# Induced Semi Riemannian structures on null submanifolds

KARIMUMURYANGO Ménédore  Applied Statistics Institute
university of Burundi
P.O.Box:5158 Bujumbura-Burundi<EMAIL_ADDRESS>,  NDAYIRUKIYE Domitien
Ecole Normale Supérieure
Section de Mathématiques
P. O. Box 6983
Bujumbura
Burundi<EMAIL_ADDRESS>,  NIBARUTA Gilbert  Ecole Normale Supérieure
Section de Mathématiques
P. O. Box 6983
Bujumbura
Burundi<EMAIL_ADDRESS>and  NIBIRANTIZA Aboubacar  Université du Burundi
Institut de Pédagogie Appliquée
Département de Mathématiques
Bujumbura-Burundi<EMAIL_ADDRESS>
###### Abstract.

In this paper, we induce a semi-Riemannian metric on the $r-$null submanifold.
We establish the links between the null geometry and basics invariants of the
associated semi-Riemannian geometry on $r$-null submanifold and semi-
Riemannian constructed from a semi-Riemannian ambient.

###### Key words and phrases:

Semi-riemannian submanifold; $r$-null submanifold; rigging; closed
normalization; associated semi-Riemannian metric.

###### 2010 Mathematics Subject Classification:

53C60, 58B20

##  1\. Introduction

One of the most fundamental problems in submanifold theory is the problem of
isometric immersibility. The embedding problem had been around since Riemann
in $1854$. Soon after Riemann introduced the notion of a manifold, Schläfli
conjectured that every Riemannian manifold could be locally considered as a
submanifold of an Euclidean space with dimension $\frac{1}{2}n(n+1)$ in
$(1873)$. This was later proved in different steps by Janet $(1926)$, E.
Cartan revised Janet’s paper with the same title in $(1927)$. This result of
Cartan-Janet implies that every Einstein $n-$manifold $(n\geq 3)$ can be
locally isometrically embedded in an Euclidean $\frac{n(n+1)}{2}$-space
$\mathbb{E}^{\frac{n(n+1)}{2}}$. In $1956$ J.Nash proved that every closed
Riemannian $n-$manifold can be isometrically embedded in an Euclidean
$m-$space $\mathbb{E}^{m}$ with $m=\frac{1}{2}n(3n+11)$ and he proved also
that every non-closed Riemannian n-manifold can be isometrically embedded in
$\mathbb{E}^{m}$ with $m=\frac{1}{2}n(n+1)(3n+11)$. R.E.Green improved Nash’s
result in $(1970)$ and proved that every non-compact Riemannian n-manifold can
be isometrically embedded in the Euclidean $m-$space $\mathbb{E}^{m}$ with
$m=2(2n+1)(3n+7)$. In $1965$, A.Friedman proved that any $n-$dimensional semi-
Riemannian manifold of index $q$ can be isometrically embedded in a semi-
Euclidean space of dimension $\frac{1}{2}n(n+1)$ and index $\geq q$.

In $1970$ Clarke and Green proved that any semi-Riemannian $n-$manifold
$M^{n}_{q}$ with index $q$ can be isometrically embedded in a semi- Euclidean
$m-$space $\mathbb{E}_{\overline{q}}^{m}$ , for $m$ and ${\overline{q}}$ large
enough. Moreover, this embedding may be taken inside any given open set in
$\mathbb{E}_{\overline{q}}^{m}$. The problem of discovering simple sharp
relationships between intrinsic and extrinsic invariants of a Riemannian
submanifold becomes one of the most fundamental problems in submanifold
theory. The main extrinsic invariant is the squared mean curvature and the
main intrinsic invariants include the classical curvature invariants namely
the scalar curvature and the Ricci curvature.

In [9], the author immersed a null hypersurface equipped with the Riemannian
metric (induced on it by the rigging) into a Riemannian manifold suitably
constructed on the Lorentzian manifold and she established the basic
relationships between the main extrinsic invariants and the main intrinsic
invariants named Chen-Ricci inequality of the null hypersurface in the
Lorentzian manifold. This rigging technique has been introduced in[6].

In the present paper, we induce a semi-Riemannian metric on the null
submanifold and we immersed isometrically the null submanifold endowed with
semi-Riemmannian metric in semi-Riemmannian manifold. We then establish links
between the null geometry and basics invariants of the associated
nondegenerate geometry such as linear connection, the curvature tensor, Ricci
curvature such that is symmetric and sectional curvature. The remaining of
this paper is organized as follows. Section (2) contains most of the
prerequisites material of null submanifolds in semi-Riemannian Manifold. The
normalization and the associated semi-Riemannian structure on a normalized
null submanifold using Rigging techniques are introduced and discussed in
Section (3). The relashionship between the null and the associated semi-
Riemannian geometry is considered in Section (4).

##  2\. Preliminaries on null submanifolds

Let $(\overline{M},\overline{g})$ be a real $(n+k)$-dimensional semi-
Riemannian manifold of constant index $q\in\\{1,\cdots,n+k-1\\}$ where
$n>1,k\geq 1$. Suppose $M$ is a $n$-dimensional submanifold of $\overline{M}$.
In case $\overline{g}_{x}$ is non-degenerate on $T_{x}M$, then $T_{x}M$ and
$T_{x}M^{\bot}$ are complementary orthogonal vector subspaces of
$T_{x}\overline{M}$, a part of the normal vector bundle $TM^{\bot}$ (the
radical distribution) lies in the tangent bundle $TM$ of a submanifold $M$ of
a semi-Riemannian manifold $\overline{M}$. Otherwise,$T_{x}M$ and
$T_{x}M^{\bot}$ are degenerate orthogonal subspaces but no longer
complementary subspaces, that is $TM\cap TM^{\bot}\neq\\{0\\}.$ Thus a basic
problem of the null submanifolds is to replace the intersecting part by a
vector subbundle whose sections are nowhere tangent to $M$. To overcome with
this problem posed by null submanifolds, the authors Bejancu and Duggal
introduced the notion of screen distribution which provides a direct sum
decomposition of $T\overline{M}$ with certain nice properties [4]. Used a
screen distribution $\mathcal{S}(TM)$ on $M$ and a screen vector bundle
$\mathcal{S}(TM^{\bot})$ over $M$ to construct a transversal bundle $tr(TM)$.
They obtained the structure equations of $M$ that relate the curvature tensor
of $\overline{M}$ with the curvature tensor of the linear connections induced
on the vector bundles involved in the study. If the mapping $Rad(TM)$ is a
smooth distribution with constant rank $r>0$, then, it is said the radical
(null, lightlike) distribution on $M$. Also, $g$ is called $r-$null
($r-$lightlike,$r-$degenerate) metric on $M$ the submanifold $M$ is said to be
$r-$lightlike ($r-$null, $r-$degenerate) submanifold of $\overline{M}$, with
nullity degree $r$ and is simply called null (lightlike) submanifold. Any
complementary (and hence orthogonal) distribution $\mathcal{S}(TM)$ of
$Rad(TM)$ in $TM$ is called a screen distribution. For a fixed screen
distribution $\mathcal{S}(TM)$ on $M$, the tangent bundle splits as

$TM=RadTM\oplus_{orth}\mathcal{S}(TM).$ (2.1)

Certainly, $\mathcal{S}(TM)$ is not unique, however it is canonically
isomorphic to the factor vector bundle $TM^{*}=TM/RadTM$ considered by Kupeli
[7]. $\mathcal{S}(TM)$ can be constructed by using the local equations of the
submanifold and therefore it enables us to obtain the main induced geometrical
objects: induced connection, second fundamental form, shap operator, $\cdots$
A screen transversal vector bundle $\mathcal{S}(TM^{\bot})$ on $M$ is any
(semi-Riemannian) complementary vector bundle of $Rad(TM)$ in $TM^{\bot}$. It
is obvious that $\mathcal{S}(TM^{\bot})$ is non-degenerate with respect to
$\overline{g}$ and $TM^{\bot}$ has the following orthogonal direct
decomposition

$TM^{\bot}=RadTM\oplus_{orth}\mathcal{S}(TM^{\bot}).$ (2.2)

$\mathcal{S}(TM)$ and $\mathcal{S}(TM^{\bot})$ are called a screen distibution
and a screen transversal vector bundle of $M$ respectively. As
$\mathcal{S}(TM)$ is not degenerate, let $\mathcal{S}(TM)^{\bot}$ be its
complementary orthogonal in $T\overline{M}\arrowvert_{M}.$ Then we have the
following decomposition

$T\overline{M}\arrowvert_{M}=\mathcal{S}(TM)\oplus_{orth}\mathcal{S}(TM)^{\bot}.$
(2.3)

Note that $\mathcal{S}(TM^{\bot})$ is a vector subbundle of
$\mathcal{S}(TM)^{\bot}$ and since both are non-degenerate we have the
following orthogonal direct decomposition

$\mathcal{S}(TM)^{\bot}=\mathcal{S}(TM^{\bot})\oplus_{orth}\mathcal{S}(TM^{\bot})^{\bot}.$

Since the theory of null submanifold M is mainly based on both
$\mathcal{S}(TM)$ and $\mathcal{S}(TM^{\bot})$, a null submanifold is denoted
by $(M,g,\mathcal{S}(TM),\mathcal{S}(TM^{\bot}))$ and we have four sub-cases
with respect to the dimension and codimension of $M$ and rank of $RadTM$:

  * (a)

$r-$null(lightlike) if $r<min(n,k)$;

  * (b)

coisotropic if $r=k<n$ (hence $\mathcal{S}(TM^{\perp})=\\{0\\}$);

  * (c)

isotropic if $r=n<k,$ (hence $\mathcal{S}(TM)=\\{0\\});$

  * (d)

totally null if $r=n=k$, (hence
$\mathcal{S}(TM)=\\{0\\}=\mathcal{S}(TM^{\perp})$).

We have seen from the above that the normal bundle $TM^{\bot}$ is orthogonal
but not a complement to $TM$, since it intersects the null tangent bundle
$RadTM$. This creates a problem as a vector of $T_{x}\overline{M}$ cannot be
decomposed uniquely into a component tangent of $T_{x}M$ and a component of
$T_{x}M^{\bot}$ . Therefore, the standard definition of second fundamental
forms and the Gauss-Weingarten formulae do not work. To deal with this
anomaly, one of the techniques used consists to split the tangent bundle
$T\overline{M}$ into four non-intersecting complementary (but not orthogonal)
vectors bundle.

######  Theorem 2.1.

[4] Let $\big{(}M,g,\mathcal{S}(TM),\mathcal{S}(TM^{\bot})\big{)}$ be an
$r-$lightlike submanifold of $(\overline{M},\overline{g})$ with $r>1.$ Suppose
$\mathcal{U}$ is a coordinate neighbourhood of $M$ and
$\\{\xi_{i}\\};i\in\\{1,\cdots,r\\}$ a basis of
$\Gamma(RadTM)\arrowvert_{\mathcal{U}}$. Then there exist smooth sections
$\\{N_{i}\\}$ of $\mathcal{S}(TM^{\bot})^{\bot}\arrowvert_{\mathcal{U}}$ such
that

$\overline{g}(N_{i},\xi_{i})=\delta_{ij},$ (2.4)

and

$\overline{g}(N_{i},N_{j})=0$ (2.5)

for any $i,j\in\\{1,...,r\\}.$

It follows that there exists a null transversal vector bundle $ltr(TM)$
locally spanned by $\\{Ni\\}$. By using (2.4) and (2.2) it is easy to check
that $B=\\{\xi_{1},\cdots,\xi_{r},N_{1},\cdots,N_{r}\\}$ is a basis of
$\Gamma{S}(TM^{\bot})^{\bot}_{|\mathcal{U}}$. The set of local sections
$\\{N_{i}\\}$ is not unique enven if one use the same vector bundle in
general. Let $tr(TM)$ be complementary called (but not orthogonal) vector
bundle to $TM$ in $T\overline{M}\arrowvert_{M}$. Then the following hold

$tr(TM)=ltr(TM)\oplus_{Orth}\mathcal{S}(TM^{\perp}),$ (2.6)
$T\overline{M}\arrowvert_{M}=\mathcal{S}(TM)\oplus_{Orth}(RadTM\oplus
ltr(TM))\oplus_{Orth}\mathcal{S}(TM^{\perp})=TM\oplus tr(TM).$ (2.7)

Let $\overline{\nabla}$ be the Levi-Civita connection on $\overline{M}$. As
$TM$ and $tr(TM)$ are complementary sub-bundles of
$T\overline{M}\arrowvert_{M}$, the Gauss and Weingarten formulae are

$\overline{\nabla}_{X}Y=\nabla_{X}Y+h(X,Y),$ (2.8)
$\overline{\nabla}_{X}V=-A_{V}X+\nabla_{X}^{t}V,$ (2.9)

$\forall X,Y\in\Gamma(TM),V\in\Gamma(tr(TM))$. $\nabla$ and $\nabla^{t}$ are
linear connections on $TM$ and the vector bundle $tr(TM)$ called the induced
linear connection and the transversal linear connection on $M$ respectively.

