(z)\,\mathrm{Id}|_{\mathfrak{v}}$ for all
$z\in\mathfrak{z}$, so that $(N,g_{2},\varphi_{2})$ is a modified $H$-type
group.  $\Diamond$

  * Example 2.example

Consider the left-invariant metric $g_{3}$ given on $\mathfrak{n}$ by

$\left\langle e_{0},e_{0}\right\rangle=\left\langle
e_{3},e_{3}\right\rangle=\left\langle e_{1},e_{2}\right\rangle=1.$

The $j$-maps with respect to the basis $\\{e_{1},e_{2}\\}$ are

$j(e_{0})=\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}\ \ \text{and}\ \
j(e_{3})=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$

so that $j(e_{0})^{2}=0$ and $j(e_{3})^{2}=\mathrm{Id}|_{\mathfrak{v}}$.
Notice that $j(e_{0}+e_{3})=j(e_{3})$, so that
$j(e_{0}+e_{3})^{2}=\mathrm{Id}|_{\mathfrak{v}}$ as well. Let $\varphi_{3}$ be
the quadratic form on $\mathfrak{z}$ satisfying

$\varphi_{3}(e_{0})=0,\ \ \varphi_{3}(e_{3})=\varphi_{3}(e_{0}+e_{3})=-1.$

The bilinear form $\left(\ ,\,\right)_{\varphi_{3}}$ satisfies

$\left(e_{0},e_{3}\right)_{\varphi_{3}}=\tfrac{1}{2}\left(\varphi_{3}(e_{0}+e_{3})-\varphi_{3}(e_{0})-\varphi_{3}(e_{3})\right)=\tfrac{1}{2}(-1-0+1)=0,$

so its matrix representation is

$\Phi_{3}=\begin{pmatrix}0&0\\\ 0&-1\end{pmatrix}.$

Therefore $(N,g_{3},\varphi_{3})$ is a modified $H$-type group.  $\Diamond$

##  3 Connection and Curvatures

In this section we give all relevant curvature formulas for modified $H$-type
groups. Let $(N,\left\langle\ ,\,\right\rangle,\varphi)$ be a modified
$H$-type Lie group. Let $\left(\ ,\,\right)_{\varphi}$ be the symmetric
bilinear form on $\mathfrak{z}$ obtained by polarizing the quadratic form
$\varphi$. Then for all $e,e^{\prime}\in\mathfrak{v}$ and all
$z,z^{\prime}\in\mathfrak{z}$,

$\displaystyle\left\langle j(z)e,j(z)e^{\prime}\right\rangle_{\mathfrak{v}}$
$\displaystyle=$ $\displaystyle\varphi(z)\left\langle
e,e^{\prime}\right\rangle_{\mathfrak{v}},$ (3) $\displaystyle\left\langle
j(z)e,j(z^{\prime})e\right\rangle_{\mathfrak{v}}$ $\displaystyle=$
$\displaystyle\left(z,z^{\prime}\right)_{\varphi}\left\langle
e,e\right\rangle_{\mathfrak{v}},\text{\ and}$ (4) $\displaystyle
j(z)j(z^{\prime})+j(z^{\prime})j(z)$ $\displaystyle=$
$\displaystyle-2\left(z,z^{\prime}\right)_{\varphi}\mathrm{Id}|_{\mathfrak{v}}.$
(5)

These identities as well as the following formulas for the Levi-Civita
connection and curvatures may be deduced from [4]. We prove only the curvature
formulas below that are unique to modified $H$-type groups.

###### Theorem 3.1

Let $(N,\left\langle\ ,\,\right\rangle)$ be a 2-step nilpotent Lie group with
left-invariant pseudo-Riemannian metric making the center nondegenerate. For
all $e,e^{\prime}\in\mathfrak{v}$ and $z,z^{\prime}\in\mathfrak{z}$, the Levi-
Civita connection on $N$ is given by

$\displaystyle\mathop{\nabla}\nolimits_{\mspace{-2.5mu}z}z^{\prime}$
$\displaystyle=$ $\displaystyle 0,$ (6)
$\displaystyle\mathop{\nabla}\nolimits_{\mspace{-2.5mu}z}e=\mathop{\nabla}\nolimits_{\mspace{-2.5mu}e}z$
$\displaystyle=$ $\displaystyle-\tfrac{1}{2}j(z)e,$ (7)
$\displaystyle\mathop{\nabla}\nolimits_{\mspace{-2.5mu}e}e^{\prime}$
$\displaystyle=$ $\displaystyle\tfrac{1}{2}\left[e,e^{\prime}\right].$ (8)

###### Theorem 3.2

Let $(N,\left\langle\ ,\,\right\rangle)$ be a 2-step nilpotent Lie group with
left-invariant pseudo-Riemannian metric making the center nondegenerate. For
all $e,e^{\prime},e^{\prime\prime}\in\mathfrak{v}$ and
$z,z^{\prime},z^{\prime\prime}\in\mathfrak{z}$, the Riemann curvature tensor
on $N$ is given by

$\displaystyle R(z,z^{\prime})z^{\prime\prime}$ $\displaystyle=$
$\displaystyle 0,$ (9) $\displaystyle R(z,z^{\prime})e$ $\displaystyle=$
$\displaystyle\tfrac{1}{4}\left[z,z^{\prime}\right]e,$ (10) $\displaystyle
R(z,e)z^{\prime}$ $\displaystyle=$
$\displaystyle\tfrac{1}{4}j(z)j(z^{\prime})e,$ (11) $\displaystyle
R(z,e)e^{\prime}$ $\displaystyle=$
$\displaystyle\tfrac{1}{4}\left[e,j(z)e^{\prime}\right],$ (12) $\displaystyle
R(e,e^{\prime})z$ $\displaystyle=$
$\displaystyle-\tfrac{1}{4}\left(\left[e,j(z)e^{\prime}\right]+\left[j(z)e,e^{\prime}\right]\right),$
(13) $\displaystyle R(e,e^{\prime})e^{\prime\prime}$ $\displaystyle=$
$\displaystyle\tfrac{1}{4}\left(j\left(\left[e,e^{\prime\prime}\right]\right)e^{\prime}-j\left(\left[e^{\prime},e^{\prime\prime}\right]\right)e\right)+\tfrac{1}{2}j\left(\left[e,e^{\prime}\right]\right)e^{\prime\prime}.$
(14)

###### Theorem 3.3

Let $(N,\left\langle\ ,\,\right\rangle,\varphi)$ be a modified $H$-type group,
and suppose that $e,e^{\prime}\in\mathfrak{v}$, $z,z^{\prime}\in\mathfrak{z}$
are pseudo-orthonormal. The sectional curvature is given by

$\displaystyle K(z,z^{\prime})$ $\displaystyle=$ $\displaystyle 0,$ (15)
$\displaystyle K(z,e)$ $\displaystyle=$
$\displaystyle\tfrac{1}{4}\varepsilon_{z}\,\varphi(z),$ (16) $\displaystyle
K(e,e^{\prime})$ $\displaystyle=$
$\displaystyle-\tfrac{3}{4}\varepsilon_{e}\,\varepsilon_{e^{\prime}}\,\left\|\left[e,e^{\prime}\right]\right\|_{\mathfrak{z}}^{2},$
(17)

where $\varepsilon_{\bullet}=\left\|\bullet\right\|^{2}=\pm 1$.

