# Parabolicity of invariant Surfaces

Andrea Del Prete  Dipartimento di Ingegneria e Scienze dell’Informazione e
Matematica, Università degli Studi dell’Aquila, Via Vetoio Loc. Coppito –
67100 L’Aquila, Italy<EMAIL_ADDRESS>and  Vicent Gimeno i
Garcia  Department of Mathematics, Universitat Jaume I-IMAC, E-12071,
Castelló, Spain<EMAIL_ADDRESS>
###### Abstract.

We present a clear and practical way to characterize the parabolicity of a
complete immersed surface that is invariant with respect to a Killing vector
field of the ambient space.

###### Key words and phrases:

Parabolicity, Invariant surfaces, isometries, Killing vector fields

###### 2023 Mathematics Subject Classification:

Primary 58J05, 31A05, 53A10,58J65,58D19

The first author was partially supported by “INdAM - GNSAGA Project”, codice
CUP_E55F22000270001, and by MCIN/AEI project PID2022-142559NB-I00. The second
author was partially supported by the Research grant PID2020-115930GA-100
funded by MCIN/ AEI /10.13039/50110001103

##  1\. Introduction

The intrinsic or extrinsic geometric conditions which guaranties that a
surface is a parabolic surface has been largely studied from the beginning of
the twentieth century. See for example [1],[16] [8], [9],[6]. This interplay
between geometry and analytical properties of functions defined on surfaces is
used to classify the surfaces in conformal types. Recall that a Riemannian
manifold is said to be parabolic if every upper bounded subharmonic function
is constant. A canonical reference on the parabolicity of manifolds is
provided by [8]. In dimension $2$, parabolicity is a conformal property.
Specifically, if the surface $(M,g)$ is parabolic, then the surface
$(M,e^{u}g)$ is also parabolic for any smooth function $u:M\to\mathbb{R}$.
Parabolic manifolds satisfy a specific maximum principle, as demonstrated in
[2], for example. Given a non-constant $C^{2}$ function $u:M\to\mathbb{R}$
with $\sup_{M}u=u^{*}<\infty$, if $M$ is a parabolic manifold, there exists a
divergent sequence $\left\\{x_{k}\right\\}_{k\in\mathbb{N}}\subset M$ such
that

$u(x_{k})>u^{*}-\frac{1}{k},\quad{\rm and},\quad\Delta u(x_{k})<0.$

Parabolicity can also be defined for manifolds $M$ with a non-empty boundary
$\partial M$, as shown in [14]. A manifold with a boundary is parabolic if and
only if every bounded harmonic function is determined by its values on the
boundary. In other words, if $f_{1},f_{2}\colon M\to\mathbb{R}$ are two
bounded harmonic functions with $f_{1}(x)=f_{2}(x)$ for any $x\in\partial M$,
then $f_{1}(y)=f_{2}(y)$ for every $y\in M$. Given a complete Riemannian
manifold $(M,g)$ and a precompact domain $\Omega\subset M$, every non-bounded
connected component of $M-\Omega$ is called an end of $M$ with respect to
$\Omega$. In this setting, $M$ is parabolic if and only if all the ends of $M$
with respect to $\Omega$ are parabolic.

In this article, we are interested in studying surfaces that are invariant
with respect to a one-parameter group of isometries of the ambient space. For
this purpose, we make use of Killing vector fields. A Killing vector field
$\xi$ is defined such that $\mathcal{L}_{\xi}g=0$, where $g$ represents the
metric tensor. We will say that a surface is invariant with respect to a
Killing field when it is invariant with respect to the one-parameter group of
isometries of the ambient manifold associated with $\xi$.

The main result of this paper is the following Theorem:

######  Main Theorem.

Let $\mathbb{E}$ be a $n$-dimensional Riemannian manifold which admits a
complete Killing vector field $\xi\in\mathfrak{X}(\mathbb{E})$. Assume that an
immersed complete regular surface $S\subset\mathbb{E}$ is invariant with
respect to the one-parameter group of isometries of $\mathbb{E}$ associated to
$\xi$. Assume that there exists $q\in S$ such that $\|\xi(q)\|\neq 0$ and
denote by $\gamma(t)\subset S$ a complete curve parameterized by arc length
satisfying

$\left\langle{\dot{\gamma}(t),\xi}\right\rangle=0.$

Then, $S$ is parabolic if and only if

  1. (1)

the curve $\gamma$ is compact, or

  2. (2)

the integral curves of $\xi$ are compact and there exists $a,b\in\mathbb{R}$,
with $a<b,$ such that
$\int_{-\infty}^{a}\frac{1}{\|\xi(\gamma(s))\|}ds=\infty$ and
$\int_{b}^{\infty}\frac{1}{\|\xi(\gamma(s))\|}ds=\infty$, or

  3. (3)

the integral curves of $\xi$ are non compact and
$\int_{-\infty}^{0}\frac{1}{\|\xi(\gamma(s))\|}ds=\infty$ and
$\int_{0}^{\infty}\frac{1}{\|\xi(\gamma(s))\|}ds=\infty.$

######  Remark 1.1.

We will see in Proposition 2.1 that the existence of a point $p\in S$ where
$\xi(p)\neq 0$ allows us to prove that there exist at most two points where
the Killing vector field $\xi$ vanishes. In particular, we will prove that, if
$\xi_{|S}$ vanishes at exactly two points, then $S$ is diffeomorphic to a
sphere and hence it is parabolic and, if $\xi_{|S}$ vanishes at only one
point, then the integral curves of $\xi$ must be compact and since
$\dot{\gamma}$ is perpendicular to $\xi$, there exist $a,b\in\mathbb{R}$, with
$a<b$, such that $\|\xi(\gamma(s))\|\neq 0$ in $s\in(-\infty,b)\cup(a,\infty)$
and
$\int_{-\infty}^{a}\frac{1}{\|\xi(\gamma(s))\|}ds=\int_{b}^{\infty}\frac{1}{\|\xi(\gamma(s))\|}ds.$

Observe that if $\xi$ vanishes on all $S$, we cannot find a curve
$\gamma\subset S$ which generates $S$, since every curve $\Gamma\subset S$
satisfies the condition $\left\langle{\dot{\Gamma}(t),\xi}\right\rangle=0$.
However, this situation does not append in low dimensions. Indeed, [10,
Theorem 5.3] assures that each connected component of the set of zeros of
$\xi$, ${\rm Zero}(\xi)$, is a closed totally geodesic submanifold of even
codimension. In particular,if $n=3$, then $\xi$ vanishes at isolated geodesics
of $\mathbb{E}$. If $n=4$, there could exist a surface $S$ where $\xi$
vanishes and its geometric properties does not depend on $\xi$ but only on the
geometry of $\mathbb{E}$; for example, we can obtain a Killing vector field in
$\mathbb{R}^{4}$ such that it vanishes identically at the totally geodesic
submanifold $\mathbb{R}^{2}\subset\mathbb{R}^{4}$, and similarly we can find a
Killing vector field in $\mathbb{H}^{4}$ such that it vanishes at the totally
geodesic submanifold $\mathbb{H}^{2}\subset\mathbb{H}^{4}$ but
$\mathbb{R}^{2}$ is parabolic and $\mathbb{H}^{2}$ is hyperbolic. If $n>4$, if
$\xi$ vanishes on a surface $S\subset\mathbb{E}$, then $S$ can be immersed
isometrically in a manifold that does not admit any Killing field a-priori.

In what follows we prove the Main Theorem and then we show a couple of
applications in some simply connected homogeneous manifold.

##  2\. Proof of the Main Theorem: Parabolicity of surfaces admiting a Killing
submersion

In order to prove the Main Theorem we need to state here that the surface only
can admit finitely many points where the Killing vector field vanishes. In
fact, $\xi$ vanishes at most on two points of $S$ and in such a case $S$ is
diffeomorphic to a sphere. If $\xi$ only vanishes at one point of $S$, then
$S$ is diffeomorphic to a plane. In both cases the integral curves of $\xi$
are diffeomorphic to $\mathbb{S}^{1}$, that is, $S$ is rotationally symmetric.
Let us summarize this in the following proposition:

######  Proposition 2.1.

Let $\mathbb{E}$ be a $n$-dimensional Riemannian manifold which admits a
complete Killing vector field $\xi\in\mathfrak{X}(\mathbb{E})$. Assume that an
immersed complete regular surface $S\subset\mathbb{E}$ is invariant with
respect to the one-parameter group of isometries of $\mathbb{E}$ associated to
$\xi$. Assume that there exists $q\in S$ such that $\|\xi(q)\|\neq 0$. Then,
there are only three options:

  1. (1)

The Killing vector field $\xi$ never vanishes on $S$.

  2. (2)

The Killing vector field $\xi$ vanishes at only one point $p\in S$. The
surface $S$ is topologically a plane and the integral curves of $\xi$ are
compact.

  3. (3)

The Killing vector field vanishes at only two points $p,p^{\prime}\in S$. The
surface $S$ is topologically an sphere and the integral curves of $\xi$ are
compact.

###### Proof.

Assume that there exists a point $p\in S$ such that $\xi(p)=0$. First of all,
we are going to prove that there are no other points with vanishing $\xi$ in a
geodesic ball of $S$ centered at $p$ and radius the injectivity radius of $p$.

Let us start joining $p$, where $\xi(p)=0$, to the point $q$ (where is assumed
that $\xi(q)\neq 0$) with a minimal geodesic segment $\gamma$. Assume that
there exists a point $p^{\prime}$, see figure 1, in $\gamma$ where
$\xi(p^{\prime})=0$.

Figure 1. If $\xi(p)=0$, $\xi(q)\neq 0$ and $\gamma$ is the minimal geodesic
segment joining $p$ to $q$, then there is no $p^{\prime}\in\gamma$ with
$\xi(p^{\prime})=0$.

Let us denote by $\\{\phi_{t}\\}$ the one-parametric group of transformations
associated to $\xi$. Let $q_{2}=\phi_{t_{0}}(q)$. Since $\xi(q)\neq 0$, we can
take $t_{0}$ sufficiently small such that $q_{2}\neq q$. Moreover as we are
assuming $\xi(p)=\xi(p^{\prime})=0$, we know that $\phi_{t_{0}}(p)=p$ and
$\phi_{t_{0}}(p^{\prime})=p^{\prime}$. Since $\xi$ is a Killing vector field,
$\phi_{t}$ are isometries and $\gamma_{1}=\phi_{t_{0}}\gamma$ is a minimal
geodesic segment as well joining $p$, $p^{\prime}$ and $q_{2}$. By using the
uniqueness of the geodesics, taking into account that $q_{2}\neq q$ we
conclude $\gamma_{1}\neq\gamma$ and hence we have two different minimal
geodesic segments joining $p$ and $p^{\prime}$, then $p^{\prime}\in{\rm
cut}(p)$ and $\gamma$ does not realizes the distance providing a
contradiction. Hence there are no points $p^{\prime}$ in $\gamma$ with
$\xi(p^{\prime})=0$.

Therefore any point $p_{1}$ in $\phi_{t}(\gamma)$, with $0<t<t_{0}$, at
distance less than the injectivity radius of $p$ satisfies that
$\xi(p_{1})\neq 0$. So, without loss of generality, we can assume that
distance between $p$ and $q$ is smaller than the injectivity radius of $p$.