$\nabla$ is torsion-free linear connection. The components $\nabla_{X}Y$ and
$-A_{V}X$ belong to $\Gamma(TM)$, $h(X,Y)$ and $\nabla_{X}^{t}V$ to
$\Gamma(tr(TM))$. Also $h$ is a $\Gamma(tr(TM))$-valued symmetric bilinear
form on $\Gamma(TM)$ called the second fundamental form of $M$ with respect to
$tr(TM)$. $A$ is a $\Gamma(TM)$-valued bilinear form defined on
$\Gamma(tr(TM))\times\Gamma(TM)$ called shape operator of $M$. From the
geometry of Riemannian submanifolds [3] and non-degenerate submanifolds [12],
it is known that the induced connection on a non-degenerate submanifold is a
Levi-Civita connection. Unfortunately, in general, this is not true for a null
submanifold.

$(\nabla_{X}g)(X,Y)=\overline{g}(h^{l}(X,Y)+\overline{g}(h^{l}(X,Z),Y)$ (2.10)

$(\nabla_{X}^{t}\overline{g})(V,V^{{}^{\prime}})=-\\{\overline{g}(A_{V}X,V^{{}^{\prime}})+\overline{g}(A_{V^{{}^{\prime}}}X,V)\\}$
(2.11)

for all $X,Y,Z\in\Gamma(TM);,V,V^{{}^{\prime}}\in\Gamma(tr(TM))$. Thus, it
follows that the induced connection $\nabla$ is not a Levi-Civita connection.

According to the decomposition (2.6), let $L$ and $S$ denote the projection
morphisms of $tr(TM)$ onto $ltr(TM)$ and $\mathcal{S}(TM^{\perp})$
respectively, $h^{l}=L\circ h$, $h^{s}=S\circ h$,
$D^{l}_{X}V=L(\nabla_{X}^{t}V)$, $D_{X}^{s}V=S(\nabla_{X}^{t}V)$. The
transformations $D^{l}$ and $D^{s}$ do not define linear connections but
Otsuki connections on $tr(TM)$ with respect to the vector bundle morphisms $L$
and $S$. Then we have

$\overline{\nabla}_{X}Y=\nabla_{X}Y+h^{l}(X,Y)+h^{s}(X,Y)$ (2.12)

(2.4).

$\overline{\nabla}_{X}N=-A_{N}X+D_{X}^{l}N+D^{s}(X,N)$ (2.13)

$\overline{\nabla}_{X}W=-A_{W}X+D^{l}(X,W)+\nabla_{X}^{s}W$ (2.14)

$\forall X,Y\in\Gamma(TM),N\in\Gamma(ltr(M))$ and $W\in\Gamma(S(TM^{\perp})).$
Since $\overline{\nabla}$ is a metric connection, using (2.12)-(2.14) we have

$\overline{g}(h^{s}(X,Y),W)+\overline{g}(Y,D^{l}(X,W))=g(A_{W}X,Y)$ (2.15)
$\overline{g}(D^{s}(X,N),W)=\overline{g}(N,A_{W}X).$ (2.16)

As $h^{l}$ and $h^{s}$ are $\Gamma(ltr(TM))$-valued and
$\Gamma(\mathcal{S}(TM^{\bot}))$-valued respectively, we call them the null
second fundamental form and the screen second fundamental form of $M$.

Suppose $M$ is either r-null with $r<min\\{m,k\\}$ or coisotropic. Then, using
the decomposition (2.1) we get

$\nabla_{X}Y=\nabla_{X}^{*}PY+h^{*}(X,PY)$ (2.17)

$\nabla_{X}\xi=-A_{\xi}^{*}X+\nabla_{X}^{*t}\xi$ (2.18)

for any $X,Y\in\Gamma(TM)$ and $\xi\in\Gamma(RadTM)$, where
$\\{\nabla_{X}^{*}PY,A_{\xi}^{*}X\\}$ and
$\\{h^{*}(X,PY),\nabla_{X}^{*t}\xi\\}$ belong to $\Gamma(\mathcal{S}(TM))$ and
$\Gamma(Rad(TM))$ respectively. It follows that $\nabla^{*}$ and $\nabla^{*t}$
are linear connections on $\mathcal{S}(TM)$ and $RadTM$ respectively. On the
other hand, $h^{*}$ and $A^{*}$ are $\Gamma(RadTM)$-valued and
$\Gamma(\mathcal{S}(TM))$-valued bilinear forms on
$\Gamma(TM)\times\Gamma(\mathcal{S}(TM))$ and $\Gamma(RadTM)\times\Gamma(TM)$
Called the second fundamental forms of distributions $\mathcal{S}(TM)$ and
$Rad(TM)$ respectively.
For any $\xi\in\Gamma(RadTM)$ consider the linear operator

$A_{\xi}:\Gamma(TM)\rightarrow\Gamma(\mathcal{S}(TM));A_{\xi}^{*}X=A^{*}(\xi,X),,\forall,X\in\Gamma(TM)$

and call it the shape operator of $\mathcal{S}(TM)$ with respect to $\xi$.
Also, call $\nabla^{*}$ and $\nabla^{*t}$ the induced connections on
$\mathcal{S}(TM)$ and $RadTM$ respectively.

The second fundamental form and shape operator of non-degenerate submanifold
of a semi-Riemannian manifold are related by means of the metric tensor field.
Contrary in case of null submanifolds there are interrelations between
geometric objects induced by $tr(TM)$ on one side and geometric objects
induced by $S(TM)$ on the other side. More precisely,

$\overline{g}(h^{l}(X,PY)=g(A_{\xi}^{*}X,PY),\
\overline{g}(h^{*}(X,PY),N)=g(A_{N}X,PY),\
\overline{g}(h^{l}(X,\xi),\xi)=0,A_{\xi}^{*}\xi=0.$ (2.19)

From (2.19) as $h^{l}$ is symetric, it follows that the shape operator of
$\mathcal{S}(TM)$ is a self-adjoint operator on $\mathcal{S}(TM)$.

Next, consider a coordinate neighbourhood $\mathcal{U}$ of $M$ and let
$\\{N_{i},W_{\alpha}\\}$ be a basis of $\Gamma(tr(TM)\arrowvert_{M})$ where
$N_{i}\in\Gamma(ltr(TM)\arrowvert_{M}),i\in\\{1,...,r\\}$ and
$W_{\alpha}\in\Gamma(\mathcal{S}(TM^{\bot})\arrowvert_{\mathcal{U}}),\alpha\in\\{r+1,...,k\\}$.
Then ( 2.12) becomes

$\overline{\nabla}_{X}Y=\nabla_{X}Y+\sum_{i=1}^{r}h_{i}^{l}(X,Y)N_{i}+\sum_{\alpha=r}^{k}h_{i}^{s}(X,Y)W_{\alpha}.$
(2.20)

We call $\\{h_{i}^{l}\\}$ and $\\{h^{s}_{\alpha}\\}$ the local null second
fundamental forms and the local screen second fundamental forms of $M$ on
$\mathcal{U}$.

We recall the equations of Gauss, Codazzi and Ricci which play an important
role in studying differential geometry of non-degenerate submanifolds(See [4]
for details). Let $\overline{R}$ and $R$ denote the Riemannian curvature
tensors of $\overline{\nabla}$ and $\nabla$ on $\overline{M}$ and $M$
respectively. The Gauss equation are given by

$\displaystyle\overline{R}(X,Y)Z$ $\displaystyle=$ $\displaystyle
R(X,Y)Z+A_{h^{l}(X,Z)}Y-A_{h^{l}(Y,Z)}X+A_{h^{s}(X,Z)}Y-A_{h^{s}(Y,Z)}X$
$\displaystyle+(\nabla_{X}h^{l})(Y,Z)-(\nabla_{Y}h^{l})(X,Z)+D^{l}(X,h^{s}(Y,Z))$
$\displaystyle-D^{l}(Y,h^{s}(X,Z))+(\nabla_{X}h^{s})(Y,Z)-(\nabla_{Y}h^{s})(X,Z)$
$\displaystyle+D^{s}(X,h^{l}(Y,Z))-D^{s}(Y,h^{s}(X,Z))$

$\forall X,Y,Z,U\in\Gamma(TM).$ Therefore

$\displaystyle\overline{R}(X,Y,PZ,PU)$ $\displaystyle=$ $\displaystyle
R(X,Y,Z,PU)+\overline{g}(h^{*}(Y,PU),h^{l}(X,Z))$ (2.21)
$\displaystyle-\overline{g}(h^{*}(X,PU),h^{l}(Y,Z))+\overline{g}(h^{s}(Y,PU),h^{s}(X,Z))$
$\displaystyle-\overline{g}(h^{s}(X,PU),h^{s}(Y,Z)),$

Throughout, we consider that the submanifold is coisotropic that is $h^{s}=0$.
Let the differential $1-$forms $\omega_{i},i=1,\cdots,r(0<r<min\\{m,n\\})$
defined by:

$\omega_{i}(X)=\overline{g}(X,N_{i}),,\forall,X\in\Gamma(TM\arrowvert_{\mathcal{U}}),i\in\\{1,...,r\\}.$
(2.22)

Then any vector $X$ on $M$ is expressed on $\mathcal{U}$ as follows

$X=PX+\sum_{i=1}^{r}\omega_{i}(X)\xi_{i}$ (2.23)

where $P$ is the projection morphism of $\Gamma((TM))$ onto $\Gamma(S(TM))$.

##  3\. Normalization and Induced semi-Riemannian metric

Let $\varpi$ denote the $1$-form wich satisfying
$\varpi(.)=\overline{g}(N,.)$. Then, take

$\omega=f^{\star}{\varpi}$ (3.1)

to be its restriction to $M$ and the map $f:M\rightarrow\overline{M}$ the
inclusion map. Throughout, a screen distribution on $M$ is denote by
$\mathcal{S}(N)$. We define the associated semi-Riemannian metric on
$\overline{M}$ as

$\underline{g}=\overline{g}-\varpi\otimes\varpi\quad\mbox{and}\quad\widetilde{g}=i^{\star}\underline{g}=g-\omega_{i=1}^{r}\otimes\omega_{i=1}^{r}.$
(3.2)

######  Lemma 3.1.

Let $(M^{n},g)$ be a r-null submanifold in semi-Riemannian manifold
($\overline{M}^{n+1},\overline{g})$. Then, $\widetilde{g}$ is nondegenerate.