  * Proof: Only equation (16) differs from the usual ones in [4]. Let $z\in\mathfrak{z}$, $e\in\mathfrak{v}$, such that $\varepsilon_{z}=\left\|z\right\|_{\mathfrak{z}}^{2}=\pm 1$ and $\varepsilon_{e}=\left\|e\right\|_{\mathfrak{v}}^{2}=\pm 1$. Then by (3),

$\displaystyle K(z,e)$ $\displaystyle=$
$\displaystyle\tfrac{1}{4}\varepsilon_{z}\,\varepsilon_{e}\,\left\langle
j(z)e,j(z)e\right\rangle_{\mathfrak{v}}$ $\displaystyle=$
$\displaystyle\tfrac{1}{4}\varepsilon_{z}\,\varepsilon_{e}\,\varphi(z)\,\left\|e\right\|_{\mathfrak{v}}^{2}$
$\displaystyle=$ $\displaystyle\tfrac{1}{4}\varepsilon_{z}\,\varphi(z).$

Let $\\{z_{1},\ldots,z_{p}\\}$ be a pseudo-orthonormal basis for the center
$(\mathfrak{z},\left\langle\ ,\,\right\rangle_{\mathfrak{z}})$, and let
$\\{e_{1},\ldots,e_{m}\\}$ be a pseudo-orthonormal basis for
$(\mathfrak{v},\left\langle\ ,\,\right\rangle_{\mathfrak{v}})$. Define

$\xi:=\sum_{k=1}^{p}\left\|z_{k}\right\|_{\mathfrak{z}}^{2}\,\varphi(z_{k}).$
(18)

This number does not depend on the choice of basis of $\mathfrak{z}$ as it is
the trace of the operator
$\mathcal{K}:z\mapsto\left\|z\right\|_{\mathfrak{z}}^{2}\varphi(z)$.

###### Theorem 3.4

Let $(N,\left\langle\ ,\,\right\rangle,\varphi)$ be a modified $H$-type group,
and let $e,e^{\prime}\in\mathfrak{v}$, $z,z^{\prime}\in\mathfrak{z}$. The
Ricci curvature is given by

$\displaystyle\mathrm{Ric}(e,z)$ $\displaystyle=$ $\displaystyle 0,$ (19)
$\displaystyle\mathrm{Ric}(e,e^{\prime})$ $\displaystyle=$
$\displaystyle-\tfrac{\xi}{2}\left\langle
e,e^{\prime}\right\rangle_{\mathfrak{v}},$ (20)
$\displaystyle\mathrm{Ric}(z,z^{\prime})$ $\displaystyle=$
$\displaystyle\tfrac{m}{4}\left(z,z^{\prime}\right)_{\varphi}.$ (21)

  * Proof: Equation (19) is true for all pseudo-Riemannian 2-step nilpotent groups with nondegenerate centers. Using the pseudo-orthonormal bases of (18) and letting $z,z^{\prime}\in\mathfrak{z}$, $e,e^{\prime}\in\mathfrak{v}$, we compute

$\displaystyle\mathrm{Ric}(e,e^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{k=1}^{p}\left\|z_{k}\right\|_{\mathfrak{z}}^{2}\left\langle
j(z_{k})e,j(z_{k})e^{\prime}\right\rangle_{\mathfrak{v}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{k=1}^{p}\left\|z_{k}\right\|_{\mathfrak{z}}^{2}\varphi(z_{k})\left\langle
e,e^{\prime}\right\rangle_{\mathfrak{v}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left\langle
e,e^{\prime}\right\rangle_{\mathfrak{v}}\sum_{k=1}^{p}\left\|z_{k}\right\|_{\mathfrak{z}}^{2}\varphi(z_{k})$
$\displaystyle=$ $\displaystyle-\frac{\xi}{2}\left\langle
e,e^{\prime}\right\rangle_{\mathfrak{v}},$

and

$\displaystyle\mathrm{Ric}(z,z^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{i=1}^{m}\left\|e_{i}\right\|_{\mathfrak{v}}^{2}\left\langle
j(z)e_{i},j(z^{\prime})e_{i}\right\rangle_{\mathfrak{v}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{i=1}^{m}\left\|e_{i}\right\|_{\mathfrak{v}}^{2}\left(z,z^{\prime}\right)_{\varphi}\left\langle
e_{i},e_{i}\right\rangle_{\mathfrak{v}}$ $\displaystyle=$
$\displaystyle\frac{m}{4}\left(z,z^{\prime}\right)_{\varphi}.$

The Ricci operator $\mathrm{Rc}:\mathfrak{n}\to\mathfrak{n}$ is defined by

$\left\langle\mathrm{Rc}\,x,y\right\rangle=\mathrm{Ric}(x,y)$

for all $x,y\in\mathfrak{n}$. By (19), the Ricci operator preserves the
splitting $\mathfrak{n}=\mathfrak{z}\oplus\mathfrak{v}$. By (21), the Ricci
operator restricted to $\mathfrak{z}$ is determined by

$\left\langle\mathrm{Rc}\,z,z^{\prime}\right\rangle_{\mathfrak{z}}=\tfrac{m}{4}\left(z,z^{\prime}\right)_{\varphi}.$
(22)

Denote by $\mathrm{Id}|^{\dagger}_{\varphi}$, the adjoint of the identity map
on $\mathfrak{z}$ with respect to the metrics $\left(\ ,\,\right)_{\varphi}$
and $\left\langle\ ,\,\right\rangle_{\mathfrak{z}}$. This adjoint is defined
by the equation

$\left\langle\mathrm{Id}|^{\dagger}_{\varphi}\,z,z^{\prime}\right\rangle_{\mathfrak{z}}=\left(z,\mathrm{Id}\,z^{\prime}\right)_{\varphi}.$
(23)

Comparing (22) and (23), $\mathrm{Rc}|_{\mathfrak{z}}$ is proportional to
$\mathrm{Id}|_{\varphi}^{\dagger}$. We deduce the following theorem regarding
the Ricci operator of modified $H$-type groups.

###### Theorem 3.5

Let $(N,\left\langle\ ,\,\right\rangle,\varphi)$ be a modified $H$-type group.
The Ricci operator preserves the splitting
$\mathfrak{n}=\mathfrak{z}\oplus\mathfrak{v}$, and is given on each factor by

$\displaystyle\mathrm{Rc}|_{\mathfrak{v}}$ $\displaystyle=$
$\displaystyle-\tfrac{\xi}{2}\,\mathrm{Id}|_{\mathfrak{v}},$ (24)
$\displaystyle\mathrm{Rc}|_{\mathfrak{z}}$ $\displaystyle=$
$\displaystyle\tfrac{m}{4}\,\mathrm{Id}|^{\dagger}_{\varphi}.$ (25)

  * Remark 3.remark

In order to better understand the operator $\mathrm{Id}|_{\varphi}^{\dagger}$
it is useful to work with matrix representations. Let $E$ be the matrix
representing $\left\langle\ ,\,\right\rangle_{\mathfrak{z}}$ with respect to
the basis $\\{z_{1},\ldots,z_{p}\\}$, so that
$E=\text{diag}\\{\varepsilon_{1},\ldots,\varepsilon_{p}\\}$. Let $\Phi$ be the
matrix representation of $\left(\ ,\,\right)_{\varphi}$ with respect to the
same basis. In general $\Phi$ is symmetric, but may be singular.