Take a point $q^{\prime}$ in the minimal geodesic joining $p$ and $q$ at
distance less than the injectivity radius of $p$. Then, $\xi(q^{\prime})\neq
0$. Take the orbit the orbit by $\phi_{t}$ of $q^{\prime}$,

$\mathcal{O}_{q^{\prime}}=\left\\{\phi_{t}(q^{\prime})\,:\,t\in\mathbb{R}\right\\}$

is a set of points of a geodesic sphere of radius ${\rm dist}(p,q^{\prime})$
centered at $p$. Indeed, $\mathcal{O}_{q^{\prime}}$ is the geodesic sphere of
distance ${\rm dist}(p,q)$ centered at $p$. Because otherwise should there
exists a point $p^{\prime}$ of accumulation of $\mathcal{O}_{q^{\prime}}$ in
the geodesic sphere of distance ${\rm dist}(p,q)$ with $\xi(p^{\prime})=0$ and
$\mathcal{O}_{q^{\prime}}$ should be a subset of the metric sphere of distance
${\rm dist}(p^{\prime},q)$ centered at $p^{\prime}$ but this is a
contradiction because $p^{\prime}$ is a point of accumulation of
$\mathcal{O}_{q^{\prime}}$ and hence ${\rm
dist}(p^{\prime},\mathcal{O}_{q^{\prime}})=0$. Therefore,

$\mathcal{O}_{q^{\prime}}=\\{r\in S\,:\,{\rm dist}(p,r)={\rm dist}(p,q)\\},$

which is compact and, as we have seen before, there are no points with $\xi=0$
in the minimal segments joining $p$ with the points of
$\mathcal{O}_{q^{\prime}}$. In particular, we proved that, if $R$ is smaller,
then the injectivity radius of $p$, then $\xi\neq 0$ in
$B_{R}(p)\setminus\left\\{p\right\\}$, where $B_{R}(p)$ is the geodesic ball
centered at $p$ with radius $R$. Furthermore, $B_{R}(p)$ is foliated by
integral curves of $\xi$. It can be proved that any point $p^{\prime}$ with
$\xi(p^{\prime})=0$ then $p^{\prime}\in{\rm cut}(p)$, see [10, Corollary 5.2].
Moreover we can prove that if $q\in{\rm cut}(p)$ then $\xi(q)=0$. Suppose, by
the contrary, that $\xi(p)=0$ and ${\rm cut}(p)\neq\emptyset$. Take a point
$q\in{\rm cut}(p)$ let us consider the minimal geodesic segment

$t\mapsto\gamma(t)=\exp_{p}(tu),\quad{\rm with}\quad\|u\|=1,$

such that $\gamma(0)=p$, $\gamma({\rm dist}(p,q))=q$ and $\gamma$ minimizes
distance for $t\in[0,{\rm dist}(p,q)]$ but it is not a minimal geodesic for
$t>{\rm dist}(p,q)$. If we suppose that $\xi(q)\neq 0$, the orbit
$\mathcal{O}_{q}$ is the metric sphere $\partial B_{{\rm dist}(p,q)}(p)$, it
has positive length and it is diffeomorfic to $\mathbb{S}^{1}$. Moreover since
$\phi_{t}$ acts by isometries every point $q^{\prime}\in\partial B_{{\rm
dist}(p,q)}(p)$ is a cut point of $p$ and there exists $u^{\prime}\in T_{p}M$
such that the geodesic segment

$t\mapsto\gamma_{1}(t)=\exp_{p}(tu^{\prime}),$

satisfies that $\gamma_{1}({\rm dist}(p,q))=q^{\prime}$ and $\gamma_{1}$
minimizes distance for $t\in[0,{\rm dist}(p,q)]$ but it is not a minimal
geodesic for $t>{\rm dist}(p,q)$. Thence ${\rm cut}(p)=\partial B_{{\rm
dist}(p,q)}(p)$ and, see for instance [18, Lemma 4.4],

$S=B_{{\rm dist}(p,q)}(p)\cup\partial B_{{\rm dist}(p,q)}(p).$

But this is a contradiction with the completeness of $S$. Therefore, we can
conclude that if $\xi(p)=0$ and $q\in{\rm cut}(p)$, then $\xi(q)=0$ and we can
state that if $\xi(p)=0$,

${\rm cut}(p)=\\{q\in S-\\{p\\}\,:\,\xi(q)=0\\}.$

Hence, if we suppose that $\xi(p)=0$ and ${\rm cut}(p)=\emptyset$, $p$ is the
only point in $S$ where the Killing vector field vanishes and $S$ is
rotationally symmetric plane.

Now suppose again that $\xi(p)=0$ and ${\rm cut}(p)\neq\emptyset$. Take a
point $p^{\prime}\in{\rm cut}(p)$, and hence with $\xi(p^{\prime})=0$, and
such that ${\rm dist}(p,p^{\prime})={\rm inj}(p)$. Thence for any point $q$
between $p$ and $p^{\prime}$ such that $d(p^{\prime},q)$ is smaller than the
injectivity radius of $p^{\prime}$, see figure 2, the orbit $\mathcal{O}_{q}$
need to be a geodesic sphere centered at $p$ and simultaneously a geodesic
sphere centered at $p^{\prime}$, namely,

$\mathcal{O}_{q}=\partial B_{{\rm dist}(p,q)}(p)=\partial B_{{\rm
dist}(p^{\prime},q)}(p^{\prime}).$

Figure 2. If $S$ has two points $p$ and $p^{\prime}$ where $\xi$ vanish, then
the points $p$ and $p^{\prime}$ are antipodals points of a rotationally
symmetric sphere.

This implies that $S$ is the connected sum

$S=\overline{B_{{\rm dist}(p,q)}(p)}\\#\overline{B_{{\rm
dist}(p^{\prime},q)}(p^{\prime})}.$

In particular, $S$ is a sphere and there exists no other point in $S$ where
$\xi$ vanishes.

Summarizing everything, if we have a point $p\in S$ with $\xi(p)=0$ there are
only two options: or ${\rm cut}(p)=\emptyset$ and $S$ is a rotationally
symmetric plane without other points with vanishing $\xi$, or ${\rm cut}(p)$
contains only one point $p^{\prime}$, where $\xi(p^{\prime})=0$ and $S$ is a
rotationally symmetric sphere. ∎

By using the above proposition We can remove a compact set $K$ of $S$ and the
Killing vector field never vanishes on $S-K$. But $\Sigma$ is parabolic if and
only if $S-K$ is parabolic (see for instance [8]). Then, in order to simplify
the discussion of the proof we are assuming that $\xi$ never vanishes on $S$,
otherwise we can do the same argument but for $S-K$.

Since the surface $S$ is invariant by the one-parameter group of isometries
$G_{\xi}=\\{\phi_{t}\\}$ associated to the Killing vector field $\xi$, _i.e._
, $G_{\xi}(S)=S$, we will also assume that $G_{\xi}$ acts freely and properly
on $S$, otherwise $G$ is not closed in $\mathrm{Iso}(S)$ with the compact-open
topology and [13, Proposition] implies that $\xi=X_{1}+X_{2}$ where $X_{1}$
and $X_{2}$ are two Killing vector fields with compact orbits satisfying
$[X_{1},X_{2}]=0$, that is, $S$ is a torus and hence it is a parabolic
surface. Then, we can endow $S/G_{\xi}$ with a smooth structure and with a
Riemannian metric tensor in such a way that the projection $\pi_{2}:S\to
S/G_{\xi}$ is a Killing submersion as well. Furthermore, since
$\left\langle{\dot{\gamma}(t),\xi}\right\rangle=0$, the curve $\gamma$ is
diffeomorfic to the set $S/G_{\xi}$ and because we are using arc-length
parametrization they are indeed isometric manifolds as well.

The Main Theorem is therefore equivalent to Theorem 2.7 where we are proving
the general case on surfaces which admits a Killing submersion. In order to
prove this theorem we will prove Lemmas 2.2, 2.3,2.4,2.5, and 2.6. In Lemma
2.2 we will prove that a conformal change on the metric tensor preserves the
parabolicity for surfaces. Indeed, parabolicity is related to the conformal
type in dimension $2$. In Lemma 2.3 we will prove that given a Killing
submersion we can perform a conformal change with a basic function in the
total space and in the base manifold in such a way that with respect to the
new metric tensors the submersion remains as a Killing submersion. In Lemma
2.4 we will prove that a Killing submersion on a surface with constant norm of
the Killing vector field has non-negative Gaussian curvature. In Lemma (2.5)
we will prove that a complete surface with non-negative Gaussian curvature is
a parabolic surface. Finally, with this previous lemmas and the technical
lemma 2.6 where is obtained the expression of the Laplacian of a basic
function we can prove Theorem 2.7 which is equivalent to the Main theorem.

######  Lemma 2.2.

In dimension $2$, parabolicity is preserved under conformal changes in the
metric tensor.

###### Proof.

Let $(M,g)$ be a $2$-dimensional Riemannian manifold and consider the
conformal change of metric $\overline{g}=f^{2}g$ given by the positive
function $f:M\to\mathbb{R}_{+}$, then, the Laplacian $\overline{\Delta}$ with
respect to the metric tensor $\overline{g}$ is related with the Laplacian
$\Delta$ with respect to the metric tensor $g$ by (see [4] for instance)

$\overline{\Delta}=\frac{\Delta}{f^{2}}.$

Then, $(M,g)$ admits non-constant bounded subharmonic functions if and only if
$(M,f^{2}g)$ admits bounded subharmonic functions. ∎

######  Lemma 2.3.

Let $\pi:M\to B$ be a surjective Killing submersion from to the complete
Riemannian manifold $(M,g_{M})$ to $(B,g_{B})$ with complete Killing vector
field $\xi\in\mathfrak{X}(M)$. Let $f:B\to\mathbb{R}$ be a smooth and positive
function. Then, $\pi:M\to B$ is also a Killing submersion from
$(M,(f\circ\pi)^{2}g_{M})$ to $(B,f^{2}g_{B})$ with the complete Killing
vector field $\xi$. Furthermore, $(M,(f\circ\pi)^{2}g_{M})$ is complete if
$(B,f^{2}g_{B})$ is complete.

###### Proof.

Since $\xi$ is a Killing vector field of $g_{M}$, the Lie derivative of the
metric tensor vanishes, _i.e._ ,

$\mathcal{L}_{\xi}g_{M}=0.$

Then

$\mathcal{L}_{\xi}\left((f\circ\pi)g_{M}\right)=\xi(f\circ\pi)\cdot
g_{M}+(f\circ\pi)\mathcal{L}_{\xi}g_{M}=0,$

and hence, $\xi$ is also a Killing vector field for $(f\circ\pi)g_{M}$.
Likewise, since $\pi:(M,g_{M})\to(B,g_{B})$ is a Riemannian submersion, for
any $v\in T_{p}M$ with $g_{M}(v,\xi)=0$,

$g_{M}(v,v)=g_{B}(d\pi(v),d\pi(v)).$

Then, for any $v\in T_{p}M$ with $(f\circ\pi)g_{M}(v,\xi)=0$,

$(f\circ\pi)g_{M}(v,v)=f(\pi(p))g_{B}(d\pi(v),d\pi(v)),$

and hence $\pi:(M,(f\circ\pi)g_{M})\to(B,fg_{B})$ is a Riemannian submersion.