Proof. Take $V_{x}=\sum_{i=1}^{r}\xi_{{i}_{|}x}$, then
$\sum_{i=1}^{r}\widetilde{g}(U_{x},\xi_{{i}_{|}x})=0=\sum_{i=1}^{r}g(u,\xi_{{i}_{|}x})-\sum_{i=1}^{r}\omega_{i}(U)\omega_{i}(\xi_{{i}_{|}x})$,
$\ \sum_{i=1}^{r}\omega_{i}(U)=0$. Then

$U\in\mathcal{S}(N).$ (3.3) $0=\widetilde{g}(U,V)=g(U,V)\Rightarrow U\in
T_{x}M^{\bot}.$ (3.4)

From $(1)$ and $(2)\Rightarrow U\in\mathcal{S}(N_{x})\cap
T_{x}M^{\bot}=\\{0\\}$ and then $U=0$ which show $\widetilde{g}$ is
nondegenerate.

With $\sum_{i=1}^{r}\omega_{i}=\overline{g}(N_{i},.)$. It is nothing to see
that

$\widetilde{g}(\xi_{i},\xi_{j})=-\delta_{ij},\omega_{i}(.)=\widetilde{g}(\xi_{i},.).$
(3.5)

######  Lemma 3.2.

Let $(M,g,\mathcal{S}(N))$ be a r-null submanifold in semi-Riemannian
manifold, then $(M,\widetilde{g})$ is a semi-Riemannian manifold of
index$~{}q-1$. In particular, $(M,\widetilde{g})$ is Riemannian if
$(\overline{M},\overline{g})$ is Lorentzian and $(M,\widetilde{g})$ is
Lorentzian if $(\overline{M},\overline{g})$ is of index 2.

Proof. Take $(e_{2},\dots,e_{n+1})$ an orthonormal basis of $\mathcal{S}(N)$
for $\overline{g}$ and note that $(\xi,e_{2},\dots,e_{n+1})$ is an orthonormal
basis of $TM$ for the rigged metic. Elsewhere, it is a fact that
$\mathcal{S}(N)$ is a non degenerate distribution of index $q-1$ (see [4]).
Since $\widetilde{g}$ agrees with ${g}$ on $\mathcal{S}(N)$ and
$\widetilde{g}(\xi,\xi)=-\delta_{ij}$, it follows that $(M,\widetilde{g})$ is
a semi-Riemannian manifold of index $~{}q-1$.

######  Lemma 3.3.

For all $X,Y,Z\in\Gamma(TM)$, we have

$\displaystyle(\nabla_{X}\widetilde{g})(Y,Z)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{r}\omega_{i}(Y)[g(A_{\xi_{i}}^{\star}X,PZ)-g(A_{N_{i}}X,PZ)]$
$\displaystyle+\sum_{i=1}^{r}\omega_{i}(Z)[g(A_{\xi_{i}}^{\star}X,PY)-g(A_{N_{i}}X,PY)]$
$\displaystyle+2\sum_{i=1}^{r}\tau^{N_{i}}(X)\omega_{i}(Y)\omega_{i}(Z).$

Let $(M,g,\mathcal{S}(N))$ be a normalized null submanifold of a semi-
Riemannian manifold, then the integral curves of the rigged vector field $\xi$
are pregeodesic but not geodesic in general. The following lemma shows that in
case the normalization is a conformal vector field, then $\xi$ is
$\overline{g}$-geodesic.

######  Lemma 3.4.

Let $(M,g,\mathcal{S}(N))$ be a conformal normalized null submanifold of a
semi-Riemannian manifold such, then $\xi_{i}$ is $\overline{g}$-geodesic and
$\sum_{i=1}^{r}\tau(\xi_{i})=0$.

Proof. Firstly, note that
$\sum_{i=1}^{r}\tau(\xi_{i})=\sum_{i=1}^{r}\overline{g}(\overline{\nabla}_{\xi_{i}}N_{i},\xi_{i})$
$N_{i}$ being conformal, there exists a function $\lambda$ such that
$\overline{g}(\overline{\nabla}_{X}N_{i},Y)+\overline{g}(\overline{\nabla}_{Y}N_{i},X)=\lambda\overline{g}(X,Y).$
Take $X=Y=\sum_{i=1}^{r}\xi_{i}$ to get
$2\sum_{i=1}^{r}\overline{g}(\overline{\nabla}_{\xi_{i}}N_{i},\xi_{i})=\sum_{i=1}^{r}\lambda\overline{g}(\xi_{i},\xi_{i})=0;$
hence
$\sum_{i=1}^{r}\tau(\xi_{i})=\sum_{i=1}^{r}\overline{g}(\overline{\nabla}_{\xi_{i}}N_{i},\xi_{i})=0$.
It follows that
$\sum_{i=1}^{r}\overline{\nabla}_{\xi_{i}}\xi_{i}=-\sum_{i=1}^{r}\tau(\xi_{i})\xi_{i}=0.$

######  Definition 3.5.

A normalized null submanifold $(M,g,N)$ of a semi-Riemannian manifold
$(\overline{M},\overline{g})$ is said to have a conformal screen [1] if there
exists a non vanishing smooth function $\varphi$ on $M$ such that
$A_{N}=\varphi A_{\xi}^{\star}$ holds.

This is equivalent to saying that $g(A_{N}X,PY)=\varphi g(A_{\xi}^{\star}X,Y)$
for all tangent vector fields $X$ and $Y$. The function $\varphi$ is called
the conformal factor.

##  4\.  Relation between the null and the associated semi-Riemannian Geometry

The main focus of this section lies on deriving jump formulas for the various
curvature quantities, that is, how the Riemann and Ricci tensor and scalar
curvature of course the reason why this is of a particular interest lies in
physics, mainly general relativity,where such formulas might find applications
due to the Einstein field equations. In this respect, using the lemma (3.3)and
as connections $\widetilde{\nabla}$ and $\nabla$ are torsion free, we can
first prove the relation between the Levi-Civita connection of the non-
degenerate associate metric $\widetilde{g}$ and the rigged connection $\nabla$
in the following.

######  Proposition 4.1.

Let $(M^{n},g)$ be a $r-$normalized coisotropic null submanifold in semi-
Riemannian Manifold with rigged vector field $\xi$. Then, for all
$X,Y\in\Gamma(TM)$, we have

$\displaystyle\widetilde{\nabla}_{X}Y$ $\displaystyle=$
$\displaystyle\nabla_{X}Y+\frac{1}{2}\sum_{i=1}^{r}\big{[}2\langle
A_{\xi_{i}}^{*}X,Y\rangle-\langle A_{N_{i}}X,Y\rangle-\langle
A_{N_{i}}Y,X\rangle$
$\displaystyle+\sum_{i=1}^{r}\omega_{i}(X)\tau^{N_{i}}(Y)+\sum_{i=1}^{r}\omega_{i}(Y)\tau^{N_{i}}(X)\big{]}\xi_{i}+\sum_{i=1}^{r}\omega_{i}(X)(i_{Y}d\omega_{i})^{\sharp_{\omega_{i}}}+\sum_{i=1}^{r}\omega_{i}(Y)(i_{X}d\omega_{i})^{\sharp_{\omega_{i}}}.$

In particular for a closed normalization,

$\displaystyle\widetilde{\nabla}_{X}Y$ $\displaystyle=$
$\displaystyle\nabla_{X}Y+\frac{1}{2}\sum_{i=1}^{r}\big{[}2\langle
A_{\xi_{i}}^{*}X,Y\rangle-\langle A_{N_{i}}X,Y\rangle-\langle
A_{N_{i}}Y,X\rangle\big{.}$
$\displaystyle+\big{.}\sum_{i=1}^{r}\omega_{i}(X)\tau^{N_{i}}(Y)+\sum_{i=1}^{r}\omega_{i}(Y)\tau^{N_{i}}(X)\big{]}\xi_{i}.$

Let $\widetilde{R}$ and $R$ denote the Riemann curvature tensors of
$\widetilde{\nabla}_{\alpha}$ and $\nabla$ respectively. Using proposition
(4.1) we prove the following.

######  Proposition 4.2.

Let $(M,g^{n},N)$ be a $r-$closed normalized coisotropic null submanifold of
semi-Riemannian Manifold with rigged vector field $\xi$ . Then, for all
$X,Y,W\in\Gamma(TM)$ and $U\in TM^{\bot}$ we have,

$\displaystyle\widetilde{g}(\widetilde{R}(X,Y)Z,PW)$ $\displaystyle=$
$\displaystyle\langle R(X,Y)Z,PW\rangle$ (4.2)
$\displaystyle+\frac{1}{2}\sum_{i=1}^{r}\\{2g(A_{\xi}^{\star}X,Z)-\sum_{i=1}^{r}\langle
A_{N_{i}}X,Z\rangle-\sum_{i=1}^{r}\langle A_{N_{i}}Z,X\rangle$
$\displaystyle+\tau^{N_{i}}(X)\sum_{i=1}^{r}\omega_{i}(Z)+\tau^{N_{i}}(Z)\sum_{i=1}^{r}\omega_{i}(X)\\}g(A_{\xi}^{\star}(Y,W)$
$\displaystyle-\\{2g(A_{\xi}^{\star}Y,Z)-\sum_{i=1}^{r}\langle
A_{N_{i}}Y,Z\rangle-\sum_{i=1}^{r}\langle A_{N_{i}}Z,Y\rangle$
$\displaystyle+\tau^{N_{i}}(Y)\sum_{i=1}^{r}\omega_{i}(Z)+\tau^{N_{i}}(Z)\sum_{i=1}^{r}\omega_{i}(Y)\\}g(A_{\xi}^{\star}X,W)\\},$
$\displaystyle\widetilde{g}(\widetilde{R}(X,Y)Z,U)$ $\displaystyle=$
$\displaystyle-\langle R(X,Y)U,PZ\rangle$ (4.3)
$\displaystyle-\frac{1}{2}\sum_{i=1}^{r}\big{[}\langle A_{N_{i}}U,Y\rangle
g(A_{\xi}^{\star}X,Z)-\langle A_{N_{i}}U,X\rangle
g(A_{\xi}^{\star}Y,Z)\big{]}$
$\displaystyle-\frac{1}{2}\sum_{i=1}^{r}\big{[}\tau^{N_{i}}(X)g(A_{\xi}^{\star}Y,Z)-\tau_{i}^{N}(Y)g(A_{\xi}^{\star}X,Z)\big{]}\omega_{i}(U),$

## References

  * [1] Atindogbe C. and K. L. Duggal, K. L., Conformal sreen on lightlike submanifolds, Int.J.of Pure and Appied Math.,11(2004), 421-442.
  * [2] Atindogbé, C., Gutiérrez, M. and Hounnonkpe, R., New Properties on Normalized null hypersurfaces,
  * [3] B. Y . Chen,Geometry of Submanifolds, Marcel Dekker, New York, (1973).
  * [4] Duggal, K. L. and Bejancu, A. Lightlike Submanifolds of semi-Riemannian Manifolds and Applications, Kluwer Academic, 364, 1996.
  * [5] Ferdinand Ngakeu and Hans Fotsing Tetsing, $\alpha$-Associated Metrics on Rigged Null Hypersurfaces arXiv:1804.09036v1[math.DG] 22 Apr 2018.
  * [6] Gutiérrez, M. and Olea, B. ,Induced Riemannian structures on null submanifolds, Math. Nachr., 289 (2015), 1219-1236.
  * [7] Kupeli D., On null Submanifolds in spacetimes, Geometry Dedicata, 23(1) (1987), 33-51.
  * [8] HONG,S.P and Tripathi M.M, On Ricci curvature of submanifolds, Int. J. Pure Appl. Math.SCi. 2(2005), no. 2, 227-245.
  * [9] Karimumuryango, M., Chen’s Ricci inequalities and topological obstructions on null submanifolds of a Lorentzian manifold, Journal of Inequalities and Applications (2018).
  * [10] K1l1ç, E.,Gülbahar, M., On the sectional curvature of lightlike submanifolds, J.of Inequal. and Appl. 2016.1(2016):57.
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  * [12] O. Neill, B., Semi-riemannian Geometry With Applications to Relativity. Copright 1983; by Academic Press.