Equation (22) may be written in matrix form as
$(\mathrm{Rc}|_{\mathfrak{z}}\,z)^{T}Ez^{\prime}=\tfrac{m}{4}z^{T}\Phi
z^{\prime}$. Taking advantage of the facts that $E^{T}=E^{-1}=E$ and
$\Phi^{T}=\Phi$, basic matrix manipulations yield

$\displaystyle(\mathrm{Rc}|_{\mathfrak{z}}\,z)^{T}E$ $\displaystyle=$
$\displaystyle\tfrac{m}{4}z^{T}\Phi$
$\displaystyle(\mathrm{Rc}|_{\mathfrak{z}}\,z)^{T}$ $\displaystyle=$
$\displaystyle\tfrac{m}{4}z^{T}\Phi E^{-1}$
$\displaystyle(\mathrm{Rc}|_{\mathfrak{z}}\,z)^{T}$ $\displaystyle=$
$\displaystyle\big{[}\tfrac{m}{4}(\Phi E^{-1})^{T}z\big{]}^{T}$

This implies that
$\mathrm{Rc}|_{\mathfrak{z}}=\tfrac{m}{4}(E^{-1})^{T}\Phi^{T}=\tfrac{m}{4}E^{-1}\Phi$.
That is, the matrix representation of $\mathrm{Rc}|_{\mathfrak{z}}$ with
respect to the orthonormal basis $\\{z_{1},\ldots,z_{p}\\}$ is

$\mathrm{Rc}|_{\mathfrak{z}}=\tfrac{m}{4}\,E^{-1}\Phi.$ (26)

Notice that $\xi:=\sum_{i}\varepsilon_{i}\,\varphi(z_{i})$ is the trace of the
matrix $E^{-1}\Phi$.

If $N$ is of (pseudo-) $H$-type, then $\Phi=E$, $E^{-1}\Phi=I_{p}$, and we
recover all of the previously known results about the Ricci operator.
$\Diamond$

  * Remark 3.remark

Clearly $\mathrm{Rc}(z)=0$ implies that either $z\in\ker\Phi$ or $\Phi z$ is
null with respect to the metric on $\mathfrak{z}$. This fact will become
useful later.  $\Diamond$

Scalar curvature is the trace of the Ricci curvature operator.

###### Theorem 3.6

Let $(N,\left\langle\ ,\,\right\rangle,\varphi)$ be a modified $H$-type group.
Since the metric $\left\langle\ ,\,\right\rangle$ is left-invariant, the
scalar curvature is constant on $N$, and is given by

$S=-\frac{m\xi}{4}.$ (27)

  * Proof: Using the pseudo-orthonormal bases of (18), we compute

$\displaystyle S$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\sum_{i=1}^{m}\sum_{k=1}^{p}\left\|z_{k}\right\|^{2}_{\mathfrak{z}}\left\|e_{i}\right\|_{\mathfrak{v}}^{2}\left\langle
j(z_{k})e_{i},j(z_{k})e_{i}\right\rangle_{\mathfrak{v}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\sum_{i=1}^{m}\sum_{k=1}^{p}\left\|z_{k}\right\|_{\mathfrak{z}}^{2}\varphi(z_{k})$
$\displaystyle=$ $\displaystyle-\frac{m\xi}{4}.$

###  3.1 Examples

We now return to the examples of Section 2. We compute the constant $\xi$ and
the matrix representation of the Ricci operator, and compute its eigenvalues
for each group.

  * Example 3.example

Consider the group $(N,g_{0},\varphi_{0})$ of Example 2. The matrices
representing $g_{0}$ and $\left(\ ,\,\right)_{\varphi_{0}}$ on $\mathfrak{z}$
are

$E=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},\ \ \text{and}\ \
\Phi_{0}=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}.$

Thus $E^{-1}\Phi_{0}=\Phi_{0}$, $\xi=\mathop{\operator@font
tr}\nolimits\left(\Phi_{0}\right)=1$, and the scalar curvature of $(N,g_{0})$
is $S=-\tfrac{1}{2}$. The Ricci operator is given by

$\displaystyle\mathrm{Rc}|_{\mathfrak{v}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},\text{\ and}$
$\displaystyle\mathrm{Rc}|_{\mathfrak{z}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}.$

It has eigenvalues $-\tfrac{1}{2}$ on $\mathfrak{v}$ and $0,\tfrac{1}{2}$ on
$\mathfrak{z}$.  $\Diamond$

  * Example 3.example

Consider the modified $H$-type group $(N,g_{1},\varphi_{1})$ of Example 2. The
matrix representation of $g_{1}$ on $\mathfrak{z}$ with respect to the ordered
basis $\\{u_{1},u_{2}\\}$ is

$E=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$

while the matrix representation of $\left(\ ,\,\right)_{\varphi_{1}}$ with
respect to the same basis is

$\Phi_{1}=\begin{pmatrix}\tfrac{1}{2}&\tfrac{1}{2}\\\\[1.07639pt]
\tfrac{1}{2}&\tfrac{1}{2}\end{pmatrix}.$

The matrix $E^{-1}\Phi_{1}$ is then

$E^{-1}\Phi_{1}=\frac{1}{2}\begin{pmatrix}1&1\\\ -1&-1\end{pmatrix}.$

Clearly $\xi=\mathop{\operator@font
tr}\nolimits\left(E^{-1}\Phi_{1}\right)=0$, so that $(N,g_{1})$ is scalar
flat. This also implies that the Ricci operator restricted to $\mathfrak{v}$
is zero. Moreover, the Ricci operator on $\mathfrak{z}$, given by the matrix

$\mathrm{Rc}|_{\mathfrak{z}}=\frac{1}{4}\begin{pmatrix}1&1\\\
-1&-1\end{pmatrix},$

is 2-step nilpotent. Therefore the Ricci operator on the entire Lie algebra
$\mathfrak{n}$ has only one eigenvalue, $\lambda=0$.

The space $(N,g_{1})$ was also found to be scalar flat but not Ricci flat in
[5].  $\Diamond$

  * Example 3.example

Next consider the group $(N,g_{2},\varphi_{2})$ of Example 2. The matrix
representations of $g_{2}$ and $\left(\ ,\,\right)_{\varphi_{2}}$ with respect
to the basis $\\{u_{1},u_{2}\\}$ are

$E=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},\ \ \text{and\ \
}\Phi_{2}=-\frac{1}{2}\begin{pmatrix}1&1\\\ 1&1\end{pmatrix}.$

The matrix $E^{-1}\Phi_{2}$ is then

$E^{-1}\Phi_{2}=-\frac{1}{2}\begin{pmatrix}1&1\\\ -1&-1\end{pmatrix}.$

As in Example 3.1, we see that $\xi=\mathop{\operator@font
tr}\nolimits\left(E^{-1}\Phi_{2}\right)=0$, so the space $(N,g_{2})$ has zero
scalar curvature. This also implies that the Ricci operator vanishes on
$\mathfrak{v}$. Again in analogy with Example 3.1, the Ricci operator
restricted to $\mathfrak{z}$ is 2-step nilpotent,

$\mathrm{Rc}|_{\mathfrak{z}}=-\frac{1}{4}\begin{pmatrix}1&1\\\
-1&-1\end{pmatrix}.$