To prove that $M$ is complete, we consider an arbitrary Cauchy sequence
$\left\\{p_{n}\right\\}_{n}$ in $M$ and prove that it is convergent. We
consider the sequence $\left\\{q_{n}=\pi(p_{n})\right\\}_{n}\subset B.$ First
notice that
$\left\langle{v,v}\right\rangle_{M}\geq\left\langle{d\pi(v),d\pi(v)}\right\rangle_{B}$
for any point $p\in M$ and any tangent vector field $v\in T_{p}M.$ Then,
$\mathop{\rm Length}\nolimits_{M}(\gamma)\geq\mathop{\rm
Length}\nolimits_{B}(\pi(\gamma))$ for any curve $\gamma\subset M$. It follows
that $\left\\{q_{n}\right\\}_{n}$ is a Cauchy sequence in $B$ and, since $BM$
is complete, $\left\\{q_{n}\right\\}_{n}$ converges to a point $q\in B$. In
particular, we can assume that $\left\\{q_{n}\right\\}_{n}$ is contained in a
compact and simply connected subset $K\subset B$ Let $F_{0}\colon K\to M$ be a
local section, then, for any $n,$ there exists $t_{n}\in\mathbb{R}$ such that
$p_{n}=\phi_{t_{n}}(q_{n}).$ Denote by $c=\min_{K}\mu$. Then, for any
$p\in\pi^{-1}(K)$ and any vector field $v\in T_{p}M$, we have
$\left\langle{v},{v}\right\rangle_{M}\geq
c\left\langle{d\pi^{\bot}(v)},{d\pi^{\bot}(v)}\right\rangle_{\mathbb{R}}$.
This implies that $\|p_{i}-p_{j}\|_{M}\geq c|t_{i}-t_{j}|$ for any
$i,j\in\mathbb{N},$ that is, $\left\\{t_{n}\right\\}_{n}$ is a Cauchy sequence
in $(\mathbb{R},g_{euc})$. Since $(\mathbb{R},g_{euc})$ is complete, we can
assume that there exist $a,b\in\mathbb{R}$ such that $t_{n}\in[a,b]$ for any
$n$. It follows that $\left\\{p_{n}\right\\}_{n}$ is contained in the compact
subset of $\pi^{-1}(K)$ delimited by $\phi_{a}(F_{0})$ and $\phi_{b}(F_{0}).$
Hence, $\left\\{p_{n}\right\\}_{n}$ is a Cauchy sequence in a compact domain,
that is convergent and this completes the proof. ∎

######  Lemma 2.4.

Let $\pi:S\to B$ be a Killing submersion with Killing vector field of constant
norm. Then, if ${\rm dim}(S)=2$, $S$ has non-negative Gaussian curvature.

###### Proof.

Given a point $p\in S$ and an horizontal vector $v\in\xi^{\perp}(p)$ with
unit-length, $\|v\|=1$, in order to obtain the Gaussian curvature, _i.e._ ,
the sectional curvature ${\rm sec}(v,\xi)$ of the plane spanned by $v$ and
$\xi$, let us consider a vector field $\overline{X}\in\mathfrak{X}(B)$ defined
in a neighborhood $U\ni\pi(p)$, such that $\overline{X}(\pi(p))=d\pi(v)$ and
with vanishing covariant derivative $\nabla^{B}_{\overline{X}}\overline{X}=0$
in $B$, _i.e._ , a geodesic vector field. Then, the lift $X\in\mathfrak{X}(S)$
of $\overline{X}$ defined in $\pi^{-1}(U)\ni p$ satisfies

(1) $\nabla_{X}X=(\nabla_{X}X)^{H}+\langle\nabla_{X}X,\xi\rangle\xi=-\langle
X,\nabla_{X}\xi\rangle=0.$

Where here the superscript $H$ denotes the horizontal part of a vector and we
have used that since $\xi$ is a Killing vector fiels $\langle
X,\nabla_{X}\xi\rangle=0$. Then,

$\displaystyle{\rm sec}(v,\xi)=$ $\displaystyle\langle
R(X,\xi)X,\xi\rangle=\langle\nabla_{\xi}\nabla_{X}X-\nabla_{X}\nabla_{\xi}X+\nabla_{[X,\xi]}X,\xi\rangle$
$\displaystyle=$
$\displaystyle\langle-\nabla_{X}\nabla_{\xi}X+\nabla_{[X,\xi]}X,\xi\rangle=\langle-\nabla_{X}\left([\xi,X]+\nabla_{X}\xi\right)+\nabla_{[X,\xi]}X,\xi\rangle$
$\displaystyle=$
$\displaystyle\langle\nabla_{X}\left([X,\xi]-\nabla_{X}\xi\right)+\nabla_{[X,\xi]}X,\xi\rangle$

In order to simplify the expression let us define the following vector fields
$Y:=\nabla_{X}\xi$ and $Z:=[X,\xi]$. Observe that both $X,Y$ are horizontal
vector fields because $\xi$ has constant norm and thence
$\langle\nabla_{X}\xi,\xi\rangle=\frac{1}{2}X\|\xi\|=0$. Moreover

$\langle
Z,\xi\rangle=\langle\nabla_{X}\xi-\nabla_{\xi}X,\xi\rangle=\langle\nabla_{X}\xi,\xi\rangle-\langle\nabla_{\xi}X,\xi\rangle=\frac{1}{2}X\langle\xi,\xi\rangle=0$

Therefore,

$\displaystyle{\rm sec}(v,\xi)=$
$\displaystyle\langle\nabla_{X}\left(Z-Y\right)+\nabla_{Z}X,\xi\rangle=\langle\nabla_{X}Z+\nabla_{Z}X,\xi\rangle-\langle\nabla_{X}Y,\xi\rangle$
$\displaystyle=$ $\displaystyle\|Y\|^{2}\geq 0.$

where we have used

$\langle\nabla_{X}Z,\xi\rangle=-\langle Z,\nabla_{X}\xi\rangle=\langle
X,\nabla_{Z}\xi\rangle=-\langle\nabla_{Z}X,\xi\rangle$

and $\langle\nabla_{X}Y,\xi\rangle=-\langle
Y,\nabla_{X}\xi\rangle=-\|Y\|^{2}$.∎

######  Lemma 2.5.

Let $S$ be a complete surface with non-negative Gaussian curvature. Then, $S$
is a parabolic manifold.

###### Proof.

This lemma is a direct consequence of a well known theorem due to Huber [9]
which states that if the negative part of the curvature $K_{-}=\max\\{-K,0\\}$
has finite integral, namely,

(2) $\int_{S}K_{-}\,dA<\infty,$

then, $\int_{M}K\,dA\leq\chi(M)$ and $M$ is conformally equivalent to a
compact Riemann surface with finitely many punctures and hence it is a
parabolic surface.∎

######  Lemma 2.6.

Let $\pi:S\to\mathbb{R}$ be a Killing submersion with never vanishing Killing
vector field $\xi\in\mathfrak{X}(M)$. Let $f:\mathbb{R}\to\mathbb{R}$ be a
smooth function. Then

$\Delta^{S}(f\circ\pi)(q)=\frac{1}{\mu(x)}\frac{d}{dx}\left(\mu(x)\frac{df}{dx}\right)_{x=\pi(q)},$

where $\mu(x)$ is the norm of the Killing vector field $\|\xi(p)\|$ for any
$p\in\pi^{-1}(x)$.

###### Proof.

Let $\pi:(M,\langle,\rangle_{M})\to(B,\langle,\rangle_{B})$ be a Rieamannian
submersion. The Laplacian of a basic function ($\widetilde{f}=f\circ\pi,\quad
f:B\to\mathbb{R}$) is given by (see [3, Lemma 3.1] for instance)

(3)
$(\triangle^{M}\widetilde{f})_{q}=(\triangle^{B}f)_{x}-\langle\nabla^{B}f_{p},\,d\pi_{q}(H_{q})\rangle_{B},$

where $x=\pi(q)$, and $H$ is the mean curvature vector field of the fiber
$\pi^{-1}(p)$. In our case taking $B=\mathbb{R}$ and $M=S$ this implies

(4)
$(\triangle^{S}\widetilde{f})_{q}=\left.\frac{d^{2}f}{dx^{2}}\right|_{x=\pi(q)}-\left.\frac{df}{dx}\right|_{x=\pi(q)}\left\langle\frac{\partial}{\partial
x},\,d\pi_{q}(H_{q})\right\rangle_{\mathbb{R}},$

where $\frac{\partial}{\partial x}$ is the unit vector tangent to
$\mathbb{R}$. Taking into account that the mean curvature vector field is
given by

$H=\nabla_{\frac{\xi}{\mu}}\frac{\xi}{\mu}=\frac{1}{\mu^{2}}\nabla_{\xi}\xi.$

Then,

(5) $\displaystyle\left\langle\frac{\partial}{\partial
x},\,d\pi_{q}(H_{q})\right\rangle_{\mathbb{R}}=$
$\displaystyle\frac{1}{\mu^{2}}\left\langle\frac{\partial}{\partial
x},\,d\pi(\nabla_{\xi}\xi)\right\rangle_{\mathbb{R}}=\frac{1}{\mu^{2}}\left\langle
X,\,\nabla_{\xi}\xi\right\rangle$ $\displaystyle=$
$\displaystyle\frac{-1}{\mu^{2}}\left\langle\xi,\,\nabla_{X}\xi\right\rangle=\frac{-1}{2\mu^{2}}X\|\xi\|^{2}=\frac{-1}{2\mu^{2}}\frac{d}{dx}\mu^{2}$

where $X$ is an horizontal lift of $\frac{\partial}{\partial x}$ and we have
used that $\xi$ is a Killing vector field. Finally equation (4) becomes

(6)
$(\triangle^{S}\widetilde{f})_{q}=\left.\frac{d^{2}f}{dx^{2}}\right|_{x=\pi(q)}+\left.\frac{df}{dx}\right|_{x=\pi(q)}\frac{1}{\mu(x)}\left.\frac{d\mu}{dx}\right|_{x=\pi(q)},$

and the proposition is proved.∎

By using the previous lemmas, instead of study the paraboilcity of a complete
surface $(S,g)$ which admits a Killing submersion $\pi:S\to M^{1}$ to the
connected $1$-dimensional Riemannian manifold $(M,g_{\rm can})$, we will study
the parabolicity of the conformally equivalent Riemannian manifold
$(S,\frac{1}{\mu^{2}}g)$ where $\mu(x)$ is the norm of the never vanishing
Killing vector field $\|\xi(p)\|$ for any $p\in\pi^{-1}(x)$. By the lemma 2.2
$(S,g)$ is parabolic if and only if $(S,\frac{1}{\mu^{2}}g)$ is parabolic.
Observe that since $S$ is complete $\pi$ needs to be a surjective map.
Moreover by lemma 2.3, the map $\pi$ induces a Riemannian submersion from
$(S,\frac{1}{\mu^{2}}g)$ to $(M,\frac{1}{\mu^{2}}g_{\rm can})$ . Since this
submersion has a Killing vector field of constant norm,
$(S,\frac{1}{\mu^{2}}g)$ is a surface with non-negative Gaussian curvature by
lemma 2.4. When $M=\mathbb{S}^{1}(R)$ or $M=\mathbb{R}^{1}$ and

$\int_{-\infty}^{0}\frac{dx}{\mu(x)}=\infty\quad{\rm
and}\quad\int_{0}^{\infty}\frac{dx}{\mu(x)}=\infty.$

we have that $(M,\frac{1}{\mu^{2}}g_{\rm can})$ is complete. Hence by using
lemma 2.3 $S$ is a complete surface with non-negative Gaussian curvature and
thus by lemma 2.5, the surface $S$ is a parabolic manifold. On the other hand,
if we assume that

$\int_{-\infty}^{0}\frac{dx}{\mu(x)}<\infty\quad\left({\rm
or}\quad\int_{0}^{\infty}\frac{dx}{\mu(x)}<\infty\right).$

By lemma 2.6 the function $F:M\to\mathbb{R}$

$F(p):=\int_{\pi(p)}^{0}\frac{dx}{\mu(x)}$

is a bounded and harmonic function (which implies that $S$ is a non parabolic
manifold). This can sumarized in the following theorem which implies the Main
theorem of the parer:

######  Theorem 2.7.

Let $\pi:S\to M^{1}$ be a Killing Submersion from a complete and
$2$-dimensional Riemannian manifold $(S,g)$ to the connected $1$-dimensional
Riemannian manifold $(M,g_{\rm can})$. Let us denote by $\mu(x)$ the norm of
the Killing vector field $\|\xi(p)\|$ for any $p\in\pi^{-1}(x)$. Then,

  1. (1)

If $M=\mathbb{S}^{1}(R)$ endowed with the canonical metric tensor, $S$ is
parabolic.