# Deep Generative Domain Adaptation with Temporal Relation Knowledge for
Cross-User Activity Recognition

Xiaozhou Ye1 Kevin I-Kai Wang2
1 2Department of Electrical, Computer, and Software Engineering, The
University of Auckland, Auckland, New Zealand
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In human activity recognition (HAR), the assumption that training and testing
data are independent and identically distributed (i.i.d.) often fails,
particularly in cross-user scenarios where data distributions vary
significantly. This discrepancy highlights the limitations of conventional
domain adaptation methods in HAR, which typically overlook the inherent
temporal relations in time-series data. To bridge this gap, our study
introduces a Conditional Variational Autoencoder with Universal Sequence
Mapping (CVAE-USM) approach, which addresses the unique challenges of time-
series domain adaptation in HAR by relaxing the i.i.d. assumption and
leveraging temporal relations to align data distributions effectively across
different users. This method combines the strengths of Variational Autoencoder
(VAE) and Universal Sequence Mapping (USM) to capture and utilize common
temporal patterns between users for improved activity recognition. Our
results, evaluated on two public HAR datasets (OPPT and PAMAP2), demonstrate
that CVAE-USM outperforms existing state-of-the-art methods, offering a more
accurate and generalizable solution for cross-user activity recognition.

##  1 Introduction

Human Activity Recognition (HAR) is an essential field within Human-Computer
Interaction, ubiquitous computing Ma et al. (2019), and the Internet of Things
Abdallah et al. (2018). It involves identifying human activities using sensor
data and contextual information Qian et al. (2019). HAR finds applications in
diverse areas like medical treatment, assisted living, fitness, security, and
home automation. Current HAR methods, especially those processing time-series
sensor data, operate under the assumption that training and testing data are
drawn from the same distribution, meaning they are independent and identically
distributed (i.i.d.) Wilson and Cook (2020). This approach assumes that a
model trained on source data will perform similarly on target data, as long as
both sets of data are from the same domain with consistent features and
distribution characteristics. However, this is often not the case in real-
world scenarios, where training and testing datasets may have different
distributions due to data heterogeneity, also known as the out-of-distribution
(o.o.d.) problem. This discrepancy can lead to reduced model performance when
applied to new, unseen data Patel et al. (2015).

Transfer learning, specifically domain adaptation Pan and Yang (2009), is a
method that addresses this data heterogeneity issue. Its core principle is to
identify and leverage common knowledge to minimize differences in data
distribution between the source and target domains. Current research in domain
adaptation primarily focuses on static data, like images, where each sample is
considered i.i.d. within its domain Patel et al. (2015). This assumption is
extended to time-series data in HAR, treating each data segment as i.i.d. in
its domain Soleimani and Nazerfard (2021). However, this approach overlooks
the inherent temporal relations in time-series data, where consecutive data
segments are often interdependent. For example, in activity recognition,
activities like walking consist of interconnected sub-activities (e.g.,
lifting the leg, moving it forward, placing the foot down), each with temporal
dependencies. The prevalent oversight in current research methods is the
insufficient attention to these temporal dependencies, which are crucial for
understanding and accurately recognizing human activities. This lack of focus
results in models that fail to capture the common sequential dynamics of
activity data, leading to suboptimal performance on domain adaptation,
especially when applied across diverse users.

This paper introduces a novel approach called Conditional Variational
Autoencoder with Universal Sequence Mapping (CVAE-USM) for time series domain
adaptation in HAR. This method focuses on capturing temporal relations in
activity data by USM encoding, enabling better adaptation between different
users’ data. By preserving these temporal relations, our approach effectively
regularizes the adaptation process, enhancing the alignment of sub-activity
distributions across users. This results in improved performance in domain
adaptation tasks. Moreover, We propose a new generative model of variational
autoencoder (VAE) based architecture that captures temporal relation knowledge
for cross-user HAR, which enhances the generalized capability of VAE
generative model. Our extensive testing on two public HAR datasets shows that
this method outperforms existing approaches in time series domain adaptation,
particularly in cross-user scenarios.

The paper is organized as follows: Section 2 reviews related work in HAR,
transfer learning, and domain adaptation. Section 3 details our proposed CVAE-
USM method, focusing on capturing and aligning temporal relations. Section 4
outlines our experimental setup and compares our method against existing
approaches. Finally, Section 5 concludes the paper and suggests directions for
future research.

##  2 Related work

###  2.1 Cross-user human activity recognition

Human Activity Recognition (HAR) is a fundamental aspect of ubiquitous
computing, playing a vital role in enhancing daily human activities. Its
primary goal is to recognize and analyze human behaviors by interpreting high-
level knowledge derived from multi-modal sensor data and contextual
information. Based on sensor modalities, HAR can be categorized into five
distinct types: Smartphones/wearable sensors-based HAR, ambient sensors-based
HAR, device-free sensors-based HAR, vision-based sensors-based HAR, and other
modality sensors-based HAR Chen et al. (2021)Lentzas and Vrakas (2020). This
paper specifically delves into HAR using wearable sensors.

Within machine learning research, sensor-based HAR is often approached as a
time series classification problem Andreas et al. (2014). Various
classification models, including ensemble learning Sekiguchi et al. (2020),
SVM Bulling et al. (2012), and HMM Amft et al. (2005), have been proposed to
solve HAR challenges. With advancements in deep learning, numerous state-of-
the-art results have been achieved for a wide range of tasks. Deep learning-
based HAR techniques excel in learning high-level features and autonomously
extracting features from extensive data sets Qian et al. (2019). However,
these approaches predominantly rely on the assumption that the training and
testing data are drawn from the same distribution, meaning the data is
considered independent and identically distributed (i.i.d.) Wilson and Cook
(2020). This assumption often fails in real-world applications, where the
collected training and testing datasets are out-of-distribution (o.o.d.). Our
focus in this paper is on addressing the challenges associated with sensor-
based HAR in the context of o.o.d. data.

There are several categories of sensor-based HAR o.o.d. challenges: First,
differences in data may result from the use of various sensor types,
platforms, manufacturers, and modalities, leading to diverse data formats and
distributions Xing et al. (2018). Second, the data pattern might change over
time, a phenomenon known as concept drift Lu et al. (2018). For example, a
person’s walking pattern could be influenced by changes in health status.
Third, behavioral differences between individuals can be significant Ye and
Wang (2023); for instance, walking pace may vary from person to person.
Fourth, the positioning of physical sensors on the body Rokni and Ghasemzadeh
(2018) or the layout of environmental sensors in smart homes Sukhija and
Krishnan (2019) can also lead to different data distributions. Our research
specifically targets sensor-based HAR o.o.d. problems stemming from behavioral
differences among individuals.

###  2.2 Transfer learning and domain adaptation

Transfer learning enables models to learn from one or more source domains and
then apply this knowledge to related target domains lacking labelled data.
This process is crucial to address the out-of-distribution (o.o.d.) problem by
minimizing distribution differences between the source and target domains.
Domain adaptation, a subset of transfer learning, specifically tackles the
o.o.d. issue while maintaining the same task across source and target domains.
This approach typically involves leveraging labelled data from the source
domain and unlabeled data from the target domain Wilson and Cook (2020).

Domain adaptation has experienced significant growth, particularly in the area
of feature-based transfer learning Pan and Yang (2009). Techniques like
Subspace Alignment (SA) Fernando et al. (2013) focus on finding similarities
between the feature subspaces of the source and target domains, using
principal component analysis and linear transformations. Optimal Transport for
Domain Adaptation (OTDA) Flamary et al. (2016), is grounded in optimal
transport theory, aiming to establish a cost-effective correspondence between
different domains. Deep domain adaptation has also seen advancements with
approaches like the Domain Adversarial Neural Network (DANN) Ganin et al.
(2016), which employs adversarial learning to train a model so that its
features cannot be used to distinguish between source and target domains. This
promotes the generation of domain-invariant features. FedMAT Shen et al.
(2022) works by treating each individual’s data as a distinct task within a
federated learning system. It uniquely combines a central shared feature
representation network with individual-specific networks equipped with
attention modules in decentralized nodes, enabling the learning of both shared
and individual-specific features from multi-modal sensor data.

These feature-based methods primarily target static data like images, often
applying the same framework to time series data Lu et al. (2021)Li et al.
(2020). However, this approach may be less effective for time series data,
where temporal relationships are crucial. Current domain adaptation strategies
often overlook these temporal relations, leading to models that may not
perform reliably, especially in cross-user HAR applications where models
trained on data from one user are applied to others. Incorporating temporal
relation knowledge could enhance the identification of commonalities between
users, improving domain adaptation in sensor-based HAR. This paper focuses on
exploring and integrating temporal relation knowledge to enhance domain
adaptation effectiveness in cross-user HAR.

##  3 Method

###  3.1 Problem formulation

In a cross-user HAR problem, a labelled source user $\displaystyle
S^{Source}=\left\\{\left(x_{i}^{Source},\
y_{i}^{Source}\right)\right\\}_{i=1}^{n^{Source}}$ drawn from a joint
probability distribution $\displaystyle P^{Source}$ and a target user
$\displaystyle S^{Target}=\left\\{\left(x_{i}^{Target},\
y_{i}^{Target}\right)\right\\}_{i=1}^{n^{Target}}$ drawn from a joint
probability distribution $\displaystyle P^{Target}$, where
$\displaystyle{n^{Source}}$ and $\displaystyle{n^{Target}}$ are the number of
source and target samples respectively. $\displaystyle S^{Source}$ and
$\displaystyle S^{Target}$ have the same feature spaces (i.e. the set of
features that describes the data from sensor readings) and label spaces (i.e.
the set of activity classes). The source and target users have different
distributions, i.e., $\displaystyle P^{Source}\neq P^{Target}$, which means
that even for the same activity, the sensor readings look different between
the two users. Given the labelled source user data and unlabelled target user
data, the goal is to obtain the labels
$\displaystyle\left\\{\left(y_{i}^{Target}\right)\right\\}_{i=1}^{n^{Target}}$
for the target user activities.

###  3.2 Conditional Variational Autoencoder with Universal Sequence Mapping

In this study, we present the Conditional Variational Autoencoder with
Universal Sequence Mapping (CVAE-USM), a novel approach for cross-user HAR.
The foundation of this method is the understanding that human physical
movements are influenced by preceding actions, and these temporal dependencies
are generally consistent across different individuals. By capturing these
shared temporal patterns in human activity time series data, our model aims to
enhance its performance and generalizability for cross-user HAR tasks.

Figure 1: CVAE-USM Learning Process.

####  3.2.1 Generative Model Architecture

Our approach utilizes a generative model framework, as depicted in the CVAE-
USM Learning Process shown in Figure 1, to achieve effective model
generalization. Generative models are advantageous in that they can produce
new data samples that are similar to the training data, aiding in data
augmentation and improving generalization capabilities. Even in scenarios
where some class labels are missing during training, generative models focus
on modelling the data distribution itself, which is more effective in
uncovering the underlying data patterns and structures Ruthotto and Haber
(2021). For cross-user HAR, where labels for target users are not available
(or only partially available), generative models are particularly useful as
they are capable of modelling user-invariant activity data distributions. They
achieve this by encoding data into a compact latent space, capturing the
underlying factors or patterns in the data Van Kasteren et al. (2010).

In implementing our method, we employ the Variational Autoencoder (VAE)
architecture Kingma and Welling (2014) as the foundational generative model.
The VAE aims to learn a structured latent space where similar data samples are
closely aligned, such as patterns found in time series data, typically
following a Gaussian distribution. In the VAE structure, input data (like
sensor features in HAR) is compressed into a lower-dimensional latent space
via an encoder. This latent space is represented probabilistically with mean
and variance parameters, encapsulating intrinsic characteristics of human
activities. The VAE employs the reparameterization trick for differentiability
during training, sampling from the latent space based on Gaussian
distributions and then using a decoder to reconstruct the original input. This
method enables the capturing of complex temporal activity patterns, making the
VAE an ideal model for cross-user HAR tasks.