Therefore $\mathrm{Rc}$ has only one eigenvalue, $\lambda=0$.  $\Diamond$

  * Example 3.example

Consider the group $(N,g_{3},\varphi_{3})$ of Example 2. The matrices for
$g_{3}$ and $\left(\ ,\,\right)_{\varphi_{3}}$ on $\mathfrak{z}$ are given by

$E=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},\ \ \text{and}\ \
\Phi_{3}=\begin{pmatrix}0&0\\\ 0&-1\end{pmatrix}.$

Thus $E^{-1}\Phi_{3}=\Phi_{3}$, and $\xi=\mathop{\operator@font
tr}\nolimits\left(\Phi_{3}\right)=-1$. Therefore the scalar curvature of
$(N,g_{3})$ is $S=+\tfrac{1}{2}$. The Ricci operator on $\mathfrak{n}$ is
given by

$\displaystyle\mathrm{Rc}|_{\mathfrak{v}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},\ \text{and}$
$\displaystyle\mathrm{Rc}|_{\mathfrak{z}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}0&0\\\ 0&-1\end{pmatrix}.$

The eigenvalues of $\mathrm{Rc}$ are then $\tfrac{1}{2}$ on $\mathfrak{v}$ and
$0,-\tfrac{1}{2}$ on $\mathfrak{z}$.  $\Diamond$

##  4 Geometry of Modified $H$-Type Groups

In this section we investigate the geometric consequences of Definition 2.
Most of these properties can be stated in terms of the Ricci operator of
Theorem 3.5.

###  4.1 Isometry groups

Let
$\mathfrak{n}^{\mathbb{C}}=\mathfrak{z}^{\mathbb{C}}\oplus\mathfrak{v}^{\mathbb{C}}$
be the complexification of the Lie algebra $\mathfrak{n}$. Since the Ricci
operator of Theorem 3.5 respects the splitting
$\mathfrak{n}=\mathfrak{z}\oplus\mathfrak{v}$, then its complexification
respects the induced splitting of $\mathfrak{n}^{\mathbb{C}}$.

Recall the following lemma of [5].

######  Lemma 4.1 ([5], Lemma 2)

Let $(N,\left\langle\ ,\,\right\rangle)$ be a 2-step nilpotent Lie group such
that $\left\langle\ ,\,\right\rangle$ is a pseudo-Riemannian left-invariant
metric for which the center is nondegenerate. Assume

$\displaystyle\mathfrak{v}^{\mathbb{C}}$ $\displaystyle=$ $\displaystyle
V_{\lambda_{1}}\oplus\cdots\oplus V_{\lambda_{j}},$
$\displaystyle\mathfrak{z}^{\mathbb{C}}$ $\displaystyle=$ $\displaystyle
V_{\lambda_{j+1}}\oplus\cdots\oplus V_{\lambda_{s}},$ (28)

for the different eigenvalues $\lambda_{1},\ldots,\lambda_{s}$ of the Ricci
operator $\mathrm{Rc}$, where $V_{\lambda_{i}}$ are the eigenspaces
corresponding to $\lambda_{i}$. Then every isometry of $N$ preserves the
splitting $TN=\mathfrak{v}N\oplus\mathfrak{z}N$; that is,
$\mathrm{Iso}^{\mathrm{spl}}(N)=\mathrm{Iso}(N)$.

Since $\mathrm{Iso}^{\mathrm{aut}}(N)=\mathrm{Iso}^{\mathrm{spl}}(N)$ for all
groups with nondegenerate centers, this actually implies that
$\mathrm{Iso}^{\mathrm{aut}}(N)=\mathrm{Iso}^{\mathrm{spl}}(N)=\mathrm{Iso}(N)$
for all such groups satisfying the hypotheses of the lemma.

For modified $H$-type groups, the restriction of $\mathrm{Rc}$ to
$\mathfrak{v}$ has only one eigenvalue, $-\tfrac{\xi}{2}$. As long as
$-\tfrac{\xi}{2}$ is not also an eigenvalue of $\mathrm{Rc}|_{\mathfrak{z}}$,
then the eigenspace decomposition of $\mathfrak{n}$ with respect to the Ricci
operator respects the splitting $\mathfrak{n}=\mathfrak{z}\oplus\mathfrak{v}$.
That is, if $V_{\lambda_{i}}$ are the eigenspaces for the different
eigenvalues $\lambda_{i}$ of $\mathrm{Rc}|_{\mathfrak{z}}$, then

$\displaystyle\mathfrak{z}^{\mathbb{C}}$ $\displaystyle=$ $\displaystyle
V_{\lambda_{1}}\oplus\cdots\oplus V_{\lambda_{s}},\text{\ and}$
$\displaystyle\mathfrak{v}^{\mathbb{C}}$ $\displaystyle=$ $\displaystyle
V_{-\frac{\xi}{2}}.$ (29)

Invoking Lemma 4.1 and recalling that modified $H$-type groups have
nondegenerate centers by definition, we obtain the following theorem about the
isometry groups of modified $H$-type groups.

###### Theorem 4.2

Let $(N,\left\langle\ ,\,\right\rangle,\varphi)$ be a modified $H$-type group.
If $-\tfrac{\xi}{2}$ is _not_ an eigenvalue of $\mathrm{Rc}|_{\mathfrak{z}}$,
then the following isometry groups coincide:
$\mathrm{Iso}^{\mathrm{aut}}(N)=\mathrm{Iso}^{\mathrm{spl}}(N)=\mathrm{Iso}(N)$.

  * Remark 4.remark

The groups $(N,g_{1})$ and $(N,g_{2})$ of Examples 2, 3.1, 2, and 3.1 do not
satisfy the hypothesis of the theorem; _viz._ [5].  $\Diamond$

###  4.2 Totally geodesic subgroups

Let $z\in\mathfrak{z}$ and $x\in\mathfrak{v}$ be non-zero vectors. In $H$-type
groups, the space
$\mathfrak{n}^{\prime}:=\text{span}\left\\{z,x,j(z)x\right\\}$ is always a
totally geodesic subalgebra of $\mathfrak{n}$, making
$N^{\prime}:=\exp{\mathfrak{n}^{\prime}}$ a totally geodesic subgroup of $N$.
Modified $H$-type groups have the following analogue.

###### Theorem 4.3

Let $(N,\left\langle\ ,\,\right\rangle,\varphi)$ be a modified $H$-type Lie
group. Let $z\in\mathfrak{z}$, $x\in\mathfrak{v}$ be non-zero vectors with
$\left\|x\right\|_{\mathfrak{v}}^{2}\neq 0$, and consider the subspace
$\mathfrak{n}^{\prime}=\text{span}\left\\{z,x,j(z)x\right\\}$. The submanifold
$N^{\prime}=\exp{\mathfrak{n}^{\prime}}$ is a totally geodesic subgroup of $N$
if and only if $z$ is not in the kernel of $\mathrm{Rc}|_{\mathfrak{z}}$ but
is an eigenvector of $\mathrm{Rc}|_{\mathfrak{z}}$.