  2. (2)

If $M=\mathbb{R}$ with its canonical metric tensor, $S$ is a parabolic
manifold iff

$\int_{-\infty}^{0}\frac{dx}{\mu(x)}=\infty\quad{\rm
and}\quad\int_{0}^{\infty}\frac{dx}{\mu(x)}=\infty.$

##  3\. Application of the Main Theorem: Parabolicity of invariant surfaces in
homogeneous 3-manifolds

In this section we use Main Theorem to study the parabolicity of invariant
surfaces with some geometric properties in 3-dimensional homogeneous
Riemannian manifold studied in different works by many authors.

###  3.1. Parabolicity of invariant surfaces with constant mean curvature in
$Sol_{3}$

In [12], Lopez and Munteanu give a description of invariant surfaces with
either constant curvature in the Thurston geometry $Sol_{3}$. Recall that
$Sol_{3}$ is isometric to $\mathbb{R}^{3}$ endowed with the metric

$e^{2z}dx^{2}+e^{-2z}dy^{2}+dz^{2}$

and the component of the identity its isometry group is generated by the
following families of isometries:
$T_{1}^{c}(x,y,z)=(x+c,y,z),\quad T_{2}^{c}(x,y,z)=(x,y+c,z),\quad
T_{3}^{c}(x,y,z)=(e^{-c}x,e^{c}y,z+c).$
In particular, in [12] they studied surfaces with constant mean curvature or
constant Gaussian curvature invariant with respect to $T_{1}$, finding a
profile curve $\gamma(s)=(0,y(s),z(s))$ parameterized by arc length. Notice
that the Killing vector field $\partial_{x}$ associated to $T_{1}$ has norm
$\|\partial_{x}\|=e^{z}$ and it is orthogonal to $\gamma^{\prime}$, so we can
easely apply our result just studyng $\int e^{-z(s)}ds$.

The classification of minimal surfaces [12, Theorem 3.1] assures that the only
$T_{1}$-invariant minimal surfaces of $Sol_{3}$ are:

  1. (1)

a leaf of the foliation
$\left\\{Q_{t}=\left\\{(x,t,z)\mid\,x,z\in\mathbb{R}\right\\}\right\\}_{t\in\mathbb{R}}$,
which are known to be isometric to the hyperbolic plane;

  2. (2)

a leaf of the foliation
$\left\\{R_{t}=\left\\{(x,y,t)\mid\,x,y\in\mathbb{R}\right\\}\right\\}_{t\in\mathbb{R}}$,
which are known to be isometric to the Euclidean plane;

  3. (3)

the surfaces $S_{\theta_{0},a}$, for
$\theta_{0}\in\mathbb{R}\setminus\left\\{k\pi:\,k\in\mathbb{Z}\right\\}$ and
$a\in\mathbb{R}$, that are generated by translating the profile curve

$\gamma(s)=\left(0,a+e^{s\sin(\theta_{0})},\log\left(\tan(\theta_{0})e^{s\sin(\theta_{0})}\right)\right).$

Obviously, the surfaces $Q_{t}$ are hyperbolic, while the surfaces $R_{t}$ are
parabolic. It remains to study the parabolicity of $S_{\theta_{0},a}.$ Since

$\int
e^{-\log\left(\tan(\theta_{0})e^{s\sin(\theta_{0})}\right)}ds=e^{-s\sin(\theta_{0})}\cot(\theta_{0})\csc(\theta_{0}),$

is a bounded function, Main Theorem implies that $S_{\theta_{0},a}$ are
hyperbolic surfaces.

For the constant mean curvature case, [12, Theorem 3.2] proves that the
$z$-coordinate of the profile curve is bounded. In particular,

$\lim_{s\to\pm\infty}\int e^{-z(s)}ds=\pm\infty$

and Main Theorem implies that the respective invariant $H$-surfaces are
parabolic.

###  3.2. Parabolicity of vertical cylinders in Killing submersion

An easy way to study surface surfaces immersed in a three-manifold which are
invariant with respect to a Killing vector field of the ambient space is by
studying vertical cylinders in Killing submersions. We recall that a Killing
submersion is a Riemannian submersion $\pi\colon\mathbb{E}\to M$ from a three-
dimensional manifold $\mathbb{E}$ onto a surface $(M,g)$, both connected and
orientable, such that the fibers of $\pi$ are integral curves of a Killing
vector field $\xi\in\mathfrak{X}(\mathbb{E}).$ These spaces have been
completely classified in terms of the Riemannian surface $(M,g)$, the length
of the Killing vector field $\mu=\|\xi\|$ and the so called bundle curvature
$\tau,$ defined such that
$\tau(p)=\frac{-1}{\mu(p)}\langle\overline{\nabla}_{u}\xi,v\rangle$, where
$\\{u,v,\xi_{p}/\mu(p)\\}$ is a positively oriented orthonormal basis of
$T_{p}\mathbb{E}$ (see [11, Section 2]). In this setting a vertical cylinder
is a surface $S$ always tanget to $\xi$. In particular, $S$ is invariant with
respect to the isometries associated to $\xi$ and it projects through $\pi$
onto a curve $\gamma\subset M$. Since $\pi$ is a Riemannian submersion, to
study $\int_{\Gamma}\frac{1}{\mu(\Gamma)}$, where $\Gamma\subset S$ is a curve
parameterized by arc length and orthogonal to $\xi$, is equivalent to study
$\int_{\gamma}\frac{1}{(\pi_{*}\mu)(\gamma)}$ where $\gamma=\pi(S)$ is
parameterezid by arc length in $(M,g)$. In particular, in this setting, the
Main Theorem, read as follows.

######  Theorem 3.1.

Let $\pi\colon\mathbb{E}\to(M,g)$ be a Killing submersion with Killing length
$\mu$. Then, for any complete curve $\gamma\subset M$, $\pi^{-1}(\gamma)$ is a
parabolic surface if and only if $\gamma$ is complete in
$(M,\tfrac{1}{\mu^{2}}g)$.

######  Remark 3.2.

It is interesting to notice that, given a curve $\gamma\subset M$, while the
conformal metric $\tfrac{1}{\mu^{2}}g$ gives us information about the
parabolicity of $S=\pi^{-1}$, the conformal metric $\mu^{2}g$ gives us
information about its mean curvature (see [5, Proposition 2.3]).

The parabolicity of surfaces of revolution (or surfaces with ends of
revolution) in spaces forms of constant sectional curvature has been studied
in [7]. Three dimensional simply connected spaces forms of constant sectional
curvature are the only manifolds with a group of isometries of dimension $6$
and can be classified (up to isometries) by their sectional curvatures. We can
label these spaces hence as $\mathbb{M}^{3}(\kappa)$. The spaces
$\mathbb{M}^{3}(\kappa)$ includes the Euclidean space for $\kappa=0$, the
spheres for $\kappa>0$ and the Hyperbolic space for $\kappa<0$.

There are no three dimensional spaces with a group of isometries of dimension
$5$. The simply-connected $3$-dimensional spaces with a group of isometries of
dimension $4$ are Killing submersions and they can be classified (up to
isometries) by the curvature $\kappa$ of the base manifold and by the torsion
$\tau$ of the fibers. We can label these spaces hence as
$\mathbb{E}^{3}(\kappa,\tau)$ (because they are endowed with the Riemannian
submersion $\pi:\mathbb{E}^{3}(\kappa,\tau)\to\mathbb{M}^{2}(\kappa)$ with
constant bundle curvature $\tau$). In what follows we give a general condition
to guarantee the parabolicity of rotational surfaces in the
$\mathbb{E}^{3}(\kappa,\tau)$-spaces.

The canonical rotational model describing the
$\mathbb{E}^{3}(\kappa,\tau)$-spaces is given by
$(\Omega\times\mathbb{R},ds^{2})$, where

$\begin{array}[]{l}\Omega=\left\\{(x,y)\in\mathbb{R}^{2}\mid\lambda(x,y)>0\right\\},\qquad\lambda(x,y)=\frac{1}{1+\tfrac{\kappa}{4}(x^{2}+y^{2})},\\\
ds^{2}=\lambda^{2}(x,y)(dx^{2}+dy^{2})+(\lambda(x,y)\tau(ydx-
xdy)+dz)^{2}.\end{array}$

Since the Killing vector field $\|\partial_{z}\|$ has unitary norm, every
surface that is invariant with respect to $\partial_{z}$ is parabolic. So, we
focus on studying the parabolicity rotational surfaces. First notice that if
$\kappa>0$, $\mathbb{E}(\kappa,\tau)$ is a Berger sphere, in particular, since
it is compact, every Killing vector field $\xi$ has bounded norm and from the
Main Theorem we deduce that every surface that is invariant with respect to
$\xi$ is parabolic. When $\kappa\leq 0$, we consider the Killing vector field
$\xi(x,y)=-y\partial_{x}+x\partial_{y}$ generating the rotation around the
$z$-axis and to describe $\mathbb{E}(\kappa,\tau)$ as a Killing submersion
with respect to $\xi$, we use the cylindrical coordinates

$x(r,\theta)=r\cos(\theta),\qquad y(r,\theta)=r\sin(\theta).$

We obtain that the space $\mathbb{E}(\kappa,\tau)$ minus its $z$-axis is
isometric to a quotient of
$(\mathbb{R}^{2}_{\kappa}\times\mathbb{R},ds^{2}_{rot})$, where
$\mathbb{R}^{2}_{\kappa}=\left\\{(r,z)\in\mathbb{R}^{2}|r>0,\,4+\kappa
r^{2}>0\right\\}$ and

$ds^{2}_{rot}=\frac{16dr^{2}}{(4+\kappa
r^{2})^{2}}+\frac{dz^{2}}{(1+r^{2}\tau^{2})}+\left(\frac{2r\sqrt{1+r^{2}\tau^{2}}}{4+\kappa
r^{2}}\right)^{2}\left(d\theta-\frac{(4+\kappa
r^{2})\tau}{4(1+r^{2}\tau^{2})}dz\right)^{2}.$

Here, $\xi=\partial\theta$ and the Killing submersion with respect to $\xi$ is
such that

$(M,g)=\left(\mathbb{R}^{2}_{\kappa},\frac{16}{(4+\kappa
r^{2})^{2}}dr^{2}+\frac{1}{(1+r^{2}\tau^{2})}dz^{2}\right)\quad\textrm{ and
}\quad\mu(r,z)=\frac{2r\sqrt{1+r^{2}\tau^{2}}}{4+\kappa r^{2}}$

and then the conformal metric tensor $g/\mu^{2}$ is given by

(7) $\frac{1}{\mu^{2}}g=\frac{4}{r^{2}(1+r^{2}\tau^{2})}dr^{2}+\frac{(4+\kappa
r^{2})^{2}}{4r^{2}(1+r^{2}\tau^{2})^{2}}dz^{2}.$

In particular, every complete curve $\gamma$ in $(M,g)$ generetes a complete
parabolic invariant surface in $\mathbb{E}^{3}(\kappa,\tau)$ if and only if
$\gamma$ is complete with respect Using the metric (7). For example, using
this tool, it is easy to see that the minimal umbrellas of the Heisenberg
group $\mathbb{E}^{3}(0,\tau)$ are hyperbolic, without computing their
extrinsic area growth (see [15]). It is sufficient to consider the curve
$\gamma(t)=(t,0)$. Its norms is in the conformal metric is
$\|\gamma(t)\|=\frac{2}{t\sqrt{1+t^{2}\tau^{2}}}<t^{-3/2}$ for
$t>\frac{2+\sqrt{4-\tau^{2}}}{\tau^{2}}$ or $\tau>2$, that is, $\gamma$ is not
complete in the conformal metric, thus $\pi^{-1}(\gamma)$ is hyperbolic.