The VAE’s sampling process offers several benefits for generalization and
domain adaptation. The randomness introduced through sampling acts as a
regularization form, preventing overfitting and enhancing generalization. By
encoding data into a distribution instead of a deterministic decision boundry
and encouraging the latent variables to approximate a Gaussian distribution
(via the Kullback–Leibler divergence), the latent space becomes more
structured and continuous Rybkin et al. (2021). In such a structured latent
space, alignment techniques like Gradient Reversal Layer (GRL) adversarial
learning, which we will discuss later, can be more effectively applied for
domain adaptation in HAR.

Moreover, we implement an adversarial learning strategy, specifically
utilizing the Gradient Reversal Layer (GRL) technique Ganin et al. (2016) as
the blue parts in Figure 1. This strategy is integral to modifying the model’s
loss functions related to domain constraints (i.e. confusing the data
distributions between source and target users). The GRL adversarial learning
operates by reversing the gradients during the training phase. The primary
goal of this technique is to ensure that the features learned by the model are
not solely tied to the specific user but also lean towards a generalized
model. Through this approach, our model can generate user-invariant
representations. Therefore, this strategy is integrated as a constraint
condition to the original CVAE for model learning.

Furthermore, after acquiring the sampled features, they are input into a
process focusing on the source user’s activity class constraint as the grey
parts in Figure 1. This involves using a linear transformation to shape how
the data distribution is generated. Specifically, this step takes into
consideration the activity class labels associated with the source user. This
means that the transformation is guided by the types of activities the source
user performed, helping to tailor the model’s understanding of the data based
on these specific activities. Therefore, this source user class constraint is
integrated as another constraint condition to the original CVAE for model
learning.

####  3.2.2 Cross-user Temporal Relations Alignment

The key element of the CVAE-USM method is the cross-user temporal relations
alignment. This CVAE constraint condition is shown in the yellow parts in
Figure 1. In this component, Gaussian Mixture Model (GMM), Universal Sequence
Mapping (USM), and Wasserstein Distance techniques are included. Each
component plays a pivotal role in capturing and adapting temporal relation
knowledge for domain adaptation. The overall process of cross-user temporal
relations alignment is described in Algorithm 1.

Algorithm 1 Cross-user Temporal Relations Alignment

Input: Mean values from CVAE encoder for source ($\mu_{s}$) and target
($\mu_{t}$) users, number of components ($K$) in GMM.

Output: Aligned temporal relation features, Wasserstein distance ($W$)
temporal relation loss.

Begin Algorithm

1\. Sub-Activity Identification with GMM:

  * •

Apply GMM to sensor data for source and target users:
$GMM_{init}(\\{\mu_{s},\mu_{t}\\},K)$.

  * •

This step is to identify and decompose complex activities into simpler sub-
activities.

2\. Temporal Relations Encoding with USM:

  * •

Transform sub-activities into feature space using USM:
$F_{s}=\mathcal{U}(Seq_{s})$, $F_{t}=\mathcal{U}(Seq_{t})$.

  * •

This step aims to capture and encode temporal relation for activities. Here,
$\mathcal{U}$ denotes the USM function, and $Seq_{s}$, $Seq_{t}$ are sequences
of encoded sub-activities for source and target, respectively, with each value
in the sequences encoded into the set $\\{0,K-1\\}$.

3\. User-Specific Temporal Feature Analysis:

  * •

Perform separate GMM clustering on the USM-encoded features for source
($F_{s}$) and target ($F_{t}$) users: $GMM_{s}(F_{s},K)$ and
$GMM_{t}(F_{t},K)$.

  * •

This step is to analyze the variations and similarities between the sub-
activities’ data distributions across users. Let $P_{s}$ and $P_{t}$ denote
the resulting distributions from $GMM_{s}$ and $GMM_{t}$ respectively,
capturing distinct temporal patterns of each user group.

4\. Data Distribution Alignment with Wasserstein Distance:

  * •

Minimize the Wasserstein distance between source and target distributions:
$W(P_{s},P_{t})=\inf_{\gamma\in\Pi(P_{s},P_{t})}\mathbb{E}{(x,y)\sim\gamma}[|x-y|]$.

  * •

This step aims to adjust model parameters to align $P{s}$ and $P_{t}$,
focusing on common temporal relations.

End Algorithm

The first step is the Gaussian Mixture Model (GMM) for sub-activity
identification. GMM Zhang et al. (2021) is a probabilistic model for
representing normally distributed subpopulations within an overall population.
It is defined as:

$p(x)=\sum_{k=1}^{K}\pi_{k}\mathcal{N}(x|\mu_{k},\Sigma_{k})$ (1)

Where $x$ is the data point. $K$ is the number of Gaussian distributions in
the mixture. $\pi_{k}$ are the mixture weights, with $\sum_{k=1}^{K}\pi_{k}=1$
and $\pi_{k}\geq 0$. $\mathcal{N}(x|\mu_{k},\Sigma_{k})$ denotes the Gaussian
distributions with mean $\mu_{k}$ and covariance $\Sigma_{k}$.

GMM identifies sub-activities hidden in the sensor data, effectively
decomposing complex activities into simpler sub-activities which represent
common knowledge across different users. This model identifies inherent sub-
structures within the dataset, irrespective of their originating domain
(source or target). This decomposition is vital for two reasons: firstly, it
simplifies the complex temporal patterns into more manageable segments, and
secondly, the common sub-activities are used for subsequent temporal relation
capture and analysis between the source and target domains.

Following the identification of sub-activities, the next step is using USM for
temporal feature encoding. This step is crucial for transforming the
identified sub-activities into a feature space that encapsulates both their
spatial structural and temporal characteristics. USM Almeida and Vinga (2002)
is a technique for encoding sequences to capture their inherent structure and
temporal relations. It can be mathematically represented as:

$F=\mathcal{U}(Seq)$ (2)

Where $F$ represents the feature space encoding the sequence. $\mathcal{U}$ is
the USM function. $Seq$ is the sequence of sub-activities.

USM plays a pivotal role in maintaining the integrity of temporal relations
within the activity sequences. This encoding not only represents the sequence
of sub-activities but also preserves the temporal dynamics associated with
each activity – that is, the temporal order and sequence of sub-activities,
particularly when analyzing complex human behaviours, thus offering a more
precise representation of human activities. This encoding respects the
chronological order of events, ensuring that the temporal dynamics—such as the
duration, sequence, and intervals between sub-activities—are accurately
captured and reflected in the feature vectors. This is particularly important
for understanding how sub-activities unfold over time.

By effectively encoding the temporal order, USM enables the identification of
recurrent temporal patterns and relationships within the data. This can
include regular sequences of activities, the typical duration of certain
activities, or the common intervals between sub-activities. Moreover, USM-
encoded features facilitate the comparison of temporal sequences across
different users. By providing a uniform method of encoding, USM allows for the
direct comparison of temporal features, supporting the data distribution
alignment of temporal relations across users. This step paves the way for the
following temporal relation alignment across users.

The next step is user-specific feature analysis. The USM-encoded features, now
rich with temporal relational information, undergo separate GMM clustering for
the source and target domains. This bifurcation is key to understanding how
temporal patterns manifest uniquely in each domain. By analyzing these user-
specific GMMs, the approach gains insights into the variations and
similarities in activity patterns across users. This comparative analysis is
essential for identifying user-specific differences in temporal activity
structures.

Finally, Wasserstein Distance is applied for data distribution alignment
across users with the above-learned temporal relation knowledge. The
Wasserstein Distance Wu et al. (2020), also known as the Earth Mover’s
Distance, measures the distance between two probability distributions. It is
defined as:

$W(P_{s},P_{t})=\inf_{\gamma\in\Pi(P_{s},P_{t})}\mathbb{E}_{(x,y)\sim\gamma}[\|x-y\|]$
(3)

Where $W(P_{s},P_{t})$ is the Wasserstein Distance between two distributions.
$P_{s}$ and $P_{t}$ are the probability distributions of source and target
users being compared. $\gamma$ represents a joint distribution with marginals
$P_{s}$ and $P_{t}$. $\Pi(P_{s},P_{t})$ denotes the set of all possible joint
distributions $\gamma(x,y)$ with marginals $P_{s}$ and $P_{t}$.
$\mathbb{E}_{(x,y)\sim\gamma}[\|x-y\|]$ is the expected value of the Euclidean
distance between sampled points $x$ and $y$ from $\gamma$.

Wasserstein distance is computed using the means of the GMMs from the source
and target users. This distance metric quantifies the disparity in the
temporal feature distributions between the two domains. Minimizing the
Wasserstein distance is used to align the source and target user distributions
in the temporal relation view based on the idea that source and target users
should have common temporal relation sub-activities. This alignment is central
to the model’s ability to generalize and accurately recognize activities
across users.

##  4 Experiments

###  4.1 Datasets and Experimental setup

For validating our cross-user domain adaptation method in sensor-based HAR, we
employed two widely recognized public datasets as outlined in Table 1. Our
focus was on the practical application scenario involving a smartwatch. To
this end, we only utilized sensor data from accelerometers and gyroscopes
positioned on the right lower arm. Below, we provide a brief overview of each
dataset, highlighting their key aspects.

Table 1: Two sensor-based HAR datasets information.

Dataset |  Subjects |  #Activities |  Common Activities
---|---|---|---
OPPT |  S1, S2, S3 |  4 |  1 standing, 2 walking, 3 sitting, 4 lying
PAMAP2 |  1, 5, 6 |  11 |  1 lying, 2 sitting, 3 standing, 4 walking, 5 running, 6 cycling, 7 Nordic walking, 8 ascending stairs, 9 descending stairs, 10 vacuum cleaning, 11 ironing

The OPPORTUNITY (OPPT) dataset Chavarriaga et al. (2013) contains recordings
of people doing morning activities in a natural way, without strict guidance,
recorded at 30 Hz. The Physical Activity Monitoring (PAMAP2) dataset Reiss and
Stricker (2012) features over 10 hours of data recorded at 100 Hz, with
subjects performing specific, pre-defined activities.

In the experimental setup for cross-user HAR, the sliding window technique is
employed for data segmentation. This widely-used approach in sensor-based HAR
involves creating fixed time intervals of 3 seconds with a 50% overlap between
windows, in line with standard practices in HAR tasks Wang et al. (2018). The
study compares four methods, divided into traditional domain adaptation and
deep domain adaptation categories. Notably, the TrC method requires a few
target domain labels for fine-tuning, unlike other methods which do not
require any labelling in the target domain.

Traditional Domain Adaptation Methods:

CORAL (Covariance Alignment) Sun et al. (2016): Aims to align the covariance
of feature layers across domains, fostering domain-agnostic features.

SOT (Substructure Optimal Transport) Lu et al. (2021): Focuses on underlying
domain structures to facilitate detailed substructure mapping, balancing broad
and specific mapping.

Deep Domain Adaptation Methods:

DANN (Domain-Adversarial Neural Network) Ganin et al. (2016): A deep learning
approach using adversarial training to produce features indistinguishable
between domains by a discriminator.

TrC (Temporal Regularized CNNs) Rokni et al. (2018): Employs CNNs for
discerning temporal features, trained on source domain data and fine-tuned on
the target domain.

Hyper-parameters tuning is implemented on these methods with the methodology
suggested by Flamary et al. Flamary et al. (2016), specifically to prevent
overfitting during testing. We divide the target user’s data into two parts: a
validation subset and a test subset. The validation subset is utilized to
fine-tune the hyper-parameters to achieve the highest accuracy. After
optimizing these hyper-parameters, we evaluate the performance of the model on
the test subset. The key metric we use for evaluation is the classification
accuracy specific to the target user.