  * Proof: The Levi-Civita connection on $\mathfrak{n}^{\prime}$ is given by

$\displaystyle\mathop{\nabla}\nolimits_{\mspace{-2.5mu}z}x=\mathop{\nabla}\nolimits_{\mspace{-2.5mu}x}z$
$\displaystyle=$ $\displaystyle-\tfrac{1}{2}j(z)x,$
$\displaystyle\mathop{\nabla}\nolimits_{\mspace{-2.5mu}z}j(z)x$
$\displaystyle=$ $\displaystyle\tfrac{1}{2}\varphi(z)\,x,\text{\ and}$ (30)
$\displaystyle\mathop{\nabla}\nolimits_{\mspace{-2.5mu}x}j(z)x$
$\displaystyle=$ $\displaystyle\tfrac{1}{2}\left[x,j(z)x\right].$

Thus, we must show that $z^{\prime}:=\left[x,j(z)x\right]$ is proportional to
$z$ if and only if $z$ is an eigenvector of $\mathrm{Rc}|_{\mathfrak{z}}$. For
$a\in\mathfrak{z}$, we compute

$\displaystyle\left\langle\left[x,j(z)x\right],a\right\rangle_{\mathfrak{z}}$
$\displaystyle=$ $\displaystyle\left\langle\mathop{\operator@font
ad}\nolimits_{x}j(z)x,a\right\rangle_{\mathfrak{z}}$ $\displaystyle=$
$\displaystyle\left\langle j(z)x,j(a)x\right\rangle_{\mathfrak{z}}$
$\displaystyle=$
$\displaystyle\left\|x\right\|_{\mathfrak{v}}^{2}\,\left(z,a\right)_{\varphi}.$

Therefore,
$z^{\prime}=\left\|x\right\|_{\mathfrak{v}}^{2}\,\mathrm{Id}|_{\varphi}^{\dagger}\,z$.
By (25),
$z^{\prime}=\left\|x\right\|_{\mathfrak{v}}^{2}\,\tfrac{4}{m}\,\mathrm{Rc}\,z$,
and $z^{\prime}$ is proportional to $z$ if and only if $z$ is an eigenvector
of $\mathrm{Rc}$.

###### Corollary 4.4

Let $(N,\left\langle\ ,\,\right\rangle,\varphi)$ be a modified $H$-type Lie
group. Let $z\in\mathfrak{z}$ and $x\in\mathfrak{v}$ be non-zero vectors.
Suppose that either $z\in\ker\mathrm{Rc}|_{\mathfrak{z}}$ or
$\left\|x\right\|_{\mathfrak{v}}^{2}=0$ (or both), and consider the subspace
$\mathfrak{n}^{\prime}=\text{span}\left\\{z,x,j(z)x\right\\}$. The submanifold
$N^{\prime}=\exp\mathfrak{n}^{\prime}$ is totally geodesic, but not a subgroup
of $N$.

  * Proof: By (30), $\mathop{\nabla}\nolimits_{\mspace{-2.5mu}z}x,\mathop{\nabla}\nolimits_{\mspace{-2.5mu}x}z,$ and $\mathop{\nabla}\nolimits_{\mspace{-2.5mu}z}j(z)x$ are all in $\mathfrak{n}^{\prime}$. By ( ‣ 4.2), $\mathop{\nabla}\nolimits_{\mspace{-2.5mu}x}j(z)x=0$. Therefore $N^{\prime}$ is totally geodesic in $N$. However, $\mathfrak{n}^{\prime}$ is not a subalgebra since $\left[x,j(z)x\right]=0$.

###  4.3 Geodesics

In this section we give explicit formulas for the geodesics of a modified
$H$-type group in terms of the $j$-maps, quadratic form $\varphi$, and the
Ricci operator on $\mathfrak{z}$. We follow the calculations of [4], making
particular use of one result that is stated here as a lemma. First, some
preliminaries.

Suppose $N$ is a connected, simply connected 2-step nilpotent Lie group. Let
$I$ be a real interval containing zero, and let $\gamma:I\to N$ be a geodesic
such that $\gamma(0)=1\in N$ and
$\dot{\gamma}(0)=z_{0}+x_{0}\in\mathfrak{z}\oplus\mathfrak{v}=\mathfrak{n}$.
Since the exponential map is a diffeomorphism for simply connected nilpotent
Lie groups, $\gamma(t)=\exp(z(t)+x(t))$.

Let $J=j(z_{0})$ denote the skew adjoint transformation of $\mathfrak{v}$
determined by the initial condition, and let $\mathfrak{v}^{1}=\ker J$. The
skew-adjointness of $J$ implies that
$\mathfrak{v}=\mathfrak{v}^{1}\oplus\mathfrak{v}^{2}$ is an orthogonal direct
sum, where $\mathfrak{v}^{2}=\mathfrak{v}/\mathfrak{v}^{1}$. Notice that $J$
is invertible on $\mathfrak{v}^{2}$. Decompose $x_{0}=x_{1}+x_{2}$, with
$x_{i}\in\mathfrak{v}^{i}$.

Now let $\\{\theta_{1},\ldots,\theta_{k}\\}$ be the distinct nonzero
eigenvalues of $J^{2}$. Decompose $\mathfrak{v}^{2}$ as the orthogonal direct
sum $\bigoplus_{j=1}^{k}\mathfrak{w}_{j}$, where $J$ leaves each
$\mathfrak{w}_{j}$ invariant and
$J^{2}|_{\mathfrak{w}_{j}}=\theta_{j}\mathrm{Id}|_{\mathfrak{v}}$. Write
$x_{2}=\sum_{j=1}^{k}w_{j}$ with each $w_{j}\in\mathfrak{w}_{j}$.

Under these assumptions, we state the aforementioned result of [4].

######  Lemma 4.5 ([4], Prop 4.15)

If $N$ is a 2-step nilpotent Lie group with nondegenerate center, and $J^{2}$
diagonalizes, then

$\displaystyle z(t)$ $\displaystyle=$ $\displaystyle tz_{1}(t)+z_{2}(t),$ (32)
$\displaystyle x(t)$ $\displaystyle=$ $\displaystyle
tx_{1}(t)+(e^{tJ}-I)J^{-1}x_{2},$ (33)

where

$\displaystyle z_{1}(t)$ $\displaystyle=$ $\displaystyle
z_{0}+\tfrac{1}{2}\left[x_{1},(e^{tJ}+I)J^{-1}x_{2}\right]+\tfrac{1}{2}\sum_{j=1}^{k}\left[J^{-1}w_{j},w_{j}\right],$
$\displaystyle z_{2}(t)$ $\displaystyle=$
$\displaystyle\left[x_{1},(I-e^{tJ})(J^{-2}x_{2})\right]+\tfrac{1}{2}\left[e^{tJ}J^{-1}x_{2},J^{-1}x_{2}\right]$
$\displaystyle-\tfrac{1}{2}\sum_{i\neq
j}\frac{1}{\theta_{j}-\theta_{i}}\left(\left[e^{tJ}Jw_{i},e^{tJ}J^{-1}w_{j}\right]-\left[e^{tJ}w_{i},e^{tJ}w_{j}\right]\right)$
$\displaystyle+\tfrac{1}{2}\sum_{i\neq
j}\frac{1}{\theta_{j}-\theta_{i}}\left(\left[Jw_{i},J^{-1}w_{j}\right]-\left[w_{i},w_{j}\right]\right).$

For modified $H$-type groups, the situation is simplified considerably. The
map $J^{2}$ is diagonalizable for all $z_{0}\in\mathfrak{z}$ and has only one
eigenvalue, $-\varphi(z_{0})$. This eliminates the need for any
$\mathfrak{w}$-decomposition, thus eliminating many of the terms in $z_{1}$
and $z_{2}$ of the lemma. Further, the maps $J^{-1}$ and $e^{tJ}$ may each be
written simply in terms of $\varphi(z_{0})$ and $j(z_{0})$. The following
lemma is straightforward to prove.