We can also use this tool to study the parabolicity of the rotational surfaces
of constant mean curvature in $\mathbb{E}^{3}(-1,\tau)$ described in [17]. In
particular, Peñafiel shows that a rotational surface of constant mean
curvature $H\in\mathbb{R}$ is parameterized by
$\gamma_{d}(t)=\left(\tanh\left(\tfrac{\sqrt{t}}{2}\right),u_{d}(t)\right),$
where

$u_{d}(t)=\int\frac{(2H\cosh(r)+d)\sqrt{1+4\tau^{2}\tanh^{2}\left(\tfrac{r}{2}\right)}}{\sqrt{\sinh^{2}(r)-(2H\cosh(r)+d)^{2}}},$

with $d\in\mathbb{R}.$ When $d=-2H$, the rotation of $\gamma_{d}(t)$ generates
an entire graph. When $H=0$, the norm of $\gamma^{\prime}(t)$ with respect to
(7) is

$\|\gamma^{\prime}(t)\|=2\sqrt{\frac{1}{\sinh^{2}(t)\left(1+\tau^{2}\tanh\left(\tfrac{t}{2}\right)\right)}}\simeq
4\sqrt{\frac{1}{1+\tau^{2}}}e^{-t}+o(e^{-2t}).$

In particular, $\lim_{t\to\infty}\int\|\gamma^{\prime}(t)\|$ is convergent and
the rotational end generated by the rotation of $\gamma$ is hyperbolic.
Furthermore, since the difference between $u_{d}$ and $u_{-2H}$ is bounded,
the surface generated by rotating any $\gamma_{d}$ is hyperbolic.

On the contrary, when $H=1/2$, we get that

$\|\gamma^{\prime}(t)\|=\left(\frac{(5+3\cosh(t))^{2}(1-4\tau^{2}+\cosh(t)+4\tau^{2}\cosh(t))}{8(1-\tau^{2}+(1+\tau^{2})\cosh(t))^{2}}+\frac{4}{\sinh^{2}(t)\left(1+4\tau^{2}\tanh\left(\tfrac{t}{2}\right)\right)}\right)^{\tfrac{1}{2}}$

which diverges for $t\to+\infty$. That is, the entire rotational graph with
critical constant mean curvature is parabolic.

## References

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# Language-guided Human Motion Synthesis with Atomic Actions

Yuanhao Zhai  University at BuffaloBuffaloNYUSA<EMAIL_ADDRESS>,  Mingzhen
Huang  University at BuffaloBuffaloNYUSA<EMAIL_ADDRESS>,  Tianyu Luan
University at BuffaloBuffaloNYUSA<EMAIL_ADDRESS>,  Lu Dong  University
at BuffaloBuffaloNYUSA<EMAIL_ADDRESS>,  Ifeoma Nwogu  University at
BuffaloBuffaloNYUSA<EMAIL_ADDRESS>,  Siwei Lyu  University at
BuffaloBuffaloNYUSA<EMAIL_ADDRESS>,  David Doermann  University at
BuffaloBuffaloNYUSA<EMAIL_ADDRESS>and  Junsong Yuan  University at
BuffaloBuffaloNYUSA<EMAIL_ADDRESS>
(2023)

###### Abstract.

Language-guided human motion synthesis has been a challenging task due to the
inherent complexity and diversity of human behaviors. Previous methods face
limitations in generalization to novel actions, often resulting in unrealistic
or incoherent motion sequences. In this paper, we propose ATOM (ATomic mOtion
Modeling) to mitigate this problem, by decomposing actions into atomic
actions, and employing a curriculum learning strategy to learn atomic action
composition. First, we disentangle complex human motions into a set of atomic
actions during learning, and then assemble novel actions using the learned
atomic actions, which offers better adaptability to new actions. Moreover, we
introduce a curriculum learning training strategy that leverages masked motion
modeling with a gradual increase in the mask ratio, and thus facilitates
atomic action assembly. This approach mitigates the overfitting problem
commonly encountered in previous methods while enforcing the model to learn
better motion representations. We demonstrate the effectiveness of ATOM
through extensive experiments, including text-to-motion and action-to-motion
synthesis tasks. We further illustrate its superiority in synthesizing
plausible and coherent text-guided human motion sequences.

language-guided human motion synthesis, atomic action, masked motion modeling,
curriculum learning

††journalyear: 2023††copyright: acmlicensed††conference: Proceedings of the
31st ACM International Conference on Multimedia; October 29-November 3, 2023;
Ottawa, ON, Canada††booktitle: Proceedings of the 31st ACM International
Conference on Multimedia (MM ’23), October 29-November 3, 2023, Ottawa, ON,
Canada††price: 15.00††doi: 10.1145/3581783.3612289††isbn:
979-8-4007-0108-5/23/10††ccs: Computing methodologies Activity recognition and
understanding Figure 1. Motion synthesis results comparison. Previous T2M (Guo
et al., 2022a) generates an unrealistic motion transition between “walk
forward” and “turn left” (see motion inside the red box), while our ATOM
generates coherent motion. The color saturation increases as the motion
progresses.

##  1\. Introduction

Language-guided human motion synthesis is a critical and challenging task,
with widespread applications in virtual reality, video games, animation, and
human-computer interaction. The ability to generate realistic and diverse
human motions based on textual descriptions can enable more intuitive control
over virtual characters, as well as seamless integration of user-generated
content in various multimedia systems. Despite previous conditional action
generation models (Zhang et al., 2020; Petrovich et al., 2021; Du et al.,
2015; Yan et al., 2018; Yu et al., 2020; Kim et al., 2020; Cai et al., 2021;
Xie et al., 2021; Dittadi et al., 2021; Wang et al., 2021) have exploited
leveraging closed-set action labels to synthesize human motion, few of them
can work beyond the training action labels, not to mention open language
descriptions.

In recent years, various methods have been proposed for the language-guided
human motion synthesis task (Ghosh et al., 2021; Goutsu and Inamura, 2021;
Delmas et al., 2022; Zhang et al., 2022; Guo et al., 2022a; Athanasiou et al.,
2022; Petrovich et al., 2022; Guo et al., 2022b; Lucas et al., 2022; Wang et
al., 2022; Hong et al., 2022; Tevet et al., 2022a, b). While these methods
have demonstrated considerable progress in generating human motions, they may
suffer from synthesizing discontinuities and unrealistic motion transitions
when dealing with actions that are not well-represented in the dataset. As
these methods rely heavily on the availability and diversity of training data,
they may struggle to generate plausible motion sequences for rare or unseen
actions, leading to abrupt transitions and incoherent movement patterns.
Furthermore, when synthesizing complex sequential motion behaviors, these
methods often struggle with capturing dependencies in motion sequences, which
is crucial for ensuring smooth and natural motion transitions between
different types of actions, as illustrated in Fig. 1.

We propose ATOM (ATomic mOtion Modeling), a novel approach for language-guided
human motion synthesis that effectively addresses the limitations of previous
methods. First, ATOM decomposes actions into atomic components, enabling the
generation of diverse and coherent motion sequences by assembling the learned
atomic actions. Additionally, by employing a masked motion modeling curriculum
learning strategy, our method learns more expressive motion representations
and effectively captures long-range dependencies in motion sequences.

In our proposed ATOM, we utilize a transformer-based conditional variational
autoencoder (CVAE) framework (Vaswani et al., 2017) to achieve atomic action
decomposition and assembly, which provides a more expressive and flexible
representation of human motions. The atomic action codebook, designed as a set
of learnable feature vectors, serves as the key and value for the cross-
attention module in the Transformer decoder. These atomic actions, learned in
an end-to-end manner, can effectively represent a wide range of short-term
basic human movements, such as raising hands and lifting legs. Fig. 2
showcases an example, where the “walking” action can be decomposed into a set
of atomic actions. To ensure the atomicity and effectiveness of the learned
actions, we apply two additional constraints: diversity and sparsity selection
constraints. The diversity constraint encourages the atomic actions to be
distinct from each other, allowing for a richer and more versatile action
representation. The sparsity selection constraint enforces atomic actions to
be compact and focused during composition, ensuring that the learned atomic
actions retain their atomicity. This constraint aids in the efficient
reconstruction of complex actions by combining a minimal set of atomic actions
while preserving their individual characteristics. The combination of the
Transformer-based architecture and these constraints allows our ATOM to
efficiently decompose and assemble complex actions for improved human motion
synthesis, ultimately generating more diverse, coherent and realistic motion
sequences based on language input.

Figure 2. Illustration of atomic action. The action “a person is walking” can
be decomposed into a set of atomic actions: from top to bottom, whole body
translation, lifting leg, hand movements and so on.

Alongside the atomic action decomposition, we incorporate a curriculum
learning strategy into our ATOM to further improve the robustness of
generating coherent and diverse motion sequences while capturing long-range
dependencies. This strategy is based on masked motion modeling and involves
gradually increasing the mask ratio during the training process. Such a
progressive learning approach provides the model with an opportunity to learn
simpler patterns and dependencies in the early stages of training, while
incrementally introducing more complex and challenging aspects of the motion
sequences as training progresses. This approach not only mitigates potential
convergence issue, but also helps the model build a more robust and
generalizable understanding of motion data, encompassing both local and long-
range motion patterns. As a result, ATOM is better equipped to synthesize
realistic and diverse human motions based on language inputs, even when
dealing with rare or unseen action labels/descriptions.

To summarize, our contributions are as follows:

  * •

We introduce a transformer-based CVAE framework that decomposes complex
actions into a set of atomic actions, enabling more effective representation
and manipulation of motion sequences. This approach allows for generating
diverse and coherent motion sequences, even for rare or unseen action
labels/descriptions.

  * •

We incorporate a curriculum learning strategy based on masked motion modeling,
which gradually increases the mask ratio during training. This strategy
enables ATOM to better capture dependencies in motion sequences, ensuring
smooth and natural motion transitions between different actions.

  * •

We provide a comprehensive evaluation of ATOM, demonstrating its effectiveness
in generating coherent and diverse motion sequences. Our method significantly
improves over previous approaches in the text-to-motion and action-to-motion
tasks.

##  2\. Related Work

Human motion synthesis Directly synthesizing human motion has always been a
challenging yet ideal task in computer vision and graphics. As the field has
progressed, methods have evolved from generating skeleton-based motion
synthesis (Du et al., 2015; Yan et al., 2018; Yu et al., 2020; Kim et al.,
2020; Cai et al., 2021) to more realistic SMPL motion synthesis (Petrovich et
al., 2021; Xie et al., 2021; Dittadi et al., 2021; Wang et al., 2021),
bringing us closer to generating authentic human motions in real-world
scenarios (Zhang et al., 2021; Luan et al., 2021, 2023). Tilmanne  _et al_.
(Tilmanne and Dutoit, 2010) employed Gaussian distributions to model the
variability of walk cycles for each emotion and the length of each cycle.
Zhang _et al_.(Zhang et al., 2020) further explored generating unbounded human
motion through a cross-conditional, two-stream variational RNN architecture.
CSGN (Yan et al., 2019) jointly modeled structures in temporal and spatial
dimensions, allowing bidirectional transforms between the latent and observed
spaces to handle semantic manipulation of action sequences. SA-GCNs (Yu et
al., 2020) proposed a variant of GCNs that leverages the self-attention
mechanism to adaptively sparsify a complete action graph in the temporal
space. ACTOR (Petrovich et al., 2021) learned an action-aware latent
representation for human motions by training a transformer-based VAE.
ActFormer (Petrovich et al., 2021) focused on multi-person interactive
actions, combining the solid spatio-temporal representation capacity of the
Transformer, the generative modeling superiority of GANs, and the inherent
temporal correlations from latent prior. INR (Cervantes et al., 2022) employs
variational implicit neural representations to generate variable-length
sequences. Recently, a growing body of work has emerged, focusing on
leveraging natural language to synthesize human motions (Ahuja and Morency,
2019; Ghosh et al., 2021; Goutsu and Inamura, 2021; Delmas et al., 2022; Zhang
et al., 2022; Guo et al., 2022a; Athanasiou et al., 2022; Petrovich et al.,
2022; Guo et al., 2022b; Lucas et al., 2022; Wang et al., 2022; Hong et al.,
2022; Tevet et al., 2022a, b). Specifically, Language2Pose (Ahuja and Morency,
2019) learns a joint embedding of language and pose decoder to generate pose
sequences. Text2gesture (Bhattacharya et al., 2021) exploits relevant
biomechanical features for body expression to create emotive body gestures.
Hier (Ghosh et al., 2021) introduces a self-supervised method for generating
long-range behaviors. T2M (Guo et al., 2022a) employs a curated language
encoder to learn crucial words and a duration estimator to synthesize human
motions of varying durations.