###  4.2 Cross-user HAR performance results

We evaluate the performance of various methods on the OPPT and PAMAP2
datasets. Each method is tested for its ability to transition between
individual users. Within each dataset, three users are selected randomly, and
a one-to-one cross-user HAR task is conducted for every possible user pair.

Figure 2: OPPT dataset classification results.

In the results of OPPT dataset as shown in Figure 2, CVAE-USM stands out by
achieving almost 100% accuracy in all testing scenarios. This exceptional
performance underscores CVAE-USM’s adaptability and effectiveness in various
conditions specific to the OPPT dataset. TrC also achieves a reasonably good
performance, consistently reaching accuracy levels in the low to mid-80%
range. This indicates its competence in the OPPT dataset, though it doesn’t
match the superior performance of CVAE-USM. Another method DANN demonstrates
moderate performance, with accuracy generally fluctuating between the mid-70%
and mid-80% range. SOT and CORAL exhibit slightly lower performance levels
compared to the others. One notable distinction contributing to the
performance discrepancies is the utilization of temporal information. CVAE-USM
effectively leverages temporal data, contributing significantly to its
superior accuracy. In contrast, TrC considers the temporal features , however
it does not exploit temporal information as effectively as CVAE-USM, which
partially explains its lower performance metrics.

Figure 3: PAMAP2 dataset classification results.

In the results of PAMAP2 dataset as shown in Figure 3, CVAE-USM again
dominates, with accuracy consistently above 70% and peaking at 82.12% in
certain scenarios. This dominance indicates CVAE-USM’s strong adaptability and
learning capabilities across different conditions in the PAMAP2 dataset. TrC,
SOT and DANN show respectable performances in specific scenarios, suggesting
that they may be better adapted to situations with minimal variance between
training and testing data or where data patterns are more consistent and less
complex. In contrast, CORAL struggles significantly, with their accuracies
mainly ranging between 30%-40%. This poor performance reflects their limited
adaptability to the PAMAP2 dataset. A critical aspect contributing to these
varied performances is the utilization of temporal relationships within the
data. CVAE-USM’s effective use of temporal information likely contributes to
its outstanding accuracy, as it can capture and learn from the dynamics and
temporal patterns inherent in the PAMAP2 dataset. In contrast, models like SOT
and DANN do not leverage temporal data, which could explain their not
outstanding results.

Overall, across both the OPPT and PAMAP2 datasets, CVAE-USM consistently
achieves the highest accuracy, proving its adaptability, robustness, and
effectiveness in handling cross-user HAR tasks. TrC shows a reasonable
performance across both datasets, indicating its potential adaptability to
different scenarios, albeit not at the level of CVAE-USM. CORAL, SOT, and DANN
exhibit varied and mixed results, suggesting that their effectiveness and
adaptability are conditional based on specific dataset characteristics and
scenarios. Some methods, particularly CORAL, show significant struggles in
adapting to these datasets, highlighting the need for further development to
enhance their effectiveness in time series data that involves temporal
relation knowledge.

###  4.3 Effect of temporal information

In this section, we analyze the impact of integrating temporal relation
knowledge into cross-user HAR, concentrating on the OPPT and PAMAP2 datasets.
The comparison involves SOT, a traditional domain adaptation method; DANN,
known for its deep domain adaptation approach; and our CVAE-USM, which is
specifically designed for time series domain adaptation.

Figure 4 displays the confusion matrices for CVAE-USM, SOT, and DANN,
reflecting their average performance across the OPPT and PAMAP2 datasets. The
matrices use color gradients, with warmer colors (closer to yellow) indicating
higher values. The x-axis represents predicted activities, while the y-axis
corresponds to the actual activities. The diagonal line from the top left to
the bottom right of these matrices signifies correct predictions, where the
predicted activity aligns with the true activity.

Figure 4: Confusion matrices of CVAE-USM, SOT, and DANN methods for the
average performance of tasks in the OPPT and PAMAP2 datasets.

In these analyses, all methods exhibit pronounced values along their
diagonals, indicating accurate recognition of most activities and underscoring
their effectiveness in cross-user scenarios. Common activities such as
’lying’, ’walking’, ’standing’ and ’cycling’ are efficiently identified by all
methods. The reason could be attributed to their distinct movement patterns,
making them easily distinguishable from other activities. However, confusion
arises in differentiating activities with inherent similarities, such as
activities like ’ascending stairs’ and ’descending stairs’.

CVAE-USM, in particular, excels in identifying activities with complex
temporal relations like ’Nordic walking’ and ’vacuum cleaning’ as the yellow
squares shown in Figure 4, with higher accuracy compared to SOT and DANN.
Additionally, while SOT and DANN show dispersed classification outcomes, CVAE-
USM exhibits a more focused and clear classification pattern due to its
effective handling of temporal states and extraction of sub-activities as the
red squares shown in Figure 4. CVAE-USM demonstrates a superior ability to
distinguish activities like ’standing’, ’sitting’, and ’walking’ as the green
squares shown in Figure 4, possibly due to its nuanced capture of temporal
relation context and subtle sensor data variations. Moreover, transitional
movements within activities, often overlooked by SOT and DANN, are effectively
captured by CVAE-USM. For example, repositioning a vacuum cleaner during
’vacuum cleaning’ is better understood and classified by CVAE-USM, thanks to
its comprehension of temporal relations.

In conclusion, the use of temporal relation knowledge in CVAE-USM effectively
reduces the variance in activity patterns between different users, especially
in dynamic activities with complex temporal relations. Extracting temporal
relation knowledge from time series data enhances the robustness and accuracy
of activity recognition. This approach is especially beneficial for activities
with rhythmic or sequential traits, allowing for a more nuanced understanding
of various activities and highlighting the importance of temporal knowledge in
improving cross-user activity recognition capabilities.

##  5 Conclusion

In this study, we present a new approach called CVAE-USM for cross-user HAR
tasks from time series data. This method is particularly effective in adapting
to different users’ data by focusing on the temporal relation knowledge of
activities. CVAE-USM uses VAE generative model and adversarial learning
techniques to align different users’ data distributions, leading to more
accurate activity recognition.

The validation is implemented on two public datasets show that CVAE-USM
outperforms existing methods in recognizing activities across various users.
This success demonstrates CVAE-USM’s potential to analyse time series data in
HAR. Overall, CVAE-USM represents its advantages in understanding temporal
relations and adapting to the complex activity data from different users. In
future, we aim to explore the application of CVAE-USM to more complex or less
structured activities, to evaluate its adaptability and robustness.

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††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Supplementary Information: Spontaneous localization at a potential saddle
point from edge state reconstruction in a quantum Hall point contact

Liam A. Cohen  Department of Physics, University of California at Santa
Barbara, Santa Barbara CA 93106, USA  Noah L. Samuelson  Department of
Physics, University of California at Santa Barbara, Santa Barbara CA 93106,
USA  Taige Wang  Department of Physics, University of California, Berkeley,
California 94720, USA  Material Science Division, Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA  Kai Klocke  Department of
Physics, University of California, Berkeley, California 94720, USA  Cian C.
Reeves  Department of Physics, University of California at Santa Barbara,
Santa Barbara CA 93106, USA  Takashi Taniguchi  International Center for
Materials Nanoarchitectonics, National Institute for Materials Science, 1-1
Namiki, Tsukuba 305-0044, Japan  Kenji Watanabe  Research Center for
Functional Materials, National Institute for Materials Science, 1-1 Namiki,
Tsukuba 305-0044, Japan  Sagar Vijay  Department of Physics, University of
California at Santa Barbara, Santa Barbara CA 93106, USA  Michael P. Zaletel
Department of Physics, University of California, Berkeley, California 94720,
USA  Material Science Division, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA  Andrea F. Young<EMAIL_ADDRESS>Department of Physics, University of California at Santa Barbara, Santa
Barbara CA 93106, USA

##  I Methods

###  I.1 Measurement

Experiments were performed in a dry dilution refrigerator with a base
temperature of 20mK. Electronic filters are used in line with transport and
gate contacts in order to lower the effective electron temperature. To improve
edge mode equilibration to the contacts most measurements are performed at
300mK unless otherwise noted. Electronic measurements were performed using
standard lock-in amplifier techniques. For the diagonal conductance
measurements an AC voltage bias at 17.77Hz is applied via a 1000x resistor
divider to (see supplementary Fig. 1 for contact references) C3-4 and the
resulting current is measured using an Ithaco 1211 trans-impedance amplifier
on C5-6 with a gain of $10^{-7}$ A/V. The voltage is measured between contacts
C1-2 and C7-8 with an SR560 voltage pre-amplifier with a gain of 1000. For two
terminal measurements the same AC bias is applied to contacts C1-4 and the
current is measured via C5-8. DC bias was added on top of the AC bias using a
passive summer.

We use transport contacts in pairs to decrease the contact resistance -
improving transport quality. The diagonal conductance is determined by
exciting an AC voltage on contacts C3/C4, and then measuring the resulting AC
current $I_{\mathrm{out}}$ on contacts C5/C6. The diagonal voltage drop,
$V_{D}$, is measured from C1/C2 to C7/C8. The differential diagonal
conductance is then defined as $G_{D}=I_{\mathrm{out}}/V_{D}$.

Figure 1: Optical Micrograph of Measured Device Transport contacts to the
monolayer are labeled C1-8, while the gates are labeled by their corresponding
control voltages $V_{i}$.

###  I.2 Thomas-Fermi Simulation

We consider a classical effective model wherein the electron density
$n(\mathbf{r})$ of the two-dimensional electron gas adjusts according to the
local electrostatic potential and compressibility of an interacting Landau-
level (LL). The classical energy functional can be decomposed into the Hartree
energy, the interaction with an externally applied potential
$\Phi(\mathbf{r})$, and the remainder,

$\displaystyle\mathcal{E}[n(\mathbf{r})]$
$\displaystyle=\frac{e^{2}}{2}\int_{{\bf{r_{1}},\bf{r_{2}}}}n({\bf{r_{1}}})V({\bf{r_{1}},\bf{r_{2}}})n({\bf{r_{2}}})$
(1)
$\displaystyle\qquad\qquad-e\int_{\mathbf{r}}\Phi(\mathbf{r})n(\mathbf{r})+\mathcal{E}_{xc}[n(\mathbf{r})],$

$\mathcal{E}_{xc}[n]$ contains not only the exchange-correlation energy but
also single-particle contributions (e.g. inter-LL interactions, disorder,
Zeeman energies, etc.). If $n(\mathbf{r})$ varies slowly compared with
$\ell_{B}$, we may neglect the dependence of the functional
$\mathcal{E}_{xc}[n(\mathbf{r})]$ on the gradient $\bm{\nabla}n$ and employ
the local density approximation (LDA),

$\mathcal{E}_{xc}[n(\mathbf{r})]=\int_{\mathbf{r}}E_{xc}(n(\mathbf{r}))$ (2)

where $E_{xc}(n)$ is determined for a system at constant density $n$. Our aim
here is to find the density configuration $n(\mathbf{r})$ corresponding to the
global minimum of the free energy $\mathcal{E}[n(\mathbf{r})]$.

The geometry we consider is shown in supplementary Fig. 3a-b. There are four
top gates a distance $d_{t}$ above the sample and one back gate a distance
$d_{b}$ below, between which the space is filled by hBN with dielectric
constant $\epsilon_{\perp}=3$ and $\epsilon_{\|}=6.6$ laturia_dielectric_2018
. The N/S gates and the E/W gates are shifted by $w_{NS}$ and $w_{EW}$,
respectively, from the center (see supplementary Fig. 3a-b). We make an
approximation by treating the cut-out “X”-shaped region to define the gates as
a metal held at fixed voltage $V=0$ (rather than as a vacuum). This allows us
to analytically solve for the electrostatic Green’s function and gate-induced
potentials without resorting to e.g. COMSOL simulations.