###### Lemma 4.6

Let $(N,\left\langle\ ,\,\right\rangle,\varphi)$ be a modified $H$-type Lie
group. Let $z_{0}\in\mathfrak{z}$ such that $\varphi_{0}:=\varphi(z_{0})\neq
0$, and denote by $J$ the transformation
$j(z_{0})\in\mathrm{End}(\mathfrak{v})$. Then

$\displaystyle J^{-1}$ $\displaystyle=$
$\displaystyle-\frac{1}{\varphi_{0}}J,\text{\ and}$ $\displaystyle e^{tJ}$
$\displaystyle=$
$\displaystyle\begin{cases}\cos\left(t\sqrt{\varphi_{0}}\right)I+\frac{1}{\sqrt{\varphi_{0}}}\sin\left(t\sqrt{\varphi_{0}}\right)J&\text{if\
}\varphi_{0}>0,\\\\[4.30554pt]
\cosh\left(t\sqrt{\left|\varphi_{0}\right|}\right)I+\frac{1}{\sqrt{\left|\varphi_{0}\right|}}\sinh\left(t\sqrt{\left|\varphi_{0}\right|}\right)J&\text{if\
}\varphi_{0}<0,\end{cases}$ $\displaystyle e^{tJ}J^{-1}$ $\displaystyle=$
$\displaystyle\begin{cases}\frac{-\cos\left(t\sqrt{\varphi_{0}}\right)}{\varphi_{0}}J+\frac{1}{\sqrt{\varphi_{0}}}\sin\left(t\sqrt{\varphi_{0}}\right)I&\text{if\
}\varphi_{0}>0,\\\\[4.30554pt]
\frac{-\cosh\left(t\sqrt{\left|\varphi_{0}\right|}\right)}{\varphi_{0}}J+\frac{1}{\sqrt{\left|\varphi_{0}\right|}}\sinh\left(t\sqrt{\left|\varphi_{0}\right|}\right)I&\text{if\
}\varphi_{0}<0.\end{cases}$

If $\varphi(z_{0})=0$, then $\ker J=\mathfrak{v}$ so that $x_{1}=x_{0}$ and
$x_{2}=0$. The formulas of Lemma 4.5 reduce to

$\displaystyle z(t)$ $\displaystyle=$ $\displaystyle tz_{0},$ $\displaystyle
x(t)$ $\displaystyle=$ $\displaystyle tx_{0}.$

If $\varphi(z_{0})\neq 0$, then $\ker J=0$ so that $x_{1}=0$ and
$x_{2}=x_{0}$. As mentioned above, this $x_{2}$ does not need to be decomposed
any further since $J^{2}$ has only one eigenvalue. In this case, the formulas
of Lemma 4.5 reduce to

$\displaystyle z(t)$ $\displaystyle=$ $\displaystyle
t\left(z_{0}+\frac{1}{2}\left[J^{-1}x_{0},x_{0}\right]\right)+\frac{1}{2}\left[e^{tJ}J^{-1}x_{0},J^{-1}x_{0}\right],$
$\displaystyle x(t)$ $\displaystyle=$ $\displaystyle(e^{tJ}-I)J^{-1}x_{0}.$

Carefully applying the formulas of Lemma 4.6, replacing $J$ with $j(z_{0})$,
and recalling from the proof of Theorem 4.3 that
$\left[x_{0},j(z_{0})x_{0}\right]=\tfrac{4}{m}\left\|x_{0}\right\|_{\mathfrak{v}}^{2}\mathrm{Rc}|_{\mathfrak{z}}z_{0}$,
one obtains the following theorem.

###### Theorem 4.7

Suppose $(N,\left\langle\ ,\,\right\rangle,\varphi)$ is a modified $H$-type
Lie group. Let $\gamma:I\to N$, $\gamma(t)=\exp(z(t)+x(t))$, be a geodesic
with $\gamma(0)=1\in N$ and $\dot{\gamma}(0)=z_{0}+x_{0}\in\mathfrak{n}$, and
suppose that both $z_{0}$ and $x_{0}$ are nonzero. Let
$\varphi_{0}=\varphi(z_{0})$.

If $\varphi(z_{0})=0$, then $\gamma$ is given by

$\displaystyle z(t)$ $\displaystyle=$ $\displaystyle tz_{0},$ (34)
$\displaystyle x(t)$ $\displaystyle=$ $\displaystyle tx_{0};$ (35)

if $\varphi(z_{0})>0$, then $\gamma$ is given by

$\displaystyle z(t)$ $\displaystyle=$ $\displaystyle
tz_{0}+\left(\frac{4\sin\left(t\sqrt{\varphi_{0}}\right)-2t\sqrt{\varphi_{0}}}{m(\varphi_{0})^{3/2}}\right)\left\|x_{0}\right\|_{\mathfrak{v}}^{2}\mathrm{Rc}|_{\mathfrak{z}}z_{0},$
(36) $\displaystyle x(t)$ $\displaystyle=$
$\displaystyle\left(\frac{1-\cos\left(t\sqrt{\varphi_{0}}\right)}{\varphi_{0}}\right)j(z_{0})x_{0}+\left(\frac{\sin\left(t\sqrt{\varphi_{0}}\right)}{\sqrt{\varphi_{0}}}\right)x_{0};$
(37)

and if $\varphi(z_{0})<0$, then $\gamma$ is given by

$\displaystyle z(t)$ $\displaystyle=$ $\displaystyle
tz_{0}+\left(\frac{4\sinh\left(t\sqrt{\left|\varphi_{0}\right|}\right)-2t\sqrt{\left|\varphi_{0}\right|}}{m\varphi_{0}\sqrt{\left|\varphi_{0}\right|}}\right)\left\|x_{0}\right\|_{\mathfrak{v}}^{2}\mathrm{Rc}|_{\mathfrak{z}}z_{0},$
(38) $\displaystyle x(t)$ $\displaystyle=$
$\displaystyle\left(\frac{1-\cosh\left(t\sqrt{\left|\varphi_{0}\right|}\right)}{\varphi_{0}}\right)j(z_{0})x_{0}+\left(\frac{\sinh\left(t\sqrt{\left|\varphi_{0}\right|}\right)}{\sqrt{\left|\varphi_{0}\right|}}\right)x_{0}.$
(39)

Recalling that it is possible for
$\left\|x_{0}\right\|_{\mathfrak{v}}^{2}\mathrm{Rc}|_{\mathfrak{z}}z_{0}$ to
vanish while neither $x_{0}$ nor $z_{0}$ are zero, we obtain the following
Corollary.