Language-guided generation Language-guided generation establishes a connection
between visual representation and semantic space, enabling more precise
control and increased creative possibilities. Synthesizing motion from
language, especially when dealing with multiple, varied actions is a
challenging task compared to generating images with specific action labels.
Numerous prior language-guided generation methods have focused on image
generation (Zhang et al., 2017; Li et al., 2020a; Nam et al., 2018; Liu et
al., 2020; Xia et al., 2021; Li et al., 2020b). Reed  _et al_. (Reed et al.,
2016) proposed using a generative adversarial network (Goodfellow et al.,
2014) (GAN) conditioned on text embeddings for image synthesis. DALL·E (Ramesh
et al., 2021) employed a discrete variational autoencoder (dVAE) to generate
diverse images based on text embeddings from GPT-3 (Brown et al., 2020). The
recently proposed CLIP (Radford et al., 2021) jointly learns a multi-modal
vision-language embedding space with impressive capabilities. Leveraging the
power of CLIP (Radford et al., 2021), StyleCLIP (Patashnik et al., 2021)
extends StyleGAN (Karras et al., 2019) to a language-driven generation model
using their proposed CLIP-guided mapper. Meanwhile, HairCLIP (Wei et al.,
2021) generates manipulated images from given CLIP (Radford et al., 2021) text
embeddings, further demonstrating the potential of language-driven generation.

##  3\. Method

Figure 3. Framework overview. Our ATOM is composed of an encoder and a
decoder. The encoder processes text embedding of the action label and masked
motion sequence, outputting a latent vector $z$. The decoder receives this
latent vector $z$ along with the text embedding. A set of learnable atomic
actions, referred to as the atomic action codebook, is fed into the decoder as
key and value for the cross-attention layers. As a result, the generated
motion sequences are assembled from the atomic actions.

###  3.1. Problem Formulation

The task of language-guided human motion synthesis aims to generate motion
sequences that accurately represent the given textual description. During the
training phase, the inputs consist of label-motion pairs
$\\{(y_{i},\bm{M}_{i})\\}$, where the label can be either a natural language
description (text-to-motion) or a discrete action class (action-to-motion).
The motion representation $\bm{M}_{i}=[\bm{p}_{1},\dots,\bm{p}_{T}]$ is a
sequence of human body representations with length $T$, where $\bm{p}_{t}$
denotes the human body representation at time $t$. The human body
representation can adopt various forms, such as 3D joint locations or SMPL
parameters (Loper et al., 2015). During the testing phase, the objective is to
synthesize motion sequences based on the input language description or action
class.

###  3.2. Conditional Transformer VAE

We utilize a CVAE-based framework, as shown in Fig. 3, which aligns motion
representations with categorical conditions. Our ATOM consists of an encoder
and a decoder, implemented using Transformer encoder and decoder (Vaswani et
al., 2017). The encoder captures the input motion sequence’s underlying
structure, transforming it into a compact latent representation. The decoder
uses the latent representation and text embedding to generate a realistic
human motion sequence corresponding to the given condition.

Encoder The encoder accepts the conditional embedding of the label $y$ and the
motion representation sequence $[\bm{p}_{t}]$, and computes the Gaussian
distribution parameters $\mu$ and $\Sigma$ for the motion latent space. A
latent variable $\bm{z}$ is then sampled from $N(\mu,\Sigma)$ using the
reparameterization trick (Kingma and Welling, 2013). We prepend two tokens,
$\mu_{0}$ and $\Sigma_{0}$, to the input, allowing their corresponding outputs
to be regarded as the Gaussian distribution parameters. These two input
tokens, $\mu_{0}$ and $\Sigma_{0}$, are derived from the input embedding
through a three-layer multilayer perceptron (MLP). To ensure compatibility,
all motion representations $\bm{p}_{t}$ are transformed to the same dimension
as $\mu_{0}$ and $\Sigma_{0}$ using a linear layer before entering the
Transformer encoder.

Decoder Given a latent vector $\bm{z}$, we first add a conditional bias to it
to incorporate the categorical information. This bias is action-specific and
learned from the input embedding through a three-layer MLP. Subsequently, the
sum is repeated $T$ times in the temporal dimension, and sinusoidal positional
encoding (Vaswani et al., 2017) is added as the query input to the Transformer
decoder. A set of learnable atomic actions is provided to the Transformer
decoder as key and value inputs, enabling the query to be reconstructed using
the atomic actions. The output of the decoder is the reconstructed motion
representation $\hat{\bm{M}}=[\hat{\bm{p}}_{t}]$, where $\hat{\bm{p}}_{t}$ the
predicted body representation at time $t$. It is worth noting that the decoder
generates the entire motion sequence in one shot, as opposed to the
autoregressive approach used in previous works (Petrovich et al., 2021).

Learning Objectives The learning objectives of the CVAE consist of two
components: a reconstruction loss $\mathcal{L}_{\text{rec}}$ and a Kullback-
Leibler (KL) divergence loss $\mathcal{L}_{\text{KL}}$. The reconstruction
loss is employed to minimize the discrepancy between the original motion
representation $\bm{M}$ and the reconstructed motion representation
$\hat{\bm{M}}$ through a mean square error loss:

(1)
$\mathcal{L}_{\text{rec}}=\frac{1}{T}\sum_{t=1}^{T}\left\|\bm{p}_{t}-\hat{\bm{p}}_{t}\right\|_{2}^{2}.$

On the other hand, the KL divergence loss $\mathcal{L}_{\text{KL}}$ minimizes
the distribution difference between the estimated posterior $N(\mu,\Sigma)$
and the prior normal distribution $N(0,I)$. Thus, the CVAE training loss
$\mathcal{L}_{\text{CVAE}}$ is a weighted sum of the two terms:

(2)
$\mathcal{L}_{\text{CVAE}}=\mathcal{L}_{\text{rec}}+w_{\text{KL}}\mathcal{L}_{\text{KL}},$

where $w_{\text{KL}}$ is a weighting hyperparameter.

###  3.3. Atomic Action Codebook

The motivation for using atomic actions in our method stems from the
observation that human motions, despite their apparent complexity, can often
be decomposed into more specific, repetitive, and atomic elements. By breaking
down complex actions into a series of atomic actions, our model can more
effectively learn the underlying structure of human motion and capture the
relationships between different actions. Furthermore, this decomposition
facilitates the generation of diverse and realistic motion sequences, as well
as the synthesis of novel actions by recombining the learned atomic elements.
It also enables our method to better generalize across different action
classes and leverage the power of textual descriptions in guiding the
synthesis process.

Building on the idea of atomic actions, we introduce a learnable atomic action
codebook, which serves as a basis for representing and generating complex
human motions within our Transformer-based architecture. This codebook
consists of a collection of atomic actions that can be combined and assembled
in various ways to produce a diverse range of motion sequences. Formally, the
codebook is implemented as a learnable matrix $\bm{A}\in\mathbb{R}^{N\times
D}$, where $N$ represents the number of atomic actions, and $D$ denotes the
hidden dimension. Each row of the matrix corresponds to an atomic action,
capturing its unique characteristics in the latent space. The codebook is
integrated into the Transformer decoder as key and value, enabling the model
to selectively attend to relevant atomic actions during the decoding process.
This design allows the input conditional embedding to be efficiently
reconstructed by combining the most appropriate atomic actions based on the
cross-attention mechanism in the decoder, leading to more accurate and diverse
motion synthesis. Our experiments in Tab. 6 and Fig. 6 show the effectiveness
of our codebook design compared to the original CVAE.

Table 1. Quantitative results comparison on the HumanML3D test set.
$\rightarrow$ indicates results are better if they are closer to the real
motion, and $\pm$ indicates 95% confidence interval.

Method | FID $\downarrow$ | Diversity $\rightarrow$ | MultiModality $\uparrow$ | R Precision (top3) $\uparrow$ | MultiModal Dist $\downarrow$
---|---|---|---|---|---
Real Motion | $0.002^{\pm.000}$ | $9.503^{\pm.065}$ | - | $0.797^{\pm.002}$ | $2.974^{\pm.008}$
Language2Pose (Ahuja and Morency, 2019) | $11.02^{\pm.046}$ | $7.676^{\pm.058}$ | - | $0.486^{\pm.002}$ | $5.296^{\pm.008}$
Text2Gesture (Bhattacharya et al., 2021) | $7.664^{\pm.030}$ | $6.409^{\pm.071}$ | - | $0.345^{\pm.002}$ | $6.030^{\pm.008}$
Hier (Ghosh et al., 2021) | $6.532^{\pm.024}$ | $8.332^{\pm.042}$ | - | $0.552^{\pm.004}$ | $5.012^{\pm.018}$
T2M (Guo et al., 2022a) | $\bm{0.455^{\pm.003}}$ | $9.175^{\pm.002}$ | $2.219^{\pm.074}$ | $\bm{0.736^{\pm.002}}$ | $\bm{3.347^{\pm.074}}$
MoCoGAN (Tulyakov et al., 2018) | $94.41^{\pm.021}$ | $0.462^{\pm.008}$ | $0.019^{\pm.000}$ | $0.106^{\pm.001}$ | $9.643^{\pm.006}$
Dance2Music (Lee et al., 2019) | $66.98^{\pm.016}$ | $0.725^{\pm.011}$ | $0.043^{\pm.001}$ | $0.097^{\pm.001}$ | $8.116^{\pm.006}$
Ours | $1.691^{\pm.031}$ | $9.312^{\pm.011}$ | $\bm{2.884^{\pm.130}}$ | $0.569^{\pm.004}$ | $5.970^{\pm.004}$

Learning Objectives To ensure the effectiveness of the atomic action codebook
in generating a wide range of motion sequences, we introduce two objectives
for its learning: diversity constraint $\mathcal{L}_{\text{div}}$ and sparsity
constraint $\mathcal{L}_{\text{spa}}$. The diversity constraint ensures that
the learned atomic actions are diverse and unique, so that the learned atomic
actions are diverse enough to represent different actions. By promoting
diversity in the codebook, our model can generate more realistic and rich
motion sequences that cover a broad spectrum of human actions. Formally, the
diversity constraint is formulated as follows:

(3) $\mathcal{L}_{\text{div}}=\|\bm{A}\bm{A}^{\top}-\bm{I}\|_{F},$

where $\bm{A}$ is the atomic action codebook, $\bm{I}$ is the identity matrix,
and $\|\cdot\|_{F}$ is the Frobenius norm. This objective encourages the
learned atomic actions to be orthogonal, promoting diversity and preventing
codebook redundancy. Our experiments in column “diversity” of Sec. 3.3, Tab.
2, Tab. 3, and Tab. 5 show that the diversity of our method is higher than the
previous approaches and closer to real motions on multiple datasets.