For a coarse-grained system with a finite resolution grid, the classical
energy functional becomes

$\displaystyle\mathcal{E}[\\{n(\mathbf{r})\\}]$
$\displaystyle=\mathcal{E}_{C}+\mathcal{E}_{xc}+\mathcal{E}_{\Phi}$ (3)
$\displaystyle\mathcal{E}_{C}$
$\displaystyle=\frac{1}{2A}\sum_{\mathbf{q}}V(\mathbf{q})n(\mathbf{q})n(-\mathbf{q})$
$\displaystyle\mathcal{E}_{xc}$
$\displaystyle=\sum_{\mathbf{r}}E_{xc}(n(\mathbf{r}))\mathop{}\\!\mathrm{d}A,$
$\displaystyle\mathcal{E}_{\Phi}$
$\displaystyle=\sum_{\mathbf{r}}\Phi(\mathbf{r})n(\mathbf{r})\mathop{}\\!\mathrm{d}A$

where $\mathop{}\\!\mathrm{d}A=\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y$
is the grid area, $A$ is the total area, and
$n(\mathbf{q})=\sum_{\mathbf{r}}e^{-i\mathbf{q}\cdot\mathbf{r}}n(\mathbf{r})\mathop{}\\!\mathrm{d}A$.
$\mathcal{E}_{C}$ is set by the gate-screened Coulomb interaction,

$V(\mathbf{q})=\frac{e^{2}}{4\pi\epsilon_{0}\epsilon_{\mathrm{hBN}}}\frac{4\pi\sinh\left(\beta
d_{t}|\mathbf{q}|\right)\sinh\left(\beta
d_{b}|\mathbf{q}|\right)}{\sinh\left(\beta(d_{t}+d_{b})|\mathbf{q}|\right)|\mathbf{q}|},$
(4)

$\epsilon_{\mathrm{hBN}}=\sqrt{\epsilon_{\perp}\epsilon_{\|}}$ and
$\beta=\sqrt{\epsilon_{\|}/\epsilon_{\perp}}$. $\mathcal{E}_{\Phi}$ is the one
body potential term arising from the potential $\Phi(\mathbf{r})$ on the
sample due to the adjacent gates,

$\Phi(\mathbf{q})=-eV_{t}(\mathbf{q})\frac{\sinh(\beta
d_{b}|\mathbf{q}|)}{\sinh(\beta(d_{t}+d_{b})|\mathbf{q}|)}-eV_{B}\frac{d_{t}}{d_{t}+d_{b}}$
(5)

where $V_{t}(\mathbf{q})$ is the top gate potential and $V_{B}$ is the back
gate potential.

The remaining energy $E_{xc}$ is defined by integrating the chemical potential
$\mu$,

$E_{xc}(n)=\int_{0}^{n}\mathop{}\\!\mathrm{d}n^{\prime}\mu(n^{\prime}).$ (6)

which encodes information about the IQH and FQH gaps and the electron
compressibility. For $\mu$ we use the experimentally measured value obtained
for monolayer graphene in the FQH regime ($B=18\,$\mathrm{T}$$) we reported in
an earlier work, Ref yang_experimental_2021 .

Physically, $n(\mathbf{r})$ can only vary on the scale of the magnetic length
$\ell_{B}$ due to the underlying quantum Hall wavefunction. To capture this
feature, we implement a square grid with periodic boundary conditions and
meshing much finer than the scale of $\ell_{B}$, and then evaluate
$\mathcal{E}$ with respect to the Gaussian convoluted density profile 111We
first use an unbounded range for $n(\mathbf{r})$ to determine the external
potential that fix bulk filling $\nu_{EW}=1$ and $\nu_{NS}=0$. Then we
constrain $n(\mathbf{r})$ to fall in between $\nu=0$ and $\nu=1$ such that any
transition from $\tilde{\nu}=0$ to $\tilde{\nu}=1$ in
$\tilde{\nu}(\mathbf{r})$ has a finite width of scale $\ell_{B}$.

$\tilde{n}(\mathbf{r})=\mathcal{N}^{-1}\sum_{\mathbf{r}^{\prime}}n(\mathbf{r}^{\prime})e^{-\left\lvert\mathbf{r}-\mathbf{r}^{\prime}\right\rvert^{2}/2\ell_{B}^{2}},$
(7)

where $\mathcal{N}$ is the corresponding normalization factor. We use a basin-
hopping global optimizer with local L-BFGS-B minimization to vary
$\\{n(\mathbf{r})\\}$ and find the lowest energy configuration.

Within this framework, we tune the E/W gate potentials such that $\nu_{EW}=1$
deep in the bulk and the N/S gate potentials such that $\nu_{NS}=0$. Various
effects of reconstruction can then be explored by appropriately tuning the
smoothness of the potential at the $\nu=0$ to $\nu=1$ interface (e.g. by
tuning the channel width $w$, gate distances $d_{t,b}$ or gate voltages
$\Phi_{g}$).

##  II Extended Information on Interaction-Driven Quantum Dots

The Coulomb blockaded resonant structure is fairly ubiquitous in this device.
Fig. S2 presents a series of several resonances which exhibit Coulomb blockade
on the electron side of the device at 9T and 20mK. Fig. S2a shows the
resonances in the ($V_{NS}$, $V_{EW}$) plane, where $\nu_{EW}\sim 3$ and
$\nu_{NS}\sim 0$. The chemical potential of the quantum dot is modulated
directly with the N/S gates as $V_{NS}$ is swept along the dashed white line
in Fig. S2a. Fig. S2b shows the differential conductance as a function of
$V_{NS}$ and $V_{SD}$ along this $V_{NS}$ trajectory along with the zero-bias
cut.

Figure 2: Coulomb blockaded resonances on the electron side. All data in this
figure was taken at B = 9T and T = 20mK. (a) Two terminal conductance across
the device plotted against $V_{NS}$ and $V_{EW}$. (b) Differential conductance
versus source-drain bias plotted along the white dashed line in (a) as well as
the corresponding zero-bias line cut. Along the white dashed line $\nu_{EW}=3$
and $\nu_{NS}=0$. (c-d) Two terminal conductance plotted against $V_{N(W)}$
and $V_{S(E)}$ for the Coulomb blockaded resonances marked by the white dashed
line in (a). The primary resonance shown in (d) is demarcated by the white dot
in (a). (e-f) Diagonal conductance across the device plotted against
$V_{N(W)}$ and $V_{S(E)}$ for the transmission step between $G_{D}=1$, and
$G_{D}=2$.

The sharp peaks in Fig. S2b have some finite curvature in the ($V_{SD}$,
$V_{NS}$) plane. Since the slope here is a direct measure of
$-C_{NS}/C_{\Sigma}$, where $C_{\Sigma}$ is the total capacitance of the dot,
this indicates that as $V_{NS}$ is increased, the capacitance of the N/S gates
to the dot is decreasing relative to $C_{\Sigma}$. This is consistent with the
dot being squeezed in the N/S direction as $V_{NS}$ is increased. This can be
further corroborated by the observation that as $V_{NS}$ is increased in Fig.
S2a, the slope of successive resonances decreases, indicating a decreased
sensitivity to modulations in $V_{NS}$ – i.e., $C_{EW}>C_{NS}$ as $V_{NS}$
increases. Additionally, Fig. S2c-d shows a representative resonance as as
function of $V_{N}$ vs. $V_{S}$ as well as $V_{E}$ vs. $V_{W}$. Much like the
analysis in the main text (Fig. 2), when $V_{N}=V_{S}$ or $V_{E}=V_{W}$, the
slope in the $V_{N}$/$V_{S}$ plane or the ($V_{E}$, $V_{W}$) plane is near
$-1$, indicating the dot is roughly centered in the QPC.

The behavior of a monotonic transition in $G_{D}$ as a function of $V_{N}$ vs.
$V_{S}$ and $V_{E}$ vs. $V_{W}$ can be seen in Fig. S2e-f. It has been well
established that monotonic steps in conductance at a quantum Hall QPC can be
described by the scattering of electrons in a magnetic field at a saddle point
potential floser_transmission_2010 . Qualitatively, the behavior of the
monotonic transmission step, and the Coulomb blockaded resonances in 2c-d are
similar; the monotonic transmission step and the resonance both have the same
curvature in the ($V_{N}$, $V_{S}$) or ($V_{E}$, $V_{W}$) plane. This further
verifies the conclusion that the quantum dots are sitting on top of an
externally applied potential which forms a saddle point, not a 2D well.

##  III Thomas-Fermi calculation of the local density in the QPC

Figure 3:  Effective model considered within the Thomas-Fermi framework. (a)
Top-view of the gate setup, which involves four square gates labeled N,S,E,
and W respectively, each of which is held at a potential denoted by
$V_{N,S,E,W}$. The back-gate (gray) is held at constant $V_{B}$. We allow for
the gate displacement from the center to be different for E/W and N/S gates.
(b) Side-view of the gates reveals two distances $d_{t}$ and $d_{b}$ for the
separation between the 2DEG and the top gate or bottom gate, respectively. (c)
The chemical potential $\mu$ and internal energy $E_{\textrm{xc}}$ derived
from Ref. yang_experimental_2021  which is taken as input data for the Thomas-
Fermi calculation carried out here.

###  III.1 Fractional Reconstruction

Figure 4:  Extended reconstructed electron density profiles in a QPC geometry
in the fractional regime. (a-b) Fractional reconstruction in an integer bulk
filling ($\nu=1$) with $E_{C}=\,46.4~{}$\mathrm{m}\mathrm{e}\mathrm{V}$$ in
the regime where the confining potential is sharper than presented in main
text Fig. 3a. (a) With $E_{V}/E_{C}=0.50$, the $\nu=\frac{1}{3}$ island
disappears and the $\nu=\frac{1}{3}$ strips become narrower. (b) With
$E_{V}/E_{C}=0.57$, the $\nu=\frac{1}{3}$ strips disappear completely.

To approximate the experimental device, we simulate a system with magnetic
field $B=13\,$\mathrm{T}$$ (magnetic length
$\ell_{B}=7\,$\mathrm{n}\mathrm{m}$$) 222In the experiment
yang_experimental_2021 , $\mu$ was obtained at $B=18\,$\mathrm{T}$$, which
introduces a difference in energy scale of $E_{C}$ by a factor of 1.19
relative to $B=13\,$\mathrm{T}$$ at which we carry out our calculations, gate
distances $d_{t}=d_{b}=30\,$\mathrm{n}\mathrm{m}$$ and channel widths
$w_{EW}=w_{NS}=35\,$\mathrm{n}\mathrm{m}$$ on a
$40\ell_{B}\,\times\,40\ell_{B}$ system with gridsize
$\mathop{}\\!\mathrm{d}x=\mathop{}\\!\mathrm{d}y=\tfrac{1}{4}\ell_{B}$ 333It
is advantageous to take a smaller $d_{t}$ here than the experimental
$60\,$\mathrm{n}\mathrm{m}$$ so that the necessary system size $L$, and hence
the number of variational parameters in the optimization, remains tractable..

For a finite system and mesh size, we find that edge reconstruction is most
easily observed when $E_{xc}$ is increased by a factor of $1-2$ (for the data
shown here, $E_{xc}$ is increased by 1.8x). One possible origin of this
discrepancy is screening from the filled LLs. In Ref. yang_experimental_2021
it was found the screening from virtual inter-LL transitions (which is
implicitly included in our $\mathcal{E}_{xc}$, since we use experimental data)
reduces the IQH gaps, and hence features in $\mu$, by about a factor of 1.4
relative to their unscreened values. However, our phenomenological model does
_not_ account for inter-LL screening in the electrostatic contributions
$\mathcal{E}_{C}$ and $\mathcal{E}_{\Phi}$. Decreasing their scale relative to
$E_{xc}$ can thus be interpreted as a very crude implementation of LL-
screening in the remaining energy functionals. Of course other approximations
are in play as well, e.g. the local density approximation, so at the outset we
do not expect more than phenomenological agreement with experiment.