###### Corollary 4.8

Suppose $(N,\left\langle\ ,\,\right\rangle,\varphi)$ is a modified $H$-type
Lie group. Let $\gamma:I\to N$, $\gamma(t)=\exp(z(t)+x(t))$, be a geodesic
with $\gamma(0)=1$ and $\dot{\gamma}(0)=z_{0}+x_{0}\in\mathfrak{n}$, supposing
that both $z_{0}$ and $x_{0}$ are nonzero. If
$\left\|x_{0}\right\|_{\mathfrak{v}}^{2}=0$ or
$z_{0}\in\ker(\mathrm{Rc}|_{\mathfrak{z}})$, then the $\mathfrak{z}$-component
of $\gamma$ is simply given by $z(t)=tz_{0}$.

Just by looking at the formulas for the geodesics in Theorem 4.7 it is clear
that every geodesic lives in a submanifold $M=\exp\mathfrak{m}$, where
$\mathfrak{m}=\text{span}\left\\{z_{0},x_{0},j(z_{0})x_{0},\mathrm{Rc}(z_{0})\right\\}$.
If $z_{0}$ is an eigenvector of $\mathrm{Rc}$ corresponding to a non-zero
eigenvalue and $x_{0}$ is non-null, then by the previous section $M$ is a
totally geodesic subgroup that is isomorphic to the Heisenberg group $H_{3}$.

If either $x_{0}$ is null or $\mathrm{Rc}(z_{0})=0$, then by Corollary 4.4,
$\left[x_{0},j(z_{0})x_{0}\right]=0=\mathrm{Rc}(z_{0})$,
$\mathfrak{m}=\text{span}\left\\{z_{0},x_{0},j(z_{0})x_{0}\right\\}$, and $M$
is totally geodesic but not a subgroup. In fact, it is a copy of
$\mathbb{R}^{3}$.

If $x_{0}$ is non-null and $z_{0}$ is not in $\ker(\mathrm{Rc})$ but is also
not an eigenvector of $\mathrm{Rc}$, then $M$ is a subgroup isomorphic to
$H_{3}(\mathbb{R})\times\mathbb{R}$. Indeed, by the previous section,
$\left[x_{0},j(z_{0})x_{0}\right]=\tfrac{4}{m}\left\|x_{0}\right\|_{\mathfrak{v}}^{2}\mathrm{Rc}z_{0}$.
Thus $z_{0}$ plays the role of the central extension.

###  4.4 Sectional curvatures of semi-central planes

In the Riemannian case, it is shown in [14] that the generalized Heisenberg
groups of Example 2 are characterized among nilpotent Lie groups by the
following property:

> Every plane $\Pi$ spanning one central and one non-central direction has
> sectional curvature $K(\Pi)=c$.

In particular, every semi-central plane $\Pi$ in an $H$-type group has
sectional curvature $K(\Pi)=\tfrac{1}{4}$. In this section we investigate the
modified $H$-type analogue to this result.

Let $(N,\left\langle\ ,\,\right\rangle,\varphi)$ be a modified $H$-type Lie
group, and $\Pi=\text{span}\left\\{z,x\right\\}$ be a nondegenerate plane
spanned by $z\in\mathfrak{z}$, $x\in\mathfrak{v}$. Since $\Pi$ is assumed to
be nondegenerate, we may assume that $\left\|z\right\|^{2}=\varepsilon_{z}=\pm
1$ and $\left\|x\right\|^{2}=\varepsilon_{x}=\pm 1$. The sectional curvature
of $\Pi$ is given in (16) by

$K(\Pi)=\tfrac{1}{4}\mathcal{K}(z)=\tfrac{1}{4}\varepsilon_{z}\varphi(z).$

If $\varphi(z)=\varepsilon_{z}4c$, $c\neq 0$, as in Example 2, then every
nondegenerate semi-central plane has curvature $K(\Pi)=c$, as in the
Riemannian case. Indeed,

$K(\Pi)=\tfrac{1}{4}\varepsilon_{z}\varphi(z)=\tfrac{1}{4}\varepsilon_{z}^{2}4c=c.$
(40)

In particular, nondegenerate semi-central planes in pseudo-$H$-type groups
have sectional curvature $K(\Pi)=\tfrac{1}{4}$.

The converse also holds.

###### Theorem 4.9

Let $(N,\left\langle\ ,\,\right\rangle)$ be a 2-step nilpotent Lie group with
left-invariant metric $\left\langle\ ,\,\right\rangle$ making the center
nondegenerate. If the sectional curvature of every nondegenerate semi-central
plane $\Pi$ is constant, $K(\Pi)=c$, then $(N,\left\langle\ ,\,\right\rangle)$
is a modified $H$-type group with quadratic form
$\varphi=4c\left\|\cdot\right\|_{\mathfrak{z}}^{2}$.

  * Proof: Let $z\in\mathfrak{z}$ and $x\in\mathfrak{v}$ such that $\varepsilon_{z}=\left\|z\right\|_{\mathfrak{z}}^{2}=\pm 1$ and $\varepsilon_{x}=\left\|x\right\|_{\mathfrak{v}}^{2}=\pm 1$, and consider the plane $\Pi=\text{span}\left\\{z,x\right\\}$. The sectional curvature of $\Pi$ is given in [4] by

$K(z,x)=\tfrac{1}{4}\varepsilon_{z}\varepsilon_{x}\left\langle
j(z)x,j(z)x\right\rangle_{\mathfrak{v}}.$

Solving for $\left\langle j(z)x,j(z)x\right\rangle_{\mathfrak{v}}$ yields

$\left\langle
j(z)x,j(z)x\right\rangle_{\mathfrak{v}}=4\varepsilon_{z}\varepsilon_{x}K(z,x).$

Recalling that $\varepsilon_{x}=\left\langle x,x\right\rangle_{\mathfrak{v}}$,
the assumption $K(\Pi)=c$ implies that

$\left\langle
j(z)x,j(z)x\right\rangle_{\mathfrak{v}}=4c\varepsilon_{z}\left\langle
x,x\right\rangle_{\mathfrak{v}}$

for all unit vectors $z\in\mathfrak{z}$, $x\in\mathfrak{v}$. We deduce that
$j(z)^{2}=-4c\varepsilon_{z}\mathrm{Id}|_{\mathfrak{v}}$ for all unit vectors
in $\mathfrak{v}$. By linearity, since $\mathfrak{z}$ is nondegenerate, we
obtain that
$j(z)^{2}=-4c\left\|z\right\|_{\mathfrak{z}}^{2}\mathrm{Id}|_{\mathfrak{v}}$
for all $z\in\mathfrak{z}$. Therefore $(N,\left\langle\
,\,\right\rangle,4c\left\|\cdot\right\|_{\mathfrak{z}}^{2})$ is a modified
$H$-type group.

###### Corollary 4.10

The $H$-type and pseudo-$H$-type groups are characterized, among 2-step
nilpotent Lie groups, by the following property: Every nondegenerate semi-
central plane has sectional curvature $K=\tfrac{1}{4}$.

In general, it is possible for modified $H$-type groups to have semi-central
planes that are flat. Indeed, if $\varphi(z^{\prime})=0$ but
$\left\|z^{\prime}\right\|^{2}\neq 0$, then any semi-central plane in the
direction of $z^{\prime}$ has $K(\Pi)=0$.