The sparsity constraint promotes the use of a sparse set of atomic actions to
represent complex motions, enhancing the atomicity and robustness of the
learned atomic actions. This ensures that generated motion sequences are
concise and meaningful while maintaining the interpretability and
generalizability of the atomic action codebook. We enforce the sparsity
constraint by maximizing the maximal attention values in each cross-attention
layer:

(4) $\mathcal{L}_{\text{spa}}=-\sum_{l}\sum_{h}\max(\bm{H}_{l,h}),$

where $\bm{H}_{l,h}$ is the attention map of the cross-attention for the
$h$-th head in layer $l$. This loss encourages the model to focus on a few
dominant atomic actions when reconstructing motion sequences, leading to a
sparser and more interpretable atomic action codebook.

By incorporating the atomic action constraints, the total loss
$\mathcal{L}_{\text{total}}$ of our ATOM is as follows:

(5)
$\mathcal{L}_{\text{total}}=\mathcal{L}_{\text{CVAE}}+w_{\text{div}}\mathcal{L}_{\text{div}}+w_{\text{spa}}\mathcal{L}_{\text{spa}},$

where $w_{\text{div}}$ and $w_{\text{spa}}$ are weighting hyperparameters.

###  3.4. Masked Motion Modeling Curriculum Learning

Drawing inspiration from the effective representation learning of masked image
autoencoders (He et al., 2022), we introduce masked motion modeling, a
technique that involves temporally masking a random portion of the input
motion sequence at a ratio $r$, and subsequently requiring the model to
reconstruct the entire motion sequence. Masked motion modeling is essential in
the motion synthesis task because it encourages the model to learn robust,
context-aware motion representations by forcing it to fill in the missing
information. This approach results in a more robust and generalized
understanding of the underlying motion structure, ultimately leading to more
realistic motion synthesis.

To further enhance the learning process, we incorporate curriculum learning by
progressively increasing the mask ratio as the training progresses according
to a growth function $g(l)\in\\{r,rl/L,r(l/L)^{2},re^{l/L-1}\\}$, where $l$ is
the current training epoch and $L$ is the total number of epochs. As a result,
in the beginning, a lower mask ratio allows the model to learn basic motion
patterns and capture fundamental structures in the motion data. As the mask
ratio increases, the model is exposed to more challenging and complex motion
sequences, promoting its ability to infer missing information and makes better
use of the learned atomic actions. This curriculum strategy enables a more
effective and stable learning experience, ultimately improving motion
synthesis performance.

##  4\. Experiments

We evaluate our method in three different settings: _text-to-motion_ ,
_action-to-motion_ , and _zero-shot action-to-motion_. Text-to-motiona and
zero-shot action-to-motion are to generate human motion given an input text
prompt; while action-to-motion generates motion given an input action class in
the form of one-hot label.

Dataset Five datasets are used for the experiments. For the text-to-motion
evaluation, we use the HumanML3D (Guo et al., 2022a) and KIT (Plappert et al.,
2016) datasets. The HumanML3D dataset (Guo et al., 2022a), a recent
development, is generated through the reannotation of AMASS (Mahmood et al.,
2019) and HumanAct12 (Guo et al., 2020) datasets. It consists of $14,616$
motions accompanied by $44,970$ textual descriptions. The KIT dataset
(Plappert et al., 2016) consists of $3,911$ motions and the corresponding
descriptions.

For the action-to-motion evaluation, we use two different datasets: HumanAct12
(Guo et al., 2020), UESTC (Ji et al., 2018), and NTU94 (Shahroudy et al.,
2016; Liu et al., 2019). The UESTC dataset (Ji et al., 2018) comprises $25$K
motion sequences spanning $40$ classes. HumanAct12 (Guo et al., 2020) contains
$1,191$ motions across $12$ categories.

We further introduce a novel setting: zero-shot action-to-motion, where the
action labels are completely unseen during training. NTU94 (Shahroudy et al.,
2016; Liu et al., 2019) is used for this purpose. Specifically, it is a
94-class single-person subset of the NTU RGB+D 120 dataset (Shahroudy et al.,
2016; Liu et al., 2019), excluding the $26$ categories of multi-person
actions. The NTU94 dataset consists of $89$K sequences of human motions, and
we randomly choose $63$ classes as the seen classes for training, and use the
remaining $31$ classes for the zero-shot action-to-motion synthesis
evaluation.

Evaluation Metrics For the text-to-motion evaluation, five metrics are
employed, as outlined in (Guo et al., 2022a): _Fréchet Inception Distance
(FID)_ quantifies the disparity between generated and ground truth motion
distributions in the latent space; _Diversity_ evaluates the variation in the
generated motion distribution; _Multimodality_ measures the average variance
given a single text prompt; _R-Precision_ and _Multimodal-Dist_ assess the
relevance of generated motion to the textual prompt. For the R-Precision, we
use the top 3 by default. As our primary goal is to improve the motion
synthesis quality, we use FID as the prior metric.

For the action-to-motion evaluation, we use four metrics: FID, Diversity,
Multimodality and classification accuracy from a pretrained motion classifier.

Table 2. Quantitative results comparison on the KIT test set.

Method | FID $\downarrow$ | Diversity $\rightarrow$ | MultiModality $\uparrow$ | R Precision (top3) $\uparrow$ | MultiModal Dist $\downarrow$
---|---|---|---|---|---
Real Motion | $0.031^{\pm.004}$ | $11.08^{\pm.097}$ | - | $0.779^{\pm.006}$ | $2.788^{\pm.012}$
Language2Pose (Ahuja and Morency, 2019) | $6.545^{\pm.072}$ | $9.073^{\pm.100}$ | - | $0.483^{\pm.005}$ | $5.147^{\pm.030}$
Text2Gesture (Bhattacharya et al., 2021) | $12.12^{\pm.183}$ | $9.334^{\pm.079}$ | - | $0.338^{\pm.005}$ | $6.964^{\pm.029}$
Hier (Ghosh et al., 2021) | $5.203^{\pm.107}$ | $9.563^{\pm.072}$ | - | $0.531^{\pm.007}$ | $4.986^{\pm.027}$
T2M (Guo et al., 2022a) | $2.770^{\pm.109}$ | $10.91^{\pm.119}$ | $1.482^{\pm.065}$ | $\bm{0.693^{\pm.007}}$ | $\bm{3.401^{\pm.008}}$
MoCoGAN (Tulyakov et al., 2018) | $82.69^{\pm.242}$ | $3.092^{\pm.043}$ | $0.250^{\pm.009}$ | $0.063^{\pm.003}$ | $10.47^{\pm.012}$
Dance2Music (Lee et al., 2019) | $115.4^{\pm.240}$ | $0.241^{\pm.004}$ | $0.062^{\pm.002}$ | $0.086^{\pm.003}$ | $10.40^{\pm.016}$
Ours | $\bm{0.472^{\pm.029}}$ | $\bm{10.957^{\pm.092}}$ | $\bm{2.049^{\pm.086}}$ | $0.390^{\pm.006}$ | $9.161^{\pm.027}$

Table 3. Quantitative results comparison on the UESTC dataset.

Method | FID (train) $\downarrow$ | FID (test) $\downarrow$ | Accuracy $\uparrow$ | Diversity $\rightarrow$ | MultiModality $\rightarrow$
---|---|---|---|---|---
Real Motion | $2.92^{\pm.26}$ | $2.79^{\pm.29}$ | $0.988^{\pm.01}$ | $33.44^{\pm.320}$ | $14.16^{\pm.06}$
Action2Motion (Guo et al., 2020) | $21.02^{\pm 2.51}$ | $24.08^{\pm 2.17}$ | $0.889^{\pm.01}$ | $30.47^{\pm.33}$ | $13.46^{\pm.03}$
ACTOR (Petrovich et al., 2021) | $20.49^{\pm 2.31}$ | $23.43^{\pm 2.20}$ | $0.911^{\pm.00}$ | $31.96^{\pm.36}$ | $\bm{14.66^{\pm.03}}$
INR (Cervantes et al., 2022) | $9.55^{\pm.06}$ | $15.00^{\pm.09}$ | $\bm{0.941^{\pm.00}}$ | $31.59^{\pm.19}$ | $14.68^{\pm.07}$
Ours | $\bm{6.68^{\pm.04}}$ | $\bm{9.67^{\pm.17}}$ | $0.934^{\pm.01}$ | $\bm{32.22^{\pm.13}}$ | $15.43^{\pm.06}$

Table 4. Quantitative results comparison on the HumanAct12 dataset.

Method | FID (train) $\downarrow$ | Accuracy $\uparrow$ | Diversity $\rightarrow$ | MultiModality $\rightarrow$
---|---|---|---|---
Real Motion | $0.09^{\pm.01}$ | $0.997^{\pm.10}$ | $6.85^{\pm.05}$ | $2.45^{\pm.04}$
Action2Motion (Guo et al., 2020) | $2.45^{\pm.08}$ | $0.923^{\pm.02}$ | $7.03^{\pm.04}$ | $2.87^{\pm.04}$
ACTOR (Petrovich et al., 2021) | $0.12^{\pm.00}$ | $0.955^{\pm.08}$ | $\bm{6.84^{\pm.03}}$ | $2.53^{\pm.02}$
INR (Cervantes et al., 2022) | $0.09^{\pm.00}$ | $0.973^{\pm.00}$ | $6.88^{\pm.05}$ | $2.57^{\pm.04}$
Ours | $\bm{0.09^{\pm.01}}$ | $\bm{0.976^{\pm.01}}$ | $6.82^{\pm.02}$ | $\bm{2.52^{\pm.03}}$

Table 5. Quantitative results comparison on the NTU94 dataset. Closed-set
(seen classes) and zero-shot (unseen classes) synthesis results are
respectively reported.

Method | Seen Classes | Unseen Classes
---|---|---
FID (train) $\downarrow$ | Acc.$\uparrow$ | FID (train) $\downarrow$ | Acc.$\uparrow$
Random Generator | $315.55^{\pm 0.16}$ | $0.016^{\pm.00}$ | $319.53^{\pm 0.92}$ | $0.032^{\pm.00}$
Action2Motion (Guo et al., 2020) | $138.62^{\pm 0.02}$ | $0.817^{\pm.00}$ | - | -
ACTOR (Petrovich et al., 2021) | $136.72^{\pm 0.03}$ | $0.823^{\pm.00}$ | - | -
Action2Motion (Guo et al., 2020) w/ CLIP | $131.48^{\pm 0.01}$ | $0.833^{\pm.00}$ | $221.57^{\pm 0.21}$ | $0.108^{\pm.00}$
ACTOR (Petrovich et al., 2021) w/ CLIP | $128.80^{\pm 0.02}$ | $0.835^{\pm.00}$ | $205.80^{\pm 0.24}$ | $0.116^{\pm.00}$
Ours | $\bm{42.30^{\pm 0.01}}$ | $\bm{0.869^{\pm.00}}$ | $\bm{112.95^{\pm 0.13}}$ | $\bm{0.153^{\pm.00}}$

Implementation Details Our ATOM is trained with the AdamW optimizer for $50$K
epochs, with an initial learning rate of $10^{-4}$ decay at the $40$K-th step
by a factor of $10$. All of the hyperparameters are determined via a grid
search: $w_{\text{KL}}=w_{\text{div}}=w_{\text{spa}}=10^{-2}$. We set the
hidden dimension $D=512$, the number of Transformer encoder layers and decoder
layers to $8$, and the number of attention head $h=8$. We set the number of
atomic actions $N=256$ for the action-to-motion task, and $N=1024$ for the
text-to-motion task. We follow existing methods (Tevet et al., 2022a) to use
CLIP (Radford et al., 2021; Tevet et al., 2022b) for the language embedding
extraction. For the text-to-motion task, we generate $120$ frames, and for the
aciton-to-motion task we generate $60$ frames. We follow existing methods (Guo
et al., 2020; Petrovich et al., 2021; Guo et al., 2022a) to select the
classifier for metric computation for each task. For the zero-shot action-to-
motion task, we use a pretrained ST-GCN classifier (Yan et al., 2018) to
compute the metrics.