At a translationally invariant boundary between $\nu=1$ to $\nu=0$, previous
variational calculations have suggested edge-state reconstruction makes it
energetically favorable to form an additional strip of $\nu=\tfrac{1}{3}$
khanna_fractional_2021 . However, in a QPC geometry where translation symmetry
is broken, the effects of reconstruction are much richer and consequently
require numerical treatment to fully characterize. In Fig. S6 of the main
text, we demonstrate the effects of edge-reconstruction in a QPC as a function
of the potential smoothness quantified by the ratio $E_{V}/E_{C}$. To
accomplish this, we fix $V_{NS}=-V_{B}$ and tune the smoothness of the
confining potential via the relative magnitudes of $V_{EW}$ and $V_{B}$. For
the smallest value of $E_{V}/E_{C}$, we find in main text Fig. 3a
reconstructed $\nu=\frac{1}{3}$ strips which extend far enough into the QPC
that they merge, corresponding to a fractional conductance across the QPC
($V_{EW}=0.51\,$\mathrm{V}$$, $V_{B}=-0.19\,$\mathrm{V}$$). With increasing
$E_{V}/E_{C}$, in main text Fig. 3b the $\nu=\tfrac{1}{3}$ strips become
narrower and an island of fractional filling is formed in the center of the
QPC ($V_{EW}=0.60\,$\mathrm{V}$$, $V_{B}=-0.10\,$\mathrm{V}$$). As shown in
Fig. S4a, we confirmed upon further increasing $E_{V}/E_{C}$
($V_{EW}=0.61\,$\mathrm{V}$$, $V_{B}=-0.09\,$\mathrm{V}$$), the
$\nu=\tfrac{1}{3}$ strips become even narrower and the island disappears. For
extremely sharp edges ($V_{EW}=0.70\,$\mathrm{V}$$,
$V_{B}=0.0\,$\mathrm{V}$$), in Fig. S4b the $\nu=\tfrac{1}{3}$ strips
disappear entirely leaving the expected density profile for a $\nu=1$ edge
everywhere in the QPC.

We note that the $\nu=\tfrac{1}{3}$ strips are smoothly connected to the bulk
(i.e. not separated by a region of $\nu=0$) even for very smooth confining
potentials, which is consistent with variational calculations which more
accurately account for the energetics of $\nu=1$ to $\tfrac{1}{3}$ edges
beyond the local-density approximation khanna_fractional_2021 . For a line cut
taken directly across the $\nu=1$ to $\nu=0$ interface (e.g. between the east
and north gates), the confining potential $\Phi_{\textrm{ext}}$ gives a
confinement energy $E_{V}\approx 0.44E_{C}$ and $0.49E_{C}$ for the two
scenarios in main text Fig. 3a-b respectively, and $0.50E_{C}$ for the
scenario where the island of fractional filling disappears (Fig. S4a). The
gate potentials reported here are slightly smaller than actual values used in
the experiment. In the experiment, the bulk filling far from the QPC can be as
high as $\nu=\pm 4$ even though the local filling at the QPC will give $\nu=1$
to $\nu=0$ interfaces as we simulate here (see Fig. 1 of the main text). Since
we simulate only the small region close to the QPC, the effective gate
voltages we employ will necessarily reflect a smaller range than that of the
experiment.

In generic quantum Hall systems, it is widely believed that disorder and
localized impurities are responsible for the formation of localized states in
the QPC which facilitate resonant tunneling and are responsible for the non-
monotinicity observed in conductance measurements across QPC transmission
steps Milliken1996 ; Ando1998 ; Baer2014 . In the device studied in this
paper, we do not expect such disorder, and the localized states were
attributed to an entirely intrinsic mechanism based purely on including the
Coulomb interaction at the soft gate-defined edges of the QPC. That picture is
supported by our Thomas-Fermi calculations, where we have shown explicitly
that the Coulomb interaction itself is sufficient to favor reconstruction of a
local island. We see then that by appropriate tuning of the gate voltages, one
can reproduce a Coulomb blockade and fractional transmission in close
approximation to the experimental findings of this report.

###  III.2 Integer Reconstruction

Figure 5:  Reconstructed electron density profiles in a QPC geometry in the
integer reconstruction regime. (a-c) Integer reconstruction in an integer bulk
filling ($\nu=1$) with $E_{C}=\,11.6~{}$\mathrm{m}\mathrm{e}\mathrm{V}$$ as
the confining potential smoothness is varied. (a) With $E_{V}/E_{C}=0.18$, the
reconstructed $\nu=1$ stripes extend along the entire edge. (b) With
$E_{V}/E_{C}=0.47$, the reconstructed $\nu=1$ stripes shrink to a single
circular dot with $\nu=1$ in the center of the QPC. (c) With
$E_{V}/E_{C}=0.55$, no reconstruction is observed.

Now we turn our attention to the possibility of reconstructed strips and
islands with integer filling. Integer reconstruction can be made more
energetically favorable in the numerics by reducing the magnitude of $E_{C}$
such that redistributing integer charge along the edge out competes the energy
gained from from forming a correlated FQH strip. Fixing $\ell_{B}$, $d_{t,b}$
and $w_{EW,NS}$ as before, we now take
$E_{C}=\,11.6~{}$\mathrm{m}\mathrm{e}\mathrm{V}$$, and work on a system grid
which is $40\ell_{B}\times 40\ell_{B}$ large.

In Fig. S5, we demonstrate edge-reconstruction as a function of the confining
potential smoothness tuned via the gate voltages. For the smallest value of
$E_{V}/E_{C}$, we find in Fig. S5a reconstructed $\nu=1$ strips that exist
throughout the QPC, ($V_{EW}=43.8\,$\mathrm{m}\mathrm{V}$$,
$V_{NS}=107\,$\mathrm{m}\mathrm{V}$$, $V_{B}=107\,$\mathrm{m}\mathrm{V}$$).
With increasing $E_{V}/E_{C}$, in Fig. S5b the formation of an island of
integer filling within the QPC ($V_{EW}=-143\,$\mathrm{m}\mathrm{V}$$,
$V_{NS}=79.6\,$\mathrm{m}\mathrm{V}$$, $V_{B}=79.6\,$\mathrm{m}\mathrm{V}$$)
is observed, and in Fig.5c, for the largest value of $E_{V}/E_{C}$, a
monotonic transition from $\nu=0$ to $\nu=1$ is recovered everywhere in the
QPC ($V_{EW}=-167\,$\mathrm{m}\mathrm{V}$$,
$V_{NS}=55.7\,$\mathrm{m}\mathrm{V}$$, $V_{B}=55.7\,$\mathrm{m}\mathrm{V}$$).

In contrast to the $\nu=\frac{1}{3}$ strips, the reconstructed $\nu=1$ strips
in Fig. S5a are separated from the bulk by fully depleted regions, which is
also consistent with variational calculations khanna_fractional_2021 . For a
line cut taken directly across the $\nu=1$ to $\nu=0$ interface (e.g. between
the east and north gates), the confining potential $\Phi_{\textrm{ext}}$ gives
a confinement energy $E_{V}\approx 0.18E_{C},0.47E_{C}$ and $0.55E_{C}$ for
three scenarios Fig. S5a-c respectively. We leave a more detailed study of the
interplay of integer and fractional reconstruction in this QPC geometry to
future work.

##  IV Additional Experimental Evidence for Fractional Reconstruction at the
QPC

Figure 6: Evidence for fractional edge modes at the boundary of a bulk integer
state: (a) $G_{D}$ at a fixed $\nu_{EW}=-1$ versus $V_{NS}+\alpha V_{B}$ and
$V_{B}$. Here $V_{NS}$ is swept in a range dependent on $V_{B}$ such that the
range $V_{NS}+\alpha V_{B}$ is kept fixed. Additionally, as $V_{B}$ is swept,
$V_{EW}$ is swept concomitantly to keep the bulk filling factor, set by
$V_{EW}+\alpha V_{B}$, fixed at $\nu=-1$, but varying $V_{EW}-V_{B}$. The
x-axis range corresponds to the full width of the $\nu=0$ plateau. (b) The
diagonal conductance map of the $\nu_{EW}=-1$ state at B=13T presented in
panel (a) shows a number of features between $G_{D}=1$ and $G_{D}=0$. The
statistical frequency of conductance values across the entire plot, reveals a
number of sharp peaks corresponding to conductance plateaus between 0 and 1.
The most prominent occurs at $G_{D}=2/3$. A broader peak is observed near 1/3,
though the quantization is less exact, and several further sharp peaks are
seen centered near 0.2, 0.5, and 0.8. The origin of these peaks is unknown,
and may be due to a more complicated reconstructed edge structure than is
accounted for by any model discussed in this work. (c) Diagonal conductance
map with the E/W regions in a fixed filling factor $\nu_{EW}=-2$ and N/S
regions entirely within the $\nu_{NS}=0$ plateau. On the y-axis, the bottom
gate and E/W gates are swept in opposite directions to vary $(V_{B}-V_{EW})$
while keeping $\nu_{EW}=-2$ fixed. On the x-axis, the N/S gates are swept
across the entire $\nu=0$ plateau. (d) Line cut given in (c) showing two
integer plateaus and an intermediate fractional plateau near $G_{D}=1.33$
demarcated by the red lines. In both (a) and (b) a correcting scale factor of
$0.95$ is applied uniformly to $G_{D}$ to correct for a parallel conduction
channel created in our device due to fringe doping from the Si gate, leading
to over-quantized plateaus. Here only one $e/3$ edge mode is observed, in
contrast to the data at B = 13T where two $e/3$ modes are observed. This is
similar to previous reports in GaAs bhattacharyya_melting_2019 , however, the
reason why one plateau is favored at low field compared to $B=13T$ remains
unclear.

In Fig. 6 we present data demonstrating additional plateaus in transmission
across the qpc while the bulk filling factor is kept in $\nu=1$. The presence
of these plateaus provide strong evidence for the existence of an additional
incompressible strip at fractional filling within the QPC at low values of
$E_{V}/E_{C}$ consistent with our simulations.

##  V Experimental Realization of the Unreconstructed Limit of $\nu=1$
Transmission at the QPC

Figure 7:  Signatures of reconstruction in $G$ between $\nu=1$ edge modes. (a)
The conductance measured across the QPC with both the East and West regions in
$\nu=1$ at B = 8T. The East and West gate voltages are swept in the opposition
direction of $V_{\text{BG}}$ along the y axis, through the range
$V_{\mathrm{EW}}\in(0.6V,-2.191V)$, to maintain a fixed filling factor while
varying the voltage difference and thereby the potential sharpness. (b) The
simulated electric potential at the monolayer, corresponding to the operating
point I. (c) Same as (b) but at the operating point II, where the potential is
much softer. (d) Simulated potential along the contours marked in grey and
black in panels B and C, respectively. The softness is quantified by the
maximum magnitude of the in-plane confining electric field, $E_{\parallel}$
(i.e. simply the gradient of the potential normal to the boundary between the
N(/S) and E(/W) regions).

In Fig. 7 we present data on a second device measured in cohen_universal_2023
which looks at the transmission as a function of potential sharpness for the
$\nu=1$ state. Due to higher gate limits in this device, by applying a large
voltage difference between the back-gate and the E/W gates, we were able to
achieve the $E_{V}/E_{C}>1$ limit. In this limit the presence of spontaneously
formed quantum dots from edge reconstruction begins to vanish and a single
monotonic step from full pinch off to $G=e^{2}/h$ is recovered.

##  VI References

## References

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