  * Remark 4.remark

The sectional curvature of nondegenerate semi-central planes depends only on
the central direction $z$ and the quadratic form $\varphi$,
$K(z,x)=\tfrac{1}{4}\varepsilon_{z}\varphi(z)$. Therefore, the quadratic form
$\varphi$ is given by

$\varphi(z)=4\left\|z\right\|_{\mathfrak{z}}^{2}K(z,\accentset{\hrulefill}{\mathfrak{v}})$
(41)

for all non-null vectors in $\mathfrak{z}$, where
$\accentset{\hrulefill}{\mathfrak{v}}$ represents all non-null vectors in
$\mathfrak{v}$. This further illustrates the close relationship between the
quadratic form $\varphi$ and the sectional curvature of $N$.  $\Diamond$

###  4.5 Nilsolitons

A pseudo-Riemannian metric on a 2-step nilpotent Lie group is said to be a
_nilsoliton_ if there exists a constant $c\in\mathbb{R}$ such that the
operator

$D:=\mathrm{Rc}+c\cdot\mathrm{Id}$ (42)

is a derivation of $\mathfrak{n}$. That is, $D$ must satisfy

$D\left[x,y\right]=\left[Dx,y\right]+\left[x,Dy\right]$

for all $x,y\in\mathfrak{n}$. By Theorem 3.5, the Ricci operator on a modified
$H$-type group $(N,\left\langle\ ,\,\right\rangle,\varphi)$ is given on
$\mathfrak{n}=\mathfrak{z}\oplus\mathfrak{v}$ by

$\mathrm{Rc}=\begin{pmatrix}\frac{m}{4}\mathrm{Id}|_{\varphi}^{\dagger}&0\\\\[2.15277pt]
0&-\frac{\xi}{2}\mathrm{Id}|_{\mathfrak{v}}\end{pmatrix}.$

For $D$ to be a derivation, the following equation must be satisfied for all
$x,y\in\mathfrak{v}$.

$\mathrm{Rc}|_{\mathfrak{z}}(\left[x,y\right])=(-\xi+c)\left[x,y\right]$ (43)

This occurs if and only if the derived algebra
$\mathfrak{n}^{\prime}=\left[\mathfrak{n},\mathfrak{n}\right]=\left[\mathfrak{v},\mathfrak{v}\right]$
is contained in a single eigenspace of $\mathrm{Rc}|_{\mathfrak{z}}$. Indeed,
if $\lambda$ is the corresponding eigenvalue of $\mathrm{Rc}|_{\mathfrak{z}}$,
then put $c=\lambda+\xi$.

We have proved the following.

###### Theorem 4.11

Let $(N,\left\langle\ ,\,\right\rangle,\varphi)$ be a modified $H$-type group.
The metric $\left\langle\ ,\,\right\rangle$ is a nilsoliton if and only if the
derived algebra $\left[\mathfrak{v},\mathfrak{v}\right]$ is contained in a
single eigenspace of the Ricci operator $\mathrm{Rc}|_{\mathfrak{z}}$.

As a corollary we recover a well known result about (pseudo-) $H$-type groups
[12, 13].

###### Corollary 4.12

Every $H$-type and pseudo-$H$-type metric is a nilsoliton.

  * Proof: The Ricci operator for an $H$-type or pseudo-$H$-type group is given by

$\mathrm{Rc}=\begin{pmatrix}\frac{m}{4}\mathrm{Id}|_{\mathfrak{z}}&0\\\
0&-\frac{p}{2}\mathrm{Id}|_{\mathfrak{v}}\end{pmatrix},$

so that every $z\in\mathfrak{z}$ is and eigenvector of
$\mathrm{Rc}|_{\mathfrak{z}}$.

Given a nilsoliton of modified $H$-type, we can construct many.

###### Corollary 4.13

If a modified $H$-type group admits a nilsoliton, then every trivial central
extension also admits a nilsoliton.

  * Proof: Let $(N,\left\langle\ ,\,\right\rangle,\varphi)$ be a modified $H$-type group such that $\left\langle\ ,\,\right\rangle$ is a nilsoliton. Let $\mathfrak{m}$ be a real vector space endowed with an inner product $\left\langle\ ,\,\right\rangle_{\mathfrak{m}}$ of any signature, and consider the trivial central extension of $\mathfrak{n}$ given by $\accentset{\hrulefill}{\mathfrak{n}}=\mathfrak{n}\oplus\mathfrak{m}$. The simply connected Lie group $\accentset{\hrulefill}{N}=\exp\accentset{\hrulefill}{\mathfrak{n}}$ endowed with the left-invariant metric $\left\langle\ ,\,\right\rangle+\left\langle\ ,\,\right\rangle_{\mathfrak{m}}$ is a modified $H$-type algebra with $\varphi(\mathfrak{m})=0$. The Ricci operator on the center remains unchanged on $\mathfrak{z}$, so $\left[\mathfrak{v},\mathfrak{v}\right]$ remains contained in a single eigenspace of $\mathrm{Rc}$.

## Acknowledgements

This work was started while the author was a postdoctoral research fellow with
CONICET at Universidad Nacional de Rosario, in Rosario, Argentina. The author
would like to thank Gabriela Ovando and Phil Parker for their guidance and
support.

## References

  * [1] P. Ciatti, Scalar products on Clifford modules and pseudo-$H$-type Lie algebras, Ann. Mat. Pura Appl. 178 (2000) 1–31.
  * [2] L. A. Cordero and P. E. Parker, Isometry Groups of PseudoRiemannian 2-Step Nilpotent Lie Groups. Houston J. Math 35 (2009) 49–72.
  * [3] L. A. Cordero and P. E. Parker, Lattices and periodic geodesics in pseudoriemannian 2-step nilpotent Lie groups, Int. J. of Geom. Meth. in Mod. Phys. 5 (2008) 79–99.
  * [4] L. A. Cordero and P. E. Parker, Pseudoriemannian 2-step nilpotent Lie groups, DGS preprint, Wichita: 2000. arXiv: math/0604298
  * [5] V. del Barco and G. P. Ovando, Isometric actions on pseudo-Riemannian nilmanifolds, Ann. Global Anal. Geom. 45 (2013) 95–110.
  * [6] P. Eberlein, Geometry of 2-step nilpotent groups with a left invariant metric, Ann. scien. de l’E.N.S. 27 (1994) 611–660.
  * [7] P. Eberlein, Geometry of 2-step nilpotent groups with a left invariant metric. II. Trans. Amer. Math. Soc. 343 (1994) 805–828.
  * [8] A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. of the A.M.S. 258 (1980) 147–153.
  * [9] A. Kaplan, Riemannian nilmanifolds attached to Clifford modules, Geom. Dedicata 11 (1981) 127–136.
  * [10] A. Kaplan, On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983) 35–42.
  * [11] J. Lauret, Modified H-type groups and symmetric-like Riemannian spaces, Diff. Geom. App. 10 (1999) 121–143.
  * [12] J. Lauret, Ricci soliton homogeneous nilmanifolds, Math. Ann, 319 (2001) 715–733.
  * [13] K. Onda and P. E. Parker, Nilsolitons of H-type in the Lorentzian setting, Wichita: 2015. arXiv: math/1205.6109
  * [14] G.T. Pripoae, Comparison theorem for generalized Heisenberg groups, Proceedings of The Conference of Geometry and Its Applications in Technology and The Workshop on Global Analysis, Differential Geometry and Lie Groups, 1999, pp. 170–5

Justin M. Ryan
3 August 2021
<EMAIL_ADDRESS>