###  4.1. Text-to-motion Evaluation

HumanML3D We compare our ATOM with existing methods on the HumanML3D dataset
in Sec. 3.3. Our approach achieves strong FID, signifying a closer match
between generated and ground truth motion distributions. Moreover, our ATOM
displays superior Diversity and MultiModality scores, showcasing its ability
to generate a wide range of motion sequences. Our method is also competitive
in terms of R Precision and MultiModal Dist, indicating good alignment between
motion and language. Collectively, these results emphasize the strengths of
ATOM in generating high-quality and diverse human motion sequences, surpassing
previous methods across various aspects.

KIT The results on the KID dataset further emphasize the superiority of our
ATOM in motion synthesis. As illustrated in Tab. 2, our method significantly
outperforms previous approaches in terms of FID, Diversity, and MultiModality,
indicating a substantial reduction in the discrepancy between generated and
ground truth motion distributions and the ability to produce diverse motions.
Although our ATOM demonstrates promising synthesis outcomes, there is still
room for improvement in aligning textual descriptions with the generated
motion, which could be addressed by incorporating more advanced language
models for language embedding. Overall, these results highlight ATOM’s
effectiveness in generating high-quality and diverse human motion sequences.

###  4.2. Action-to-motion Evaluation

UESTC Our experimental results on the UESTC dataset provide a compelling
demonstration of ATOM’s effectiveness in the action-to-motion synthesis task,
as shown in Tab. 3. Among the four metrics, FID is the most important
indicator in evaluating the overall synthesis quality. Our ATOM achieves the
lowest FID on the training and testing subsets, significantly outperforming
the best results from previous methods. This demonstrates that our method
generates more realistic motion sequences compared to the competing methods.
In terms of accuracy, ATOM attains a strong accuracy of $0.934$, indicating
that our method is accurate in generating motion sequences that correspond to
the given action label. Regarding the diversity of the generated motions, ATOM
outperforms previous methods in diversity and achieves competitive multi-
modality score. These results illustrate our method’s capability to generate
realistic, diverse and rich motion sequences.

HumanAct12 The experimental results on the HumanAct12 dataset in Tab. 5
further validate the effectiveness of our ATOM in the action-to-motion
synthesis task. Our method and INR (Cervantes et al., 2022) achieve the same
lowest FID of $0.09$, closely approximating the real motion score of $0.09$.
This consistency between the UESTC and HumanAct12 datasets highlights the
robustness of our method in generating realistic motion sequences. Our ATOM
also achieves the highest classification accuracy, showing its ability to
generate motions that correspond to the action labels across different
datasets. We also achieve promising diversity and multi-modality scores.
Overall, the consistent and superior performance of ATOM across both UESTC and
HumanAct12 datasets emphasizes its effectiveness and robustness in the action-
to-motion synthesis task.

(a) Seen classes. (b) Unseen classes.

Figure 4. Zero-shot action-to-motion qualitative results comparison with ACTOR
(Petrovich et al., 2021) on the NTU94 dataset.

###  4.3. Zero-shot Action-to-motion Synthesis

Apart from the common text-to-motion and action-to-motion settings, we
introduce a novel setting: zero-shot action-to-motion synthesis, where the
action labels for inference are entirely unseen during training. This new
setting aims to evaluate the model’s ability to generalize to unseen actions,
crucial for practical applications with novel action descriptions. Assessing
performance in the zero-shot action-to-motion setting enables a better
understanding of the model’s potential to synthesize realistic and diverse
human motion sequences in real-world scenarios.

We evaluate our zero-shot action-to-motion synthesis performance on the NTU94
dataset, and the results are listed in Table 5. Two previous methods are
included for comparison, _i.e_., ACTOR (Petrovich et al., 2021) and
Action2Motion (Guo et al., 2020). Note they take as input one-hot label
vectors to output motion sequences of certain classes, thus they cannot
synthesize actions of unseen classes (denoted as “$-$” in Table 5). To achieve
zero-shot synthesis, we replace the one-hot label vector input in with CLIP
text embedding, denoted as “w/ CLIP”. We make the following observations from
the results: (1) Our ATOM significantly outperforms previous methods in both
seen class and unseen class synthesis tasks. Surprisingly, our FID on unseen
classes is even lower than the ACTOR’s FID on seen classes, which demonstrates
the effectiveness of our method. (2) Our method exhibits lower variance
compared with previous methods, indicating a more stable synthesis results.
(3) Though previous methods show promising results on small-scale human motion
datasets (_e.g_., NTU13 and HumanAct12), they show poor capability in large-
scale datasets such as NTU94.

We further compare the qualitative results with ACTOR (Petrovich et al., 2021)
w/ CLIP in Fig. 4. First, our ATOM can successfully generate seen classes,
while ACTOR shows poor qualitative results, coinciding previous findings (Song
et al., 2022) that several previous methods only work well on simple human
motion datasets consisting of $\sim 10$ classes. The high FID and high
accuracy of previous methods (Guo et al., 2020; Petrovich et al., 2021) show
that they focus on learning trivial motions for classification, instead of
human-recognizable motions. Second, we observe our method generates realistic
motions on unseen classes when there are certain body parts in the textual
description, _e.g_., “touch chest”, “brush hair”, or “cross hands in front”.
In contrast, ACTOR cannot generate meaningful human motions in such cases.

Figure 5. Visualization of learned atomic actions on the HumanML3D dataset.
For each textual prompt, we visualize the atomic action that has the highest
attention value to it. Table 6. Ablation study on the atomic action codebook
on the KIT dataset. The variant of model w/o codebook is realized by
implementing the decoder with a Transformer encoder, where the query, key and
value are the same.

#Atom | $\mathcal{{L}_{\text{{div}}}}$ | $\mathcal{{L}_{\text{{spa}}}}$ | FID $\downarrow$ | Diversity $\rightarrow$ | R Precision $\uparrow$
---|---|---|---|---|---
CVAE baseline | $6.87^{\pm.021}$ | $9.102^{\pm.072}$ | $0.280^{\pm.024}$
256 | - | - | $4.57^{\pm.031}$ | $9.204^{\pm.038}$ | $0.295^{\pm.039}$
512 | - | - | $2.42^{\pm.023}$ | $9.673^{\pm.057}$ | $0.303^{\pm.034}$
1024 | - | - | $1.43^{\pm.038}$ | $10.327^{\pm.052}$ | $0.337^{\pm.014}$
2048 | - | - | $1.47^{\pm.033}$ | $10.311^{\pm.062}$ | $0.356^{\pm.023}$
1024 | $\checkmark$ | - | $0.873^{\pm.021}$ | $10.937^{\pm.069}$ | $0.381^{\pm.003}$
1024 | - | $\checkmark$ | $0.732^{\pm.039}$ | $10.688^{\pm.073}$ | $0.384^{\pm.012}$
1024 | $\checkmark$ | $\checkmark$ | $\bm{0.472^{\pm.029}}$ | $\bm{10.957^{\pm.092}}$ | $0.390^{\pm.006}$

Table 7. Ablation study masked motion modeling curriculum learning on the KIT
dataset.

Mask ratio $r$ | Learning Scheme | FID $\downarrow$ | Diversity $\rightarrow$ | R Precision $\uparrow$
---|---|---|---|---
w/o masked modeling | $2.48^{\pm.031}$ | $10.342^{\pm.031}$ | $0.345^{\pm.021}$
25% | $g(l)=r$ | $1.22^{\pm.030}$ | $10.473^{\pm.037}$ | $0.344^{\pm.008}$
50% | $0.76^{\pm.049}$ | $10.659^{\pm.051}$ | $0.372^{\pm.009}$
75% | $1.48^{\pm.033}$ | $10.551^{\pm.034}$ | $0.388^{\pm.013}$
50% | $g(l)=rl/L$ | $\bm{0.472^{\pm.029}}$ | $\bm{10.957^{\pm.092}}$ | $\bm{0.390^{\pm.006}}$
$g(l)=r(l/L)^{2}$ | $0.621^{\pm.031}$ | $10.683^{\pm.041}$ | $0.373^{\pm.009}$
$g(l)=re^{l/L-1}$ | $0.583^{\pm.024}$ | $10.590^{\pm.057}$ | $0.387^{\pm.007}$

###  4.4. Ablation Study

Atomic Action First, we quantitatively analyze the effect of atomic actions in
Tab. 6. As the number of atomic actions increases, the FID decreases, and both
the Diversity and R Precision improve, indicating better motion generation
quality. However, further increasing the number of atomic actions to $2048$
does not yield substantial improvements. The combination of diversity and
sparsity constraints leads to the best performances, suggesting that they are
essential components for generating high-quality, diverse human motion
sequences in our model.

We further visually analyze the learned atomic actions in Fig. 5. Upon
examination, we find that these atomic actions primarily represent two types
of motion: body translation and specific body part movements. Body translation
atomic actions typically involve motions like running, or actions involve
directional changes, where the entire body is engaged in coordinated movement,
see the first two examples of Fig. 5. On the other hand, specific body part
movements focus on the motion of a particular body part, such as waving a
hand, or sitting down, see the last two examples of Fig. 5. This distinction
highlights the versatility of the learned atomic actions, as they capture both
global and local motion patterns. As a result, our approach can effectively
synthesize complex human motions by combining these diverse atomic actions in
a meaningful and contextually appropriate manner.

Masked Motion Modeling Curriculum Learning The results of the masked motion
modeling curriculum learning are illustrated in Tab. 7. When compared to the
baseline without masked modeling, incorporating masked modeling consistently
leads to a reduction in FID and an improvement in Diversity and R Precision,
and achieves the best results at a mask ratio of $50\%$, indicating the
benefit of this approach for motion generation quality. Moreover, the learning
scheme that follows a linear progression $g(l)=rl/L$ demonstrates the best
performance across all metrics, suggesting that a gradual exposure to the
complexity of motion reconstruction during training is an effective strategy
for enhancing the model’s synthesis capabilities.

Qualitative Results We showcase qualitative results in Fig. 6, highlighting
the differences between our ATOM and the CVAE baseline. In the first example,
the human figure transitioning between ”sitting” and ”running” appears tilted
in the CVAE result. For the second CVAE example, the person continues to
rotate even after sitting down. In contrast, our final ATOM generates a more
natural and realistic motion sequences, demonstrating the effectiveness of our
proposed atomic action and curriculum learning.

Figure 6. Qualitative comparison between our CVAE baseline and our ATOM.

##  5\. Conclusion

We present ATOM, a novel approach for language-guided human motion synthesis
that addresses the limitations of previous methods by leveraging atomic action
decomposition and a curriculum learning strategy. ATOM’s transformer-based
CVAE framework effectively decomposes complex actions into atomic components,
allowing for the generation of diverse and coherent motion sequences. The
incorporation of curriculum learning further enhances the model’s ability to
capture dependencies in motion sequences, resulting in smooth and natural
motion transitions between different actions. Our comprehensive evaluation
demonstrates ATOM’s superior performance over existing approaches in the text-
to-motion and action-to-motion synthesis tasks.

###### Acknowledgements.

This work is supported in part by the Defense Advanced Research Projects
Agency (DARPA) under Contract No. HR001120C0124, the National Science
Foundation (NSF) IIS-2008532, NSF Award #1846076, and the Institute of
Education Sciences (IES), U.S. Department of Education (ED) through Award
#2229873. Any opinions, findings and conclusions or recommendations expressed
in this material are those of the author(s) and do not necessarily reflect the
views of the DARPA, the NFS, the IES, or the ED.